Introduction 

Welcome aboard to the crown jewel of this site, the biggest passion project of anything in Pointless Large Number Stuff, one of the most ambitious things I have ever made: the Pointless Gigantic List of Numbers! I first created this list as a Google Doc in February 2014 or so, before I even started this site, and it didn't take long till I got carried away with it. Here's a few older versions of the list, so you can see how much it's grown:

• June 7, 2014 

• June 23, 2014

• November 4, 2014

• October 30, 2015

• October 21, 2021

• July 5, 2022

All numbers big and small, notable and obscure, sharply defined and dubious, are welcome in this list. It features the famous landmark googolisms (e.g. gongulus), a variety of obscure googolisms (e.g. coogol), numbers with interesting properties (e.g. 211), numbers significant in humanity or science (e.g. the world population), various numbers I made (e.g. meroogol), the unnamed outputs of plugging in numbers into systems (e.g. f3(3)), and even numbers with personal significance, like 17. The list as a whole has over 2100 entries and counting! Its inclusion or exclusion of specific numbers is a little biased, since this is my list. :-)

Like Sbiis Saibian's list, my list is divided into ranges of numbers (color-coded to see the progression), which are themselves divided into entries for numbers. The entries include how the number can be expressed and/or an approximation for the number, unless neither is possible to compactly write, as well as the name of the number if it has one. The entries usually contain descriptions of what is notable about their number, except in later parts of the list.

I have decided to start this number list with zero and end with the largest finite numbers known to man. There are spinoff lists outside this range: if you're interested in some notable negative, imaginary, or complex numbers look here, and if you're interested in looking at some infinite numbers look here.

Making a list of numbers on a site on large numbers was not my own idea. There are two other lists which serve as inspiration for this list:

1. Sbiis Saibian's Numbers Sorted by Size, a number list heavily focused on numbers notable to googology

2. Robert Munafo's Notable Properties of Specific Numbers, a very comprehensive list of any and all numbers with notable properties

Since I have now said what needs to be said about this list, we are now ready to begin our journey through the large numbers, starting with part 1, any numbers from zero to a million.

PART 1: THE TINY DUST BITS

0 ~ 1,000,000

You can find any numbers that I find to be interesting below a million on this part of the list. It has been argued that any number below a million shouldn't be a large number, so in some sense these numbers aren't large at all. They aren't scary at all either ... ok maybe a little.

The Zero Range
0
Entries: 1

Zero
0

We open our list of positive numbers with the important number 0, which I consider to be the second most important number in existence, behind the number 1. Although zero is neither positive nor negative, it still serves as a natural starting point for a list like this. If you want to learn about notable negative numbers and miscellany look here.

Zero's importance stems from being the additive identity, meaning that adding 0 to a number gives the original number. In fact, in formal mathematics 0 is often defined as the number x such that any number a plus x is still a. Therefore it's an inherently special number in all of mathematics and everywhere, which is also why 0 gets a range of numbers on this list all to itself.

While all other numbers represent something, 0 represents nothing, making it special. But 0 can represent more than nothingness. It can also represent the very beginning or before the first, or a placeholder digit in numbers such as 50630005 with 0s in it. Additionally, when accounting for negative numbers, zero represents pure neutrality or stability—a lack of either extreme.

Zero also very often leads to degenerate cases in functions, i.e. calculations that do not result in large values. For example, x+0 = x, x*0 = 0, x0 = 1, x^^...^^0 with as many ^s as we want is 1, and f0(x) is equal to x+1 in the fast-growing hierarchy.

However, since 0 does not normally represent a quantity of something, it took a very long time for zero to be recognized as a number like 1, 2, 3, etc. were thought of as numbers (see my large number timeline for more). The earliest known record of zero as a number was around 250 BC, and in Europe this happened around the 1500s. Before then it was only recognized as a digit, and even that took a while, largely because many numeral systems like Egyptian and Roman numerals didn't need zero.

Because of zero's unique history, the name "zero" cannot be traced back to Indo-European roots like 1, 2, 3, and so on can be; rather, it's traced back to Arabic "sifr". That became "zefiro" in Italian, and was later shortened to "zero". For the name of zero, English is actually a special case among Germanic languages (like English, German, and Dutch): in most Germanic languages the word for "zero" resembles "null", and that's also true for Slavic languages such as Russian. However, in other Indo-European languages (such as Romance languages like French, Spanish, and Italian), the word for zero does resemble "zero".

For an interesting bit of trivia relating to zero, see the entry for 1010^10 in part 2, a number equal to one followed by ten billion zeros.

The Microscopic Range
0 ~ 0.001
Entries: 17

Reciprocal of a croutonillion

click here (thanks Cloudy176)

Reciprocal of BIG FOOT
1/FOOT10(10100)

This is the reciprocal of BIG FOOT, once viewed as the largest named number by the googology community, but now seen as one of several ill-defined attempts to meaningfully outdo Rayo's number.

In any case, if the definition of BIG FOOT were fixed in some way, this would be an insanely small number, unfathomably close to 0 - and yet there's infinitely many numbers between this number and 0, and infinitely many numbers between this and the next number or the reciprocal of any really big number.

Reciprocal of Rayo’s number
1/Rayo(10100)

This number is the reciprocal of one of the largest numbers ever known, known as Rayo's number. Once again this is inconceivably close to the number 0, but there's still infinitely many numbers between this and the next or previous numbers, or whatever else. This is a mind-bending truth that comes from working with small numbers if you've gotten accustomed to the behavior of large numbers.

Reciprocal of meameamealokkapoowa oompa
1/{LLLL........LLLL, 10}10,10 with a meameamealokkapoowa-sized array of L’s

The reciprocal of Jonathan Bowers' largest named number, called meameamealokkapoowa oompa. Since meameamealokkapoowa oompa is a ridiculously huge power of 10, its reciprocal's decimal expansion is a bunch of zeros and then 1. The number of 0's here might seem to be a lot smaller than meameamealokkapoowa oompa, but on a scale with numbers this large, it’s very far from any meaningful difference because with this number we are FAR past exponentiation.

Reciprocal of a beengulus
1/{10,100(0,0,4)2}

Just pulling out the reciprocal of a random Bowerism. The reciprocal of this number is a superdimensional array of 100(4*1002) entries, much bigger than the array used to represent a gongulus.

Reciprocal of a tristo-threagol
1/E100###100###100#100#100

Now pulling out the reciprocal a random Saibianism, which is approximately {10,101,2,1,3} using Bowers' notation. This number is, once again, 0.0000.....0001 with a shit load of 0s, like the previous two numbers. The number of 0s is still almost exactly the same as a tristo-threagol, relatively speaking. This is approximable as a 5-entry array, so it’s in the low mid-level googolisms.

Reciprocal of Graham's number
1/G(64)

Because if I don't mention it, someone else will.

Googolminex
1/1010^100 or 10-10^100

Also known as the reciprocal of a googolplex. This is 0.000....0001 with a googol minus one zeros after the decimal point. There’s no way we could write all those zeros out, even if we put a 0 on every atom in the observable universe.

The name "googolminex" was coined by Conway and Guy in their Book of Numbers. They said that since googolplex = 10googol, x-plex can be thought of as 10n (though it could have been anyone who first came up with this), and by analogy (plus:plex::minus:?) x-minex should be 10-n, aka 1/10n.

Interestingly, googolminex, despite being less than 1, can be considered a googolism due to its unique googological name, although Sbiis Saibian suggests that a name for a small number be called a "micronym" - he has named a few "micronyms" (example). See 21 and grangolplex/googolcentiplex for further discussion on the term "googolism".

Googol-minutia-speck
10^-110

This is currently Sbiis Saibian's smallest googolism. It is formed by combining the name "googol" with the suffix -minutia meaning the reciprocal of x with the suffix -speck which divides a number by ten billion. Sbiis Saibian suggests the term "micronym" for googolisms for numbers less than 1.

To get an idea of how small this number is, imagine the volume of the observable universe, 3.5*1080 m3. Then imagine a 10^110th of the observable universe, aka a googol-minutia-speck fraction of the universe. That would be 3.5*10-30 m3, aka a sphere 94 picometers in diameter. You may wonder, how small is 94 picometers? 94 picometers is approximately the diameter of an oxygen atom. That means that we can imagine an oxygen atom as being approximately a googol-minutia-speck portion of the universe. Now that's a pretty small number.

Googol-minutia
10^-100 or 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000001

This teeny tiny number is the reciprocal of a googol, called "googol-minutia" by Sbiis Saibian. It can be easily written out, and it's close to 0 but not quite inconceivably close. It can be called a googolth, or by the -minex idea it could be called "hundredminex" (see googolminex). Since a googol can be considered to be on the (very high end) universe scale, a googol can be considered to be on the fraction of the universe scale. A small biological virus would be a googolth of the volume of the observable universe.

Planck time in seconds
5.390*10^-44

Planck units, the smallest units theoretically measurable, are home to some of the smallest numbers known to the real world, and to be meaningful as something other than the reciprocal of a large (greater than 1) number. This is the length of the Planck time in seconds - about 2*10^43 of those times fit in a second! This is the smallest positive number Robert Munafo puts on his number list.

See also the Planck length and the Planck temperature.

Planck length in meters
1.619*10^-35

This is the length of the Planck length in terms of meters. The Planck length is the amount of length light travels in a Planck time, and it's the smallest physically meaningful length. Likewise, a Planck area is a square Planck length and a Planck volume is a cubic Planck length.

See also the Planck time and the Planck temperature.

One nonillionth
10^-30 or 0.000000000000000000000000000001

One nonillionth is notable because it represents the smallest official SI prefix, quecto-. That and ronto-, referring to an octillionth or 10^-27, are the newest additions to the small SI prefixes, introduced in 2022. There are virtually no practical purposes for these microscopic, or should I say quectoscopic units; they were created primarily as counterparts to the new large prefixes, ronna- and quetta-, which were devised because of computer science.

There are many unofficial extensions to those prefixes, mostly devised for the sake of novelty. I discuss some of these extensions on this list, and on this page where I propose my own extension from 2014.

One septillionth
10^-24 or 0.000000000000000000000001

From 1991 to 2022, one septillionth represented the smallest official SI prefix, yocto-. Its name is derived from octo-, which obviously means eight. It’s very small of course, so small that smaller prefixes have very few practical purposes—and yet, this didn't stop the International System of Units (SI for short) from adding two smaller ones to the family.

Here are some examples of the yocto- prefix: A proton weighs about a yoctogram, A yoctometer may be the size of a neutrino. The half-life of hydrogen-7 is 23 yoctoseconds. As you can see, yocto- makes things very very small. Zepto- is a thousand times more than yocto-, and then we have atto-, femto-, pico-, nano-, micro-, and milli-, each being 1000 times more than the previous.

Proportion of integers such that π(x) > li(x)
~ 0.00000026

This number was shown in 1994 to be approximately the proportion of all positive integers such that π(x) > li(x) - see Skewes' number for details. This proportion is surprisingly large, given that the smallest such number with this property isn't until at least 10^14.

One millionth
10^-6 or 0.000001

A millionth of something is represented with the prefix micro-, one of the more well-known SI prefixes. Its name literally means "small". Some examples of it: Microbes can be measured in micrometers. Light takes a microsecond to travel 300 meters in a vacuum, but that’s more of an example of how fast light is. For one more example, the 2011 Japanese earthquake, known for its sheer destruction, decreased the duration of Earth’s day by 2 microseconds, which much better shows how stable and large Earth is rather than how short a microsecond is.

Percentages this small are usually not noted as 0.0001%, but as the easier to understand 1 ppm (part per million). The ppm unit can be used to denote, for instance, 0.00000334 as 3.34 ppm. Also used is ppb (parts per billion), the less common ppt (parts per trillion), and the still less common ppq (parts per quadrillion).

Eyelash mite-speck
0.000002

This number is among Sbiis Saibian's smallest googolisms. See here and here for how it's formed. It's equal to the reciprocal of 500,000 or two parts per million. Because it's a googolism for a small number, it's an example of a micronym (see also googolminex).

The Not-Too-Small Range
0.001 ~ 0.99999
Entries: 29

One hundredth / one percent
1/100 or 0.01

This is 1/100, or one percent (1%). 1% is figuratively used to represent almost none, but just as 100 is typically not a giant number, 0.01 isn't always a "tiny number". And just as 100 is an easy number to imagine in your head, 1% is an easy percentage to picture: just take one part or something out, leaving 99 parts behind. It's a sizable proportion when dealing with large numbers of entities—for instance, if a country has 50 million people and 1% of them have a certain disease, then that's a respectably large 500,000 people. A 1% chance is a probability that it would often be unwise to dismiss as rare.

This number is very commonly subtracted from prices to make them look cheaper than they really are. For example, objects that are worth two dollars are usually priced as $1.99 so that it looks like it costs one dollar and a bit. It shouldn't work, but it does. This trick is called psychological pricing. (Until 2024, this information was in the entry for 99.)

One twentieth
1/20, 5/100, or 0.05

This 1/20, often referred to as 5%. 5% is easily perceivable as more than 1% but less than 10%. If 1% of bottles of milk at a grocery store were spoiled, I assume most people wouldn't worry too much, but they'd be weary if 5% of milk bottles were spoiled.

As with 0.01, this number is sometimes subtracted from prices to make them appear much smaller. And in statistics, this number is typically used as a maximum threshold for a p-value that is statistically significant. The phrase "p-value" refers to how likely it is that a result occurred by chance.

(1/e)e
0.065988...

This is the reciprocal of the mathematical constant e, raised to the power of e. It is the smallest number x such that xx^x^x^x.... (an infinite power tower of x's) converges to a finite number. Any smaller number will fail to converge and instead alternate between numbers close to 0 and numbers close to 1 as the power tower gets taller.

See also e1/e.

One tenth
1/10 or 0.1

One tenth, or 10%, is figuratively used to represent a bit of something. It is easier to visualize than 1%. Imagine if 10% of your coworkers had caught a nasty disease and have to stay home for the next week. The office would be noticeably smaller.

One ninth
1/9 or 0.1111..... with infinite 1’s

One ninth has the simplest repeating decimal expansion of all fractions. Since 9 is one less than the base of our numeral system, its reciprocal when written out has infinite 1’s. It’s slightly larger than one tenth and hard to distinguish from it.

Champernowne constant
0.12345678910111213141516...

This is an interesting irrational number called the Champernowne constant, formed by combining the digits of all the positive integers in base 10 into a decimal. It's one of the very few numbers that have been proven to be a normal number in base 10, i.e. a number where all strings of digits of any length n appear with equal probability - for example in this numbers' digits you're just as likely to find 302 in a random spot as you are to find 734, and you're as likely to find 164835 as you are to find 430798. Other well-known irrational constants like the square root of 2, e, or pi, are conjectured to be normal but this has not been proven.

The Champernowne constant's continued fraction (see 292 for a discussion of continued fractions) is an unlikely source of large numbers. The first few numbers to appear in its continued fraction are 0, 8, 9, 1, 149,083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, and 15, and the next one has 166 digits, so it's larger than a googol. The terms of the sequence have very erratic behavior, and the next very large term is the 41st term, which has 2504 digits. Even weirder, people have found patterns of terms in the Champernowne constant's continued fractions.

One can define the Champernowne constant in any base—for example, in binary it would read as 0.11011100101110111... (that's not equal to the base 10 Champernowne constant)—however, in other bases the Champernowne constant has not been proven to be normal.

One eighth
1/8 or 0.125

One eighth’s full expansion has three decimal places, so it terminates. This is because 8’s prime factorization (23) consists only of factors of 10. If we were in an odd-numbered base, the positional expansion of 1/8 (a base-neutral term for "decimal expansion") would not terminate.

One seventh
1/7 or 0.142857.... where the underlined digits repeat infinitely

This number’s expansion is more complicated than that of nearby numbers - in fact, it's the simplest fraction with a repeating pattern of more than one digit (see 7). This number is connected to the interesting number 142,857, called integral-megaseptile.

One sixth
1/6 or 0.16666..... with infinite 6’s

This number's digits also repeat, because its prime factorization contains 3, which is not a factor of 10. It’s reasonably distinguishable from a tenth and a fifth, but not that easy to tell apart from a seventh.

One fifth
1/5 or 0.2

Another simple decimal, which is easy to visualize from anything. Not that much to say about it, but since 5’s prime factorization is simply 5 and 5 is a factor of 10, ⅕ has a terminating decimal expansion. It’s twice as big as a tenth. By now each fraction (not necessarily each number) is easily distinguishable from its neighbors in the list.

i^i
0.207879576...

When you raise i (the unit of imaginary numbers) to its own power, you strangely end up with a real number that falls between a fifth and a fourth. It's equal to e-π/2 - the reason you can use complex powers like this is because you can extrapolate the infinite series to calculate powers to complex inputs, which will often give you unusual results such as this one.

Natural density of abundant numbers
0.2477±0.0003

The natural density of a certain set of positive integers (such as square or composite numbers) is the percent of all positive integers that are in that set. If the percent of numbers within that set approaches 0 as the range of numbers reaches infinity (such as if the set is the powers of 2), then the natural density of that set is 0.

Abundant numbers, numbers whose divisors add up to a number greater than to the original number, have a natural density between 0.2474 and 0.2480. This means that about one out of four positive integers is abundant.

See also 0.607927102...

One fourth / one quarter
1/4 or 0.25

A particularly commonly used fraction. One fourth has its normal name as well as the more special name one quarter. Fourth is used more for denoting parts of a tangible object, while quarter is used more for larger areas or units that are less tangible. For instance, the term “quarter” is used as the name of a 25-cent coin, or a quarter of a dollar.

Common logarithm of 2
log10(2)
0.301029996...

This is the number that, when you raise 10 to its power, becomes two. Despite being less than 1, this number actually has some significance to googology. Here's how:

Consider a number like 22^500 (that's raising 2 to the power of "2 to the power of 500"). Naturally we want to estimate this number in base 10, as a power of 10. However, no conventional calculator can support values this big. Instead, we can convert the 2 in the base to 10log(2):

22^500
= (10log(2))2^500

Then we can just use the laws of exponents, specifically (ab)c = ab*c:

= 10log(2)*2^500

All that's left now is calculating log(2)*2500 - putting the equation in a calculator gives us 9.8539*10149.

Therefore 22^500 is about:

109.8539*10^149

This means that making use of this logarithm (and any logarithm numbers for that matter) is useful for determining where powers that are not powers of 10 fall amongst the googolisms. See also log(log(2)), a negative number with a similar use in googology.

One third
1/3 or 0.333.... with infinite 3’s

This number is the result of dividing one into three, notable for being extremely simple yet impossible to express exactly in decimal without expressing some kind of repetition. This is because 3 cannot be divided into our base, 10. This leads to some properties that grade schoolers who have learned about fractions may find mentally distorting. For example, take Sbiis Saibian's recount of his disconcerting childhood experiences with the digits of 1/3 (source):

My first falling out with infinity, so to speak, had to do with a certain fraction. You may recall in grade school learning that 1/3 is "point 3 repeating". This struck me however as very disconcerting. Mathematics, the one absolutely exact science, the paragon of precision, now contained a number that could not be expressed "exactly". Yes I know that 1/3 can be expressed "exactly" as a fraction, but as a kid I didn't see it that way. The decimal representation was the "true" form of number, and yet 1/3 had no such representation. Stubbornly I refused to accept this as fact. So I performed the long division on 1/3 thinking I'd prove them wrong. Well the 3's kept coming until it became abundantly clear that it could never end. Here was infinity, not as some distant entity, or abstraction for the whole of mathematics, but as something immediately present in a finite number! After this I became more suspicious of infinity, and my attention turned directly to it.

This shows that it kind of goes against the intuitive notion of the precision of mathematics that one third can't be expressed exactly in decimal, yet it is a real and well-defined number. This is also the case with some very large numbers like megafuga-four.

Even stranger, this leads to some utterly counter-intuitive properties. For example, observe here:

1/3 = 0.333...
1/3*3 = 0.333...*3
1 = 0.999...

It is infamously counterintuitive that this extremely simple and sound proof, shows that 0.999... with infinite 9's equals one! When learning of that equality, people often attempt to defy the laws of mathematics to "show" that 0.999... cannot equal one, because it's so counterintuitive. This is what my brother once when we were teenagers and I told him about the equality: he tried to redefine "numbers" in some crazy way that makes absolutely no sense and is mathematically unsound just so that 0.999... does not equal one! However, he eventually accepted his intentions as naive.

The name “one third” for this number isn’t special, but the word “third” somewhat is. It’s one of only three ordinal numbers that has a name other than a root based on the English number’s name followed by -th, such as fourth or fiftieth, except for numbers whose names end in one, two, or three. Even those aren’t special because it’s simply things like “twenty-third” and “one hundred fifty second”. Third is somewhat special but is still based on the word "three". The two truly special ordinal numbers are first and second, both of which clearly have their own origin.

Aarex's Funny Number
0.42513153135

One old page of Aarex's old large numbers website has a table with the numbers 0 through 20 and their names, and lists a property of each of them. But between 0 and 1, Aarex puts 0.42513153135 and says its name is "42513153135 hundred billionth" and its property is that it's a "funny number".

Common logarithm of three
log(3)
~ 0.47712125...

This is the number that, when you raise 10 to its power, becomes three. Similarly to log(2), this number has use in googology when you're estimating large powers of 3 (such as 3^3^3^3) in terms of base 10. But more interestingly, you can use this and log(2) to mentally estimate the logarithms of some of the composite numbers, specifically the 3-smooth numbers (numbers that have no prime factors larger than 3). This is because of the identity, log(a*b) = log(a)+log(b). Here's some examples of logarithms we can estimate using these two logarithms:

log(6) = log(2*3) = log(2)+log(3) ~ 0.301+0.477 ~ 0.778
log(9) = log(3*3) = log(3)+log(3) ~ 0.477+0.477 ~ 0.954
log(18) = log(2*3*3) = log(2)+log(3)+log(3) ~ 0.301+0.477+0.477 ~ 1.255

To estimate the logarithms of any 5-smooth number you only need to know one more logarithm, the logarithm of five (about 0.699).

One half
1/2 or 0.5

Also known as half, fifty percent, zero point five, one over two, one out of two, or five out of ten. This number is so important in life that it gets the special name “one half”, not one second or the silly-sounding one twoth. There’s no other figurative way to describe 50% of something than half. One half is sometimes used figuratively to mean most, and in some cases it may as well mean most. If a room with 100 people has 50 people named "John" and the other 50 people each have a different name, saying "half of them are named John" wouldn't be that different from saying "most of them are named John"—both would be equally odd scenarios. Cases like this demonstrate how much humans' interpretation of probabilities can vary depending on context.

Euler-Mascheroni constant
0.57721566...

This is the Euler-Mascheroni constant, denoted with the Greek letter gamma (γ). Don't let the name fool you—this number has nothing to do with macaroni! It was first used in 1734 by Leonhard Euler, one of the most important mathematicians to have ever lived. Here's how this number is defined:

Consider the sum:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ... + 1/n

The first few values of this sum for n = 1, 2, 3 4 ... = 1, 1.5, 1.8333..., 2.08333... As n approaches infinity (yes the sum diverges to infinity, unlike what you might expect for a sequence like this) the difference between the sum and ln(n) approaches γ.

The Euler-Mascheroni constant crops up in several places in mathematics, but it's not as well known as constants like pi and e.

Natural density of squarefree numbers
6/π2
~ 0.607927102

Squarefree numbers are numbers that aren't divisible by a square number (other than 1 of course) (see also 49). Their natural density is 6/π2, meaning that about 61% of all positive integers are squarefree. This is one of the many examples of pi cropping up in a place you don't expect it to.

See also 0.2477±0.0003 (another natural density number) and 1.64493..., this number's reciprocal.

Two thirds
2/3 or 0.666...... with infinite 6’s

The simplest and most common non-unit fraction. It is very familiar and easily visualized. Two thirds is used figuratively to mean more than half.

2/3 is important enough that the ancient Egyptians had a special symbol for it - in fact it's the only non-unit fraction that got its own symbol. Other non-unit fractions were expressed as the sum of unit fractions, e.g. 1/2+1/4 in place of 3/4.

Natural logarithm of two
ln(2)
0.693147181...

Another logarithmic number. This number crops up several different places, like when working with half-lives of radioactive substances, or with doubling and the rule of 72. e to the power of this number is 2.

Three fourths / three quarters
3/4 or 0.75

Another common and tangible non-unit fraction. This is also so common that it’s used figuratively to mean most, both as three fourths/quarters and 75%.

Four fifths
4/5, 8/10, or 0.8

Once again, 8/10 is more tangible than 4/5, and therefore more often used. We’re close to wrapping up the numbers below 1.

Nine tenths
9/10 or 0.9

Most commonly referred to as 90%. This number is just as tangible as 10% is, and 90% is used as a figurative term for most, almost “almost all”.

Nineteen twentieths
19/20 or 0.95

Most commonly called 95%. This is about a lower limit for “almost all” and it’s used figuratively (as 95%) for almost all.

Ninety-nine hundredths
99/100 or 0.99

Ninety-nine hundredths, also known as 99%—a phrase commonly used to mean "almost all", even if it means much less than 99%. Visualizing 99% of something is almost like visualizing the whole thing.

But often, 99% just isn't close enough to the whole thing. For example, imagine a machine to detect whether someone was a criminal with 99% accuracy. This may seem like a good accuracy, but imagine that machine was used on the entire population of the United States. That would give 3 million false results! Therefore, when dealing with a large scale 99% accuracy isn't even close to good accuracy.

One minus a millionth / nine hundred ninety nine thousand nine hundred ninety nine millionths
999,999/1,000,000 or 0.999999

Almost indistinguishable from 1 in general terms, but still can be told apart when dealing with a small scale. You may encounter numbers like this when dealing with purity of materials, for example when collecting elements of the periodic table.

One minus the reciprocal of Rayo’s number
1-1/Rayo(10100)

Because I can.

The One Range
1
Entries: 1

One
1

One is arguably the most important number in existence, defined as the number x such that any number a multiplied by x is still a. It has many unique properties (mostly for trivial reasons) and it also plays a unique role among the numbers (READ MORE)

The Superunit Range
1 ~ 1.999999
Entries: 24

One plus the reciprocal of Rayo’s number
1+1/Rayo(10100)

Yep, we are beginning our journey through large numbers very very slowly, with an INSANELY small large number - remember that in the entry for 1 I talked about how Sbiis Saibian proposed that a number between 0 and 1 should be considered a small number and a number between 1 and infinity should be considered a large number (1 is neither large nor small). This number is 1, a decimal point, a shit ton of zeros, and a sequence of digits whose beginning we will never know. We don't know whether it's a terminating or repeating decimal—all we know is that the sequence of digits eventually will cycle. It's not an irrational number like pi.

Even if you raised this number to the power of something like Loader's number you wouldn't make it close to the next entry ...

One plus the reciprocal of meameamealokkapoowa oompa
1+1/{LLLL........LLLL, 10}10,10 with a meameamealokkapoowa-sized array of L’s

This is one plus the reciprocal of Jonathan Bowers' largest googolism, a number called meameamealokkapoowa oompa. As you probably guessed, this is 1.000.....00001 with just under a meameamealokkapoowa oompa zeros, another example of an insanely small large number. Although meameamealokkapoowa oompa is extremely ambiguously defined, I'm still including this as an example of how small we can get.

One plus the reciprocal of a tethrathoth
1+1/E100#^^#100

A tethrathoth is a googolism by Sbiis Saibian, which, like meameamealokkapoowa oompa, is an insanely huge power of 10. This is one plus its reciprocal - like the previous number this is 1.000.....00001 with an unfathomable huge amount of zeros - however the amount is much smaller than the previous entry's amount of zeros.

Graham's root of googolminex and one
(1+10-10^100)-1/G(64)

This number was given by Sbiis Saibian as an example of an extremely small large number - see Graham's number * googolplex.

One plus the reciprocal of Graham's number
1+1/G(64)

Another number virtually equal to one, which is 1.0000.......00000????????........ where ? is an unknown digit. When you raise this to the power of Graham's number, you get a number almost exactly equal to the mathematical constant e. It's much larger than the number two entries before this one in that you need to raise it to a number almost exactly (on a googological scale) equal to a tethrathoth to get this number. And yet, both of those numbers are virtually equal to one.

One and a googolminex
1+1/1010^100

One plus the reciprocal of a googoplex, aka one, a decimal point, a googol-1 zeros, and another 1. This number is still inconceivably close to 1, but at least the number of zeros before we hit a nonzero digits is not completely unfathomable, as we can use the entire observable universe to get a feel of how big a googol is. But you still need to square this number over a googol times to exceed two, so this is still a really really small large number, and much more of one than the next entry.

One and a googolth
1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000001

This number looks very close to one, and true, it is. But it's so far from one that I can actually write all the zeros that show the difference between this and 1, as the number of zeros is only 99! In contrast with the previous number, you need to square this number only 332 times to exceed two, so it's really a very big jump from the previous number, but quite tiny in comparison to ...

One and a millionth
1.000001

Still very close to one, but not quite inconceivably close to one. It may seem insanely close, and as a ratio it really is, e.g. when you compare the volumes of two stars. But you only need to square this number 20 times to exceed two. How about:

One point oh one
1.01

Also known as a 1% increase. This is a big jump from the previous number, as objects with a size ratio of this can at least be distinguished if you look closely, something that you could never do with one and a millionth. You only need to square this 7 times to exceed two.

A 1% increase looks like a slight increase, and generally is one. For example, if something increases by 1% each year it would take about 70 years to double. This may seem like a slow increase or a fairly fast increase depending on the concept, but hey, we're now at numbers where an increase of this ratio is generally quite tangible. Though this increase with doubling time may seem unimpressive, with only a few more percentage points such an increase becomes absolutely drastic (see 1.07).

Twelfth root of two
1.05946309...

This is the number such that when you multiply it by itself twelve times, you get exactly 2 - yes that's how amazingly big this number is. It's notable as the pitch ratio of a half step in music, e.g. the pitch ratio of C and C-sharp, or the ratio of E and F. This number comes from the fact that two notes that are an octave apart have a pitch ratio of precisely two, and that there are twelve half-steps in an octave - it can be thought of as a number so big that pitches with a ratio of this can be easily distinguished.

Other interesting things about pitch ratios include:

A perfect fifth (like C and G) has a pitch ratio of exactly (or very nearly) 1.5, and perfect fifths are known for sounding natural and harmonizing well.

A tritone (half an octave, like C and F-sharp) has a pitch ratio of exactly the square root of 2, and tritones are known for their dissonance, though they find plenty of use in seventh chords. Not to say dissonance doesn't have its own merits of course.

One point oh seven
1.07

This is an example of a number that looks like a slight increase - however in reality that increase is quite staggering. If the world population were to multiply by this much annually (which it doesn't), that means it increases by 7% annually. That sounds quite small, but it actually means that the world population will double in only ten years - in twenty years it would quadruple and in only thirty it would multiply eightfold! Now that's a staggering increase, but pales in comparison to what you'd get by adding three more percentage points (see next entry).

One point one
1.1

This is 1.1, an increase by 10% from one. It's a number that can be thought of as so big that it's an easy-to-notice ratio almost regardless of what is involved. If you saw someone 10% taller than you, you'll definitely notice the difference in height right away (e.g. 6' is very easy to tell from 6'7" which is ten percent taller). An object 10% bigger than an object next to it is quite a bit bigger. If your computer has a 10% bigger hard drive than your friend’s, you’d be able to fit in quite a bit more stuff.

If something were to increase by 10% annually, the increase would be quite horrifying even compared to the already staggering 7%. If that increase were to happen to something, say the world population, then in only eight years the population doubles, in fifteen it quadruples, in twenty-two it multiplies eightfold, and in thirty years it would be 20 times larger! That increase easily crushes a 7% annual increase.

One point two
1.2

An approximation of the factor which the coolness rating of Rainbow Dash's dress should be multiplied by, helpfully provided by the spunky rainbow horse herself in her famous line, "it needs to be about 20% cooler". Yes, I shoved a reference to My Little Pony: Friendship Is Magic into this large number list, and no, I'm not sorry for it. I did at one point have a picture of the so-called "Mane 6" ponies on the entry for 6 after all.

Apéry's constant
1.20205

This is ζ(3) with the Riemann zeta function, an irrational constant which is the value of the infinite sum 1/13+1/23+1/33+1/43 ... It is an important constant in mathematics and also in physics, and it's known as Apéry's constant because Roger Apéry proved it to be irrational. Unlike some other values of the zeta function (e.g. ζ(2)), this one is not known to have a nice compact expression (aside from ζ(3) of course) and needs to be expressed with infinite sums. It isn't even known whether this is a transcendental number or not.

For more on the Riemann zeta function see ζ(2).

One and a fourth/quarter
1.25

This number is nearly always a sizable increase from 1. For example, if you are a man 6 feet (183 cm) tall you would be considered quite tall but not by any means unusually tall, but being 25% taller than that (i.e. 7 feet 6 inches / 229 cm) is considered extraordinarily tall.

Square root of 2
1.41421356....

The square root of two is one of the most important and well-known irrational numbers. It is defined as the number x such that x multiplied by itself is 2. It isn't too hard to prove that the square root of two is irrational, as it was one of the first numbers proven to be irrational. This number, like constants such the square root of 3, e, and pi, appears very often in mathematics. For example, the length of the diagonal of a square is always √(2)*side length.

e1/e
1.444667...

This is the mathematical constant e raised to the power of its reciprocal. It has a number of interesting properties. It's the highest value of the function y = x1/x. It's also the solution to the equation xe = e, which sounds impossible but it's not: (e1/e)e = e(1/e)*e = e1 = e. This is also the largest number x such that xx^x^x^x....... (infinite power tower of x's) converges to a finite value, a property discovered by Euler. If a number a is between this one and (1/e)e, an infinite power tower of a's will converge to the number b (b≤e) such that b1/b = a. For example, an infinite power tower of √2's will converge to the solution of b1/b = √(2), aka 2. An infinite power tower of any number larger than this will diverge to infinity.

See also (1/e)e.

One and a half
1.5

Probably the most notable increase from 1, and is a significant increase from 1. For example, if you're an adult and see someone 50% taller than you, they would look like an absolute monster, and if you noticed an object 50% bigger than its neighbor you would easily notice the difference.

Triangular root of 2
1.5615528... ≈ Δ√2 = (-1 + √17)/2

This YouTube video by Dr. Barker tells you about something triangular roots. Just as square roots are the inverse of the sequence of square numbers, triangular roots are the inverse of triangular numbers. For instance, 10 is the fourth triangular number, so the triangular root of 10, Δ√10, is 4.

To calculate the nth triangular number, you use the formula n * (n + 1) / 2, or ½n2 + ½n. If you plug those values into the quadratic formula, you get this formula for triangular roots: (-1 ± √(1+8x))/2. This means the triangular root of 2 is about 1.5615, slightly greater than the square root of 2. As numbers get higher, their triangular roots get close to their square root times √2, as you'll see when I calculate Δ√100. The ± sign in the formula means that numbers also have a negative triangular root. The "negative triangular root" of n is not -Δ√n, but rather -Δ√n - 1. See this entry in the non-positive number list.

Golden ratio / phi
1.618033... = [1+√(5)]/2

The golden ratio, sometimes denoted with the Greek letter phi (φ), is an important constant in mathematics, defined as the number x such that 1/x = x-1 and equal to half the sum of one and the square root of 5. It has several interesting properties. For example, it is the solution to the equation x = 1+1/x, and also the solution to the equation x2 = x+1 - this means that φ (1.618033...), 1/φ (0.618033...), and φ2 (2.618033...) all end with the same digits. Also, as you go higher and higher through the Fibonacci sequence the ratio between consecutive terms approaches the golden ratio.

There is also the "golden rectangle", a rectangle whose sides have a ratio of the golden ratio. It has the notable property that if you chop off the a square of side length a from the golden rectangle, where a is the length of the shorter side of the rectangle, you end up with another smaller golden rectangle. Additionally golden rectangles are commonly considered to be the most aesthetically pleasing, and thus many pieces of architecture are designed to resemble that rectangle.

π2/6
1.64493...

This is the value of the infinite sum, 1/12+1/22+1/32+1/42 ..., aka the value of ζ(2) with the Riemann zeta function. The zeta function is an extremely famous function in mathematics, noted ζ(n) with the Greek letter zeta and defined as the sum of the series 1/1n+1/2n+1/3n+1/4n ... and it's known for its interesting behavior and important properties. Some values of the function, such as this one, are known to have a nice compact expression, and some, such as ζ(3), are not.

One reason why this number is interesting is because it gives an example of an occurrence of pi in a place you won't expect it to, in the nice compact expression π2/6. This number has several interesting properties - for example, if you pick two random integers, the odds of them having no divisors in common is 1 in 1.64493..., and this number's reciprocal is the natural density of the squarefree numbers.

Square root of 3
1.732...

Another very important irrational number that crops up largely in geometry. For example, the height of an equilateral triangle with side length a is equal to a*√(3)/2.

Square root of pi
1.77245...

This number is the square root of pi, a number that crops sometimes in mathematics. It is the area underneath a bell-curve of height 1 and standard deviation 1. It is also equal to Γ(1/2) using the gamma function, a generalization of the factorial to any real or complex inputs. Since Γ(n+1) for positive integer n = n!, this number can be thought of as -1/2!, the factorial of -1/2. Both of these are examples of pi cropping up in a place you don't expect it to.

One point eight
1.8

1.8 is the ratio between an increase of one degree Fahrenheit and an increase of one degree Celsius, and therefore it (along with 32) shows up in the formula for converting between the two scales.

The Instantly Perceptible Range
2 ~ 6.999
Entries: 15

Two
2

Like one, two is another extremely important number, and the first integer that can only be defined from existing numbers. It also has many unique properties (READ MORE)

For a discussion of the importance of the powers of 2, see 1024.

Square root of 5
2.236...

The square root of five crops up sometimes in mathematics. It isn't as common as the square roots of 2 and 3, but still important, most notable in the definition of the golden ratio.

ln(10)
2.302585...

This is the number x such that ex = 10. It is the ratio of ln(x) (the natural logarithm) to log(x) (the common logarithm), and it therefore makes calculating common logarithms easier if you have an algorithm to calculate ln(x).

Two point five four
2.54

2.54 is defined by the SI as the exact length of an inch in centimeters. As a kid I always thought 2.54 was just an approximation, but in today's world it's important to have simple, precise definitions for units outside the metric system. An inch has been exactly 2.54 centimeters since 1930 in the United Kingdom, and 1934 in the United States.

e
2.71828...

We're going to talk about e! The big, famous constant e! OK, there's more to e than memetic thumbnail of that Numberphile video, but James Grime's enthusiasm about the number is as infectious as it is well-deserved.

This is a well-known mathematical constant that competes with the square root of two as the second best known irrational constant, behind pi. It is also called Euler's number since this important constant was made famous by Leonhard Euler. However, he was not the first person to use the constant—that would be Jacob Bernoulli, who would be really annoyed about this if he didn't also have plenty of mathematical concepts to his name.

e can be defined in multiple ways: the limit of (1+1/x)x as x approaches infinity, the number a such that the derivative of ax is itself (that means that at any point, the slope of the graph of ex is the same as the x-coordinate of the point), the number a such that the function ax has slope 1 at x = 0, and more. It varies greatly what people first learn e to be, and I first learned it as the number a such that the function ax has slope 1 at x = 0.

Another way to calculate e is with the sum:

1 + 1/1! + 1/2! + 1/3! + 1/4! ... (sum of the reciprocals of the factorials)

In fact you can calculate ex for any x with the sum:

1 + x/1! + x2/2! + x3/3! + x4/4! ...

e appears most often in calculus when working with functions, and there it crops up again and again. It even appears a few times in googology, like when working with infinite power towers, estimating large factorials like the contrived googolbang, and in the definition of Skewes' number.

Three
3

Three, the result of adding one to two, is typically the minimum number of instances needed to recognize a pattern, so it's no wonder that sets of three are so common... (READ MORE)

π / Pi
3.14159265...

Pi is the most well-known of all the irrational numbers. It's most commonly defined as the ratio of the circumference of a circle to its diameter. People knew about it since ancient times; ironically, despite ancient Greek mathematicians knowing about the number, it was not represented with the Greek letter pi until the 1700s.

Pi shows up in mathematics when working with circles and other geometric functions such as the trigonometric ratios, but it also shows up in a lot of places you wouldn't expect it to (example). This is part of why pi has been given a lot of cult significance as a number - people have memorized pi to insane amounts just for the sake of it, and the current record for digits of pi memorized is 100,000! Many people have pointed out some unusual coincidences regarding the digits of pi as well, as is often the case with other cult numbers.

Because pi's digits go on forever, it is a common misconception that pi is an infinite number; therefore, pi has sometimes been mistakenly thrown into the large number discussion as an example of a really large number. Well guess what: an even more incredible number is 3.2, or better yet, the unfathomably large 4! Something some might consider a little more clever would be pi with the decimal point removed, but since such a "number" is not finite, it's the same thing as throwing infinity into the large number discussion. As I said in the introduction to this site, throwing infinity into the large number discussion is cheating, and nobody likes cheaters.

However, pi does indeed legitimately show up a few times in googology, most notably in Stirling's factorial approximation, a method to estimate large factorials like the googolbang. It also appears in googolisms like Ballium's number and piplex, but those don't really reflect pi's use in googology as much as its popularity as a number.

Mìlǜ (密率)
355/113
3.14159292035...

This is an approximation for pi that has been known in China since, believe it or not, the fifth century. It was discovered by the ancient Chinese mathematician Zu Chongzhi. The Chinese name for his approximation means "close ratio". If you're wondering what the Chinese call pi, it's "yuánzhōulǜ"(圓周率), which means "circle ratio".

It's quite fortunate that such a simple approximation is accurate to six decimal places. See 292 for why this is.

Yuēlǜ (约率)
22/7
3.142857... where the underlined digits repeat infinitely

The most common approximation for pi, often used in computer programs or rough calculations. This one has some personal significance to me. First of all, I noticed by myself as a kid that 3 1/7 is close to pi. Later, one time in third grade I was at school talking about pi with a friend of mine who was also interested in mathematics. He used long division to calculate 22/7, and said that was equal to pi. I knew it was an approximation, and told him of that (I think). This shows that people often mistake approximations for a number for the actual value.

The Chinese name for his approximation means "approximate ratio". Zu Chongzhi gave Chinese names for both 22/7 and 355/113.

Square root of 10
3.16227...

Another less accurate approximation for pi, but a somewhat convenient one in ancient times because it's the square root of our numeral base. 

Square root of 12
3.46410....

Once as a child (first or second grade), for whatever reason I thought pi was the square root of 12, until my mom told me that it wasn't. I have no idea why I ever thought such a ridiculous thing, but nonetheless I was surprised when I later learned that the square root of 10 (not 12) was sometimes used as an approximation for pi.

Four
4

Four is the smallest composite, the smallest non-trivial perfect power, and a common occurrence with twos in googological functions. It is also, among other things, a number things are very often grouped in (READ MORE)

Five
5

Five is a prime, Fibonacci, Fermat number, part of the only twin prime triplet, the number of Platonic solids, and more. A human hand typically has five fingers, which is why it's half of the base of our numeral system (READ MORE)

Six
6

Some of six's properties is that it's the smallest perfect number, a triangular number, a factorial, primorial, the number of faces on a cube, the number of trigonometric functions, etc. (READ MORE)

Six is also the boundary between class 0 and class 1 numbers, and the base of Robert Munafo's entire idea of classes - class 0 numbers are numbers small enough to recognize immediately, and they range from 1 to 6. Then class 1 numbers are numbers too large to recognize immediately but still physically perceivable, and they go range from 7 to a million. Read my page on 6 (link above) for more on that. 

Tau / τ

6.28318531... = 2*π

This number is two times the famous number pi. It's the ratio of a circle's circumference to its radius (not diameter), and it's often represented with the Greek letter tau. This number is seen often enough that it's sometimes treated as a mathematical constant in its own right, and some people argue that tau deserves to have all the recognition and fame that pi has.

The Secondary-Sense Range
7 ~ 19.999
Entries: 15

Seven
7

Seven is a prime number, a Mersenne number, a factorial prime, the smallest number of sides of a regular polygon not constructible with compass and straightedge, and more. It is a number that has a famous cult significance, given much spiritual significance since ancient times (READ MORE)

Eight
8

Eight is the second and smallest non-trivial cubic number (2^3), a power of 2 (2^3), and the largest cubic Fibonacci number. It's also the number of vertices on a cube, as well as the number of cardinal directions most people would choose from when specifying on a map (north, northeast, east, southeast, south, southwest, west, northwest). 

8 is also the number of bits in a byte (stands for "by eight") in computing, giving it a lot of digital significance. Since a bit can store 0 or 1, a byte can store 256 different values. Bytes are also multiplied by powers of 1024 (or 1000, confusingly enough) to give kilobytes, megabytes, gigabytes, etc. For more on that see the entry for 1024, a general discsussion of the convenience of powers of 2 to us Earthlings.

Eight occurs commonly in nature, in creatures such as spiders or octopi (the name "octopus" even means "eight feet"). A set of eight also often has the sense of completeness a set of four has.

The prefix octa- (both Latin and Greek, such as October, octagon, octahedron, octopus, octave) is used for eight, and it is one of the more commonly used number prefixes.

Nine
9

Nine is 3^2, the third square number. It is the smallest odd composite number. It is also the maximum number of cubes needed to sum up to any positive integer, as well as the third expofactorial (3^2^1, see 262,144) and the fourth subfactorial (see 44). 

9 and 8 are a unique pair of numbers. They are the only pair of perfect powers that neighbor: 8 = 23 and 9 = 32. They're also the only positive integer pair of xy and yx, x < y, where xy > yx. For more oddities with exponents see the entry for 16, the only nontrivial integer case of xy = yx, and for more on close perfect powers see 26, the only number to fall right between a square and a cube. Also, 8 and 9 add up to 17 :)

9 is the largest number that is normally represented with a single glyph - beyond this point multiple digits are combined using place-value notation, which is today the dominant way to denote numbers. Place value notation has been used since the time of the Sumerians (see my large number timeline for more on that).

Because 9 is the largest single-digit number, it's easy to test for divisibility by 9. Just add the digits of the number repeatedly until you end up with a single-digit number - if that number is 9 then it's divisible by 9. For example, take 76,347: 

7+6+3+4+7 = 27 

2+7 = 9 - therefore 76,347 is divisible by 9. This method is known as "casting out nines", and works for the largest single-digit number in whatever base you're using.

Prefixes for 9 include nona-/nov- (Latin, like in November) and ennea- (Greek, like in enneagon and enneagram). They’re relatively uncommon, but not obscure.

(A pet peeve of mine is when Greek-based naming systems (mono-, di-, tri-, tetra-, penta ... ) use nona- instead of ennea-, even though nona- is Latin. For example, a 9-sided polygon is often called a nonagon. I hate that usage with a passion, and think nonagon should never be used and instead have the word enneagon.)

Ten
10

Ten is a number that is both triangular and tetrahedral, and it also has a special property discussed at the entry for 39 (sum of prime numbers 2 to 5 and the product of 2 and 5). But its importance in humanity definitely comes from being the base of our numeral system (READ MORE)

Booga-e (by Nayuta Ito's proposal)
~ 10.30495187

Up-arrow notation is a famous way to make insanely large numbers - if you're not familiar with it read here. However, a notable gap in the notation is its lack of support for arguments that are not nonnegative integers, such as 1.5, -3, or pi.

To solve this problem, Nayuta Ito of Googology Wiki made a proposal to define up-arrows for any positive real arguments - for example, in that system we can define values like pi^^^e. But better yet, the system allows us to define fractional up-arrows - for example we can have 3{1.5}3 ({x} means x up-arrows), which is equal to about 144.023. The system has these three rules:

b>1 and n>1: a{n}b = a{n-1}b-1

0≤b≤1 and n>1: a{n}b = ab

0≤b≤1 and 0≤n≤1: a{n}b = abn*b1-n

This particular number is defined with Sbiis Saibian's booga- prefix, a prefix defined as booga-x = n{n-2}n. For example, booga-four = 4{4-2}4 = 4{2}4 = 4^^4, a number with about 8*10^153 digits. For more on the booga- prefix read here.

With that generalization of up-arrows, we can define booga-x for any positive number x>2. This number is booga-e in that system, equal to about 2.718{0.718}2.718. The number is equal to exactly eee-2*ee-3, or about 10.30495.

Another number you can define with that system is booga-pi, which isn't too much bigger.

Eleven
11

Eleven is the fifth prime number, and the smallest two-digit prime. It's the smallest repunit prime, and the next is equal to 1,111,111,111,111,111,111 - see that number's entry for more.

Multiples/powers of eleven tend to be interesting: for example, 121, 1331, and 14,641, the square, cube, and fourth power of eleven, coincide with the first few rows of Pascal's triangle, a very important triangle of numbers in mathematics. Another property is that when you take a multiple of 11, reversing it gives another multiple of 11. All this stems from 11 being one more than the base of our numeral system, and is thus something specific to base 10. This kind of stuff is also reflected in the School House Rock song "Good Eleven", which talks about multiplying by eleven and how easy it is.

It's also easy to test for divisibility by 11 - my favorite way is to subtract the last digit from the rest of the digits, and repeat that process - if you get 0 then it's divisible by 11, and if you don't it isn't. For example:

Test 45,056:

4505 - 6 = 4499
449 - 9 = 440
44 - 0 = 44
4 - 4 = 0 - therefore it's divisible by 11.

This makes eleven the fourth easiest prime number to test for divisibility in base 10: the easiest are 2 and 5 (check the last digit), followed by three (sum of digits is divisible by 3). The rest of the prime numbers are harder to test for divisibility in base 10.

Even better, this divisibility test also gives you the quotient when dividing a number by 11 - the quotient is each digit you subtracted, read in reverse order - in our case, the digits subtracted to test 45,056 for divisibility by 11 and 6, 9, 0, and 4, and therefore the quotient 45,056/11 = 4096.

The name "eleven" literally means "one left" - like twelve it doesn't match with the pattern of names "thirteen" to "nineteen".

Because things often go on a scale from one (or zero) to ten (since ten is our numeral base), "up to eleven" is a common idiom for taking something beyond the extreme - the idiom is usually credited to have originated from the movie "This is Spinal Tap". It was once the only thing I know about that movie, until I learned The Simpsons' voice actor Harry Shearer was involved in it.

Twelve / dozen
12

Twelve is a highly composite number - it has more divisors than any smaller number (for more on highly composite numbers see 36). Because of that, twelve has a special numerological significance, as sets of twelve are notably common in the human world. Some examples are:

• 12 months in a year

• 12 zodiac signs

• clocks in America and sometimes in Europe are 12 hours

• 12 inches in a foot

• 12 eggs in a typical package

• 12 days of Christmas

• the 12 Olympic Greek gods

• 12 function buttons on a keyboard

• 12 notes in a scale in Western music

In fact, sets of twelve are so common that such a set gets a special term called a dozen (for example, a box of a dozen donuts) - the name comes from French "douzaine" meaning twelve of something.

Twelve has the property of being the smallest abundant number. An abundant number is a number whose factors add up to a number larger than the original. Twelve is abundant since its factors (1, 2, 3, 4, and 6) add to 16, which is larger than 12. Additionally, a perfect number is a number whose factors add up to exactly the original number (they're very rare, for more about them see 496) and a deficient number is a number whose factors add up tot a number smaller than the original. An interesting property of abundant numbers is that if a number is perfect of abundant, all of its multiples will be abundant. Therefore abundant numbers are common, as around 25% of numbers are abundant.

Twelve is also the number of pentominoes (see also 35), and the fourth "weak factorial" (see that entry for more), i.e. the smallest number divisible by 1 through 4.

The main prefix for twelve is dodeca-, which is Greek. It is found in words like dodecagon and dodecahedron. Its Latin prefix, duodeci-, is found in a few words like duodecimal, the usage of twelve as a numeral base instead of ten.

(God, imagine how awesome it would be if we used a duodecimal (dozenal) system instead of a decimal system. A base divisible by 2, 3, 4, AND 6! Though seximal, base six, is also an appealing option—thanks YouTuber jan Misali.)

Thirteen / baker’s dozen
13

Thirteen has a number of mathematical properties: it is the sixth prime, a Fibonacci number (next is 21) and the fourth busy beaver number, the largest fully known value of the sequence. The first three busy beaver numbers are 1, 4, and 6 - for more on them see 107 and 4098. The prefixes for thirteen, fourteen, fifteen, etc. are trideca-, tetradeca-, pentadeca-, etc.

13 is the number of Archimedean solids - those are 3-D figures that are fully symmetric like the five Platonic solids, but have two or more different types of regular-polygon faces. Those 13 solids are:

1. the truncated tetrahedron, the simplest which is just a tetrahedron with its corners cut off

2. the cuboctahedron, as its name suggests it combines properties of the cube and the octahedron with 8 triangular and 6 square faces

3. the truncated cube, a cube with the corners cut off which has 8 triangular and 6 octagonal faces

4. the truncated octahedron, an octahedron with the corners cut off which has 6 square and 8 hexagonal faces

5. the rhombicuboctahedron, this is my favorite one, which I like to think of as the octagon's 3-dimensional equivalent

6. the truncated cuboctahedron, a cool shape which looks kind of like the rhombicuboctahedron's more complex big brother

7. the snub cube, this one looks like a smoothed-out cube and has 6 square and 32 triangular faces

8. the icosidodcecahedron, this one combines the icosahedron and the dodecahedron with 20 triangular and 12 pentagonal faces

9. the truncated dodecahedron, which has 12 decagonal (10-sided) faces and 20 small triangular faces that look "chipped-off"

10. the truncated icosahedron, this one's shape is familar because it looks very much like a soccer ball

11. the rhombicosidodecahedron, this is has 62 faces and is somewhat reminiscent of the rhombicuboctahedron

12. the truncated icosidodecahedron, this one is to the rhombicosidodecahedron as the truncated cuboctahedron is to the rhombicuboctahedron

13. the snub dodecahedron, this is the most complex and the smoothest-looking one with 92 faces

13 is also the smallest emirp in decimal. An emirp is a prime number whose digits, when reversed, produce a different prime, in this case 31. See also 17.

Thirteen is perhaps most notable for having a connotation of bad luck in Western culture. This means that often, house numbers or elevator floors of 13 are omitted, and Friday the 13th is considered a very unlucky day. In fact, to keep an even-odd-even-odd pattern in numbering, sometimes even 14 is omitted in such numberings - for example, I have seen airplane seats that are numbered 1, 2, 3, 4, ... 10, 11, 12, 15, 16, 17, etc.

13 of something is sometimes known as a baker's dozen, similar to the term "dozen" to twelve, but the term is a lot less widespread.

It's also the current best lower bound to the problem where Graham's number arose - see my page on Graham's number for details.

Triangular root of 100
13.650971... ≈ Δ√100 = (-1 + √801)/2

Using the formula (-1 ± √(1+8x))/2, you can calculate the triangular root of 100 as this. The 13th triangular number is 91 and the 14th is 105, so you could say the 13.650971th triangular number would then be 100. Note that this number is quite close to 10√2 ≈ 14.14121.

Fourteen / Poulter's dozen
14

14 is an example of a semiprime, a number that is the product of two prime numbers, in this case the product of the primes 2 and 7. Semiprimes have no more than two non-trivial divisors (i.e. divisors that are not one and itself), and if the semiprime is a square then it has only one non-trivial divisor.

Fourteen is the first even number that doesn’t really have a feel of completeness due to being the double of the somewhat irregular seven - it can't be divided by very many numbers, as its only factors are 1, 2, 7, and 14. 

14 is the smallest Keith number (see 197).

I have heard of 14 of something being called a "poulter's dozen", though that usage is quite rare.

Bignum Bakeoff was a competition held in 2001 where the goal was to make a C program with no more than 512 characters to produce as big of a number as you can. 20 entries were submitted, and 14 of them produced a value that was not 1. Of the other six, three of them (carnahan.c, pete.c, pete-2.c) didn't terminate, and three of them (dovey.c, edelson.c, f.c) only produced 1. The fourteen programs that did produce a large number were, in order of output (smallest to largest):

pete-3.c, pete-9.c, pete-8.c, harper.c, ioannis.c, chan-2.c, chan-3.c, pete-4.c, chan.c, pete-5.c, pete-6.c, pete-7.c, marxen.c, loader.c

Click on any one of the links above to see an entry on the list describing each of the programs submitted.

Fifteen
15

Fifteen is an example of a triangular number, a number that is equal to the sum of the first x integers - in 15's case, 15 is 1+2+3+4+5. The nth triangular number can be calculated with the well-known formula, n*(n+1)/2. Triangular numbers are named that because a triangular number of things can be arranged as a triangle. For example, with 15 we have:

    o
   o o
  o o o
 o o o o
o o o o o

The first 20 triangular numbers are:

1  3  6  10  15  21  28  36  45  55
66 78 91 105 120 136 153 171 190 210

15 is the magic constant of the smallest and best-known magic square, a square of distinct numbers (none is used more than once) whose rows, columns, and diagonals add up to the same number. The smallest is the only (not counting rotations and reflections as distinct) 3x3 magic square, which is:

4 9 2
3 5 7
8 1 6

15 is halfway between the numbers 10 and 20, and therefore one of the more common numbers from 11 to 19. Like 14, 15 is also an example of a semiprime. 14 and 15 are the first pair of semiprimes that are neighboring numbers - I call such numbers twin semiprimes. See also 33.

(the last thing you'd want on your large number site is someone not being motivated to update it, but as it turns out... I'm actually updating it now!)

Sixteen
16

Sixteen is an example of a perfect power, a number that can be expressed exactly with exponents and integers. Of perfect powers, numbers that can be expressed as x^2 are called square numbers/perfect squares, numbers expressible as x^3 are called cubic numbers, numbers expressible as x^4 can be called tesseract or quartic numbers, continue with penteract, hexeract, etc, with the Greek number prefixes.

16 is 2^4 (second tesseract) and 4^2 (fourth square). Actually, it's a particularly interesting perfect power. It's the first number expressible as a perfect power in more than one way (see also 64, 81, and 4096). Better yet, it has the special property of being the only non-trivial solution of x^y = y^x where x and y are both integers. See also square numbers at 25.

Sixteen is part of an interesting sequence of numbers of the form x^x^x. It is the second member of the sequence, preceded by the trivial 1 and succeeded by an astronomical number equal to about 7.6 trillion. For more on that sequence see my entry on 10^10^10.

16 is another more common number because it’s a power of 2. Its prefix, hexadeca-, is fairly common, such as in the term "hexadecimal", the usage of 16 as a numeral base - that connects to computers' connection with powers of 2 (see my entry on 2 for more on that).

In the googo- naming system, 16 can be named googoij. See also googoi (2) and googoiji (216).

Seventeen
17

Seventeen is both one of my childhood favorite and current favorite numbers, and one with so long of an entry that it has its own page (READ MORE)

Eighteen
18

18 is this list's example of a composite number, which you probably already know, is a number that can be evenly divided by numbers other than 1 and itself. In 18's case, 18 can be divided by 2, 3, 6, and 9. Its prime factorization is 2*3^2. Why did I choose to give an "example entry" for composite numbers? Because many of my list's entries are mainly examples for certain classes of numbers (Kaprekar numbers for 45, highly composite for 36, primorial primes for 29, sums of primes for 58), and thus I've decided to add example entries for composite and prime numbers for completeness's sake - Robert Munafo does this to a greater extent on his list.

I discussed friendly and solitary numbers in my entry for 10: 18 is the smallest nontrivial case of a number known to be solitary, which is to say no other integer shares the ratio between the sum of its divisors (including itself) and itself—13/6 in this case. I say nontrivial because if a number and the sum of its divisors have no factors in common, then we automatically know it's solitary. This is the case for all prime numbers and some composite numbers.

Because 18 is quite highly divisible for its size, it's a fairly commonly used number for grouping things. It's the second smallest abundant number, followed by 20, 24, 30, 36, 40, 42, 48....

18 has the property of being the only positive integer that is exactly twice the sum of its digits. This is quite easy to see if you consider that if a number has three or more digits the sum of the digits will always be necessarily much less than half of the number, and then you can check each number below 100 individually to see this.

18 is the most common adulthood age in the world, as in most countries the age you are considered an adult and the voting age is 18. Other common adulthood ages are 16 and 21.

Nineteen
19

Nineteen is an example of a prime number, a number that can't be divided evenly by any numbers other than 1 and itself. Primes are significant in mathematics for zillions of reasons as I discuss on this list, and the first few are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37...

Nineteen is the eighth prime number. It's also twin primes (primes two numbers apart) with 17, cousin primes (primes four numbers apart) with 23, and sexy primes (primes six numbers apart) with 13. This makes it an example of a number in all three well-known prime pairs. Such numbers are uncommon, and another example is 11. See also 23, 53, and 211.

19 and 17 are an interesting prime pair; they're twin primes that are both exponents of Mersenne primes (131,071 and 524,287 respectively). 17 and 19 appear to be the largest twin prime pair to have this property.

The Higher-Secondary Range
20 ~ 49
Entries: 33

Twenty
20

o o o o o
o o o o o
o o o o o
o o o o o

^ For visualization, that above is twenty o's.

Twenty is an example of a tetrahedral number - a tetrahedral number is a number that is equal to the sum of the first n triangular numbers (see 15). The first few are 1, 4, 10, 20, 35, 56, 84, 120, 165, 220... and they're called tetrahedral numebrs because a tetrahedral number of spherical objects (e.g. oranges) can be arranged in a tetrahedron (aka a triangular pyramid):

layer 1  layer 2  layer 3  layer 4
   o              
  o o       o
 o o o     o o       o
o o o o   o o o     o o       o

Like with triangular numbers, tetrahedral numbers have a special formula: the nth tetrahedral number is equal to n*(n+1)*(n+2)/6.

Twenty has sometimes been used as a base of number naming. This may be because twenty is the total count of all fingers and toes on a human being. For example, the Mayans used twenty as their base, with five as a sub-base, and 20 is also used in traditional Welsh and in some African languages, and in Danish numerals from 50-99 (which the Swedes and Norwegians love to tease them about) and in French numerals from 60-99.

Twenty of something is sometimes called a score (see the entry for 87 for an example of this usage), though this usage is somewhat dated nowadays.

The most common prefix for 20 is icosa- (from Greek), like in icosahedron. The Latin prefix for 20, viginti-, is sometimes used in terms like vigintillion, a favorite large number of mine.

20 is the number of entries that were submitted in Bignum Bakeoff, a large number competition in 2001 where the goal was to make a program in C to generate the largest number you can. For more on it see 14.

And finally, 20 is the number of possible opening moves one can make in chess. There are 16 possible pawn moves, and four possible knight moves.

Twenty-one
21

21 is notable in the English numeral system for being the first number name constructed from existing names. This is somewhat important in googology because the term "googolism", albeit a broad term, more often than not refers to a specific unique name for a number, and twenty-one is the smallest nonnegative integer that does not get a unique name in English - therefore in some sense twenty-one is the smallest nonnegative integer name in English that is not a googolism. Sbiis Saibian has further discussed the meaning of the term "googolism" on some of his Googology Wiki blog posts. See also grangolplex/googolcentiplex.

21 is also the 7th Fibonacci number, preceded by 13 and followed by 34 - for more on them see 89. It is also the value of S(3) in the frantic frog function, a variant of the busy beaver function in googology. The S function is noted S(n) (for more on it see 107), while the busy beaver function is noted Σ(n) or BB(n).

Also worth noting that 21 is the legal drinking age in the United States, although in most of the world it's 18.

Twenty-two / Dumevalka
22

22 is the smallest Lychrel number in base 2, i.e. the smallest number in binary that will never become a palindrome if you take the number, add its reverse, and repeat the process - see 196 for details. In bases that are powers of two it's easy to see that some numbers are Lychrel numbers. In base 10 that has not been proven, though there are some candidate base-10 Lychrel numbers, the smallest and best known of which is 196.

Twenty-two is the exact value of a very small googolism coined by Googology Wiki user SpongeTechX using a notation he invented called copy notation. The number is called dumevalka, and it's defined as 2[2,2]. This solves solves to 2[[2]] (a[b,c] = a[bc]b, where [b is [[..[[ with b [s) = 2[2] (a[bc]b = a[b-1a[b-1a[b-1.......[b-1a]b-1.....]b-1 nested c times, so this is 2[2] with two nestings) = 22 (a[b] = aaa.....aaa written as a number, with b copies of a, so this is two copies of two). It's an example of degenerate cases produced by two in googological functions, and in fact it mirrors the degenerate case of 4 in up-arrows, Conway chain arrows, and Bowers' arrays.

22 is also (as of August 2014) the smallest value for which the Busy Beaver function is known to surpass Graham's number - that is, BB(22) is known to be greater than G, but BB(21) and below may or may not be. Previously the smallest such value was 24; before that it was 25, and before that it was 64.

Twenty-three
23 

23 is the ninth prime number. It's notable for being the first one not to be part of a pair of twin primes (primes two numbers apart) or closer, since 21 and 25 are both composite. Such prime numbers are called isolated primes, but once you get to large enough numbers, the majority of primes will be isolated primes.

Also, 23 is an example of a factorial prime, a prime one more or less than a factorial number, similar to primorial primes (see 29). The sequence of factorial primes begins 2, 3, 5, 7, 23, 719, 5039, 39,916,801....

23 is the smallest prime number with the property that you can remove any digit and it's still prime - see also my entry for 137.

Additionally 23 is the number of chromosome pairs in a normal human cell (see also 46).

23 is also a cult number noted largely for Illuminati associations and conspiracy theories, and a number I seem to randomly produce pretty often. For example, 23 is randomly chosen in the entry for 1, and a draft version of this list had -23 on the list but specifically excluded 23 (sorry people who like the number 23). Therefore 23 is my second favorite cult number behind 17.

Gaz
23 ⅔

See 10^74.

Twenty-four
24

I've always liked the number 24—it can be divided in many different elegant ways. Two twelves, three eights, four sixes, six fours, eight threes, twelve twos. I especially have a soft spot for this number because of the card game 24, discussed at the end of this entry. An older version of this list had the passage: "For a time, when I didn't have a solid decision on my "favorite number", 24 was my favorite number (probably because of the game) - I switched back and forth between several numbers before settling on a childhood favorite as my "favorite number", 7." And as of this writing (the year 2024), I think I'm warming up more to 24 again.

24 is an example of a factorial number, a number that is the product of the first n positive integers. It's the fourth factorial, equal to 4*3*2*1. The nth factorial is denoted n!, and the sequence of factorials grows pretty quickly, starting with 1, 2, 6, 24, 120, 720, 5040, 40,320 ... Factorials have many applications in mathematics, in combinatorics and elsewhere. The most well-known is that n! denotes the number of ways you can arrange n objects in order - for example here are the 24 (4!) ways to arrange 4 objects:

ABCD ABDC ACBD ACDB ADBC ADCB
BACD BADC BCAD BCDA BDAC BDCA
CABD CADB CBAD CBDA CDAB CDBA
DABC DACB DBAC DBCA DCAB DCBA

The factorial is a classic example of what the non-googologists see as a fast-growing function, and therefore it's part of the layman's toolbox for generating large numbers. Some googolisms are defined with factorials like the googolbang, and Lawrence Hollom has developed a powerful (yes, powerful even by googology standards) notation to extend the factorial function into the realm of googology.

24 is also a highly composite number since it has more divisors than any smaller number. Its non-trivial divisors are 2, 3, 4, 6, 8, and 12.

There are quite a lot of things that are grouped into 24, similarly to twelve. For example, there are 24 hours in a day, food and bottles are often packaged in groups of 24, sports teams often come in groups of 24, and purity of gold is measured on a scale of 0 to 24 carats.

24 is the name of a popular game where you are given four single digit numbers and you need to use three steps with addition, multiplication, subtraction, and division to turn them into 24. The choice of 24 was probably motivated by the fact that 24 has a lot of divisors for its size.

Twenty-five
25

25 is the fifth square number (5^2), and this list's example of a square number. Square numbers are numbers that can be expressed as a number multiplied by itself, or as x^2 (pronounced x squared), and are named like that because square numbers of objects can be arranged in a perfect square - for example here's a square of 25 o's:

o o o o o
o o o o o
o o o o o
o o o o o
o o o o o

The first 20 square numbers are:

1    4    9    16   25   36   49   64   81   100
121  144  169  196  225  256  289  324  361  400

See also cubic numbers at 125 and perfect powers in general at 16.

25 shows up quite often in life, being a quarter of 100 (see also 50). For example, many currencies have a 25-cent coin, and 25% is one of the most used percentages (see 1/4).

25 is the smallest square that is a sum of two squares: 25 = 52 = 32+42. However, this property isn't too special - it's more interesting that 25 is a square that is a sum of two consecutive squares because that property is rare. The first few squares with that property are 25, 841, 28,561, 970,225....

25 is also the smallest (and only two-digit) Friedman number. A Friedman number is a number whose digits can be rearranged with mathematical operators to produce the original number. In 25's case, 25 = 5^2.

Twenty-six
26

26 is the only number that falls directly between a perfect square (25) and a perfect cube (27). It also might be the one and only number that falls directly between a pair of perfect powers, though this hasn't been proven - proving that kind of thing is quite difficult. For example, not until 2002 was it proven that 8 and 9 are the only pair of perfect powers that neighbor. It's been conjectured that for any integer n there's only a finite number of pairs of perfect powers that differ by n, though as of yet that hasn't been proven.

26 is also the number of letters in the English alphabet. They are abcdefghijklmnopqrstuvwxyz, making 26 somewhat culturally significant, such as in combinatorics with things like codebreaking. Powers of 26 also appear sometimes in everyday combinations, e.g. 263 = 17,576 possible three-letter acronyms.

Because there are 26 letters in the alphabet, 26 had some significance to me as a kid, since when I was 1 to about 6-7 years old the alphabet was pretty much my favorite thing in the world.

(pointless fact: 26 is called "lel" in Sbiis Saibian's experimental unique name system for numbers, which pretty much explains why such systems don't work. This connects to me because for a while I tended to use "lel" in place of "lol". By the way, 28 is called "lol" in that system.)

Twenty-seven / Booga-three / Fzthree
27

27 is equal to 3^3, the third cube. It can be defined with tetration as 3^^2 but is much too small to even be a small tetrational number, not even a good exponential number. Related to being the third cube, it's the smallest odd composite number that is not a semiprime.

27 has an interesting property relating to cubes - the digits of 27^3 (19,683) add up to 27 (1+9+6+8+3 = 27). 27 is the largest number with this property, and 8 is the smallest (not counting the trivial cases of 0 and 1). - 17, 18, and 26 also have this property. It's especially interesting that 8 and 27, the smallest and largest number with this property, are themselves cubes.

27 used to have a notable property in Bowers' array notation that is now held by a googolism called tritri - {3,3,3} in Bowers' array notation once was 3^3, but it's now equal to 3^^^3 since Chris Bird suggested to make {a,b} solve to ab instead of a+b. See ultatri for an example of 27's occurrence in Bowers' arrays. 27 is still is a very prominent number when working with threes in googology (example: 3^^^3 is a power of 3^27 threes), especially often in chained arrow notation (see part 2 of this list for examples). This is because the quirks of chained arrow notation give rise to exponents in the middle of each other - for example, 3->3->2->2 solves to {3,3,27} in Bowers' arrays. See my article on Conway chain arrows for further information.

With the fz- prefix which takes a number to the power of itself, 27 can be named fzthree. Sbiis Saibian coined a prefix called booga-n, meaning n^n-2n, or n n-ated to n in terms of hyper-operators. With that function, 27 can be represented as booga-three. For more on the booga- prefix read here. See also booga-four.

27 is also another cult number, and it's Robert Munafo's favorite as he discusses on his number list - he says that he "notices it a lot more than he 'should'" and puts some 27-based numbers on his list just for the sake of 27-based numbers. These three links are a few 27-based cult number websites, and this page (which gives a far far better covering of 27 than my entry does) might as well count as one. An example of the cult of 27 is the 27 club in music - the 27 Club consists of the many musicians (like Jimi Hendrix, Jim Morrison, Kurt Cobain, or Amy Winehouse) who died at the age of 27, giving rise to some speculation. 27's cult status actually dates all the way back to Roman times, or so I'm told at least.

Personal: 27 is part of the product 41*27

Twenty-eight
28

28 is the second perfect number. It's a perfect number because it's equal to the sum of its factors, which are 1, 2, 4, 7, and 14. The previous perfect number is 6 and the next is 496 (see that entry for more on perfect numbers).

28 is also used as an approximation for the number of days in a lunar month, which matches nicely with the designation of 7 days in a week.

Twenty-nine
29

29 is the tenth prime number. It's twin primes with the Mersenne prime 31, and it's a primorial prime as well. A primorial prime is a number one more or less than a primorial (see 30 for more). If you add 1 or -1 to a primorial (the product of the first n prime numbers), then you automatically know that number is prime.

29 is the number of notches on the Lebombo Bone, a 35,000-year-old bone discovered at the border of South Africa or Eswatini that was the first known documentation of numbers. It was clearly used for counting something from the notches in the bone. For more on the bone see my large number timeline.

It's also the smallest number x such that Graham's number cannot be written as a power tower of x's (not)

Thirty
30

o o o o o o
o o o o o o
o o o o o o
o o o o o o
o o o o o o

Thirty is the product of the first three prime numbers (2*3*5). Products of the first n prime numbers are known as primorials, and n primorial (the product of all primes less than or equal to n) is sometimes denoted n#.

30 is a common number for dividing things evenly, since it can be divided evenly into twos, threes, fives, sixes, and tens, and it forms two primorial primes (29 and 31). However it's not a highly composite number.

Thirty is also the largest number with the property that all smaller numbers that are relatively prime to it (i.e. have no factors in common with 30) are prime.

Thirty-one
31

Thirty-one is a prime number with several interesting properties. It is perhaps most notable as the third Mersenne prime (M5 = 2^5-1), since Mersenne primes are known to be the easiest way to find very large prime numbers. It's also a primorial prime, along with 29.

31 is a member of the primeth recursion sequence, a sequence defined as PR(1) = 1 and PR(n+1) = the nth prime number. The first few values of the primeth recursion sequence are 1, 2, 3, 5, 11, 31, 127, and 709. The sequence is the nth term sequence for the prime numbers - see my page on the Ackermann function (which is closely related to such sequences) for details. Interestingly, two consecutive members of that sequence (31 and 127) are both Mersenne primes.

31 is the starting member of a sequence of similar-looking prime numbers that starts with 31, 331, and 3331, continues like this, and ends with 33,333,331. A similar sequence notable for being pretty long is discussed in the entry for 43.

31 also one of only two numbers that is a more-than-2-digit repunit prime in more than one base: it is 11111 in binary and 111 in base 5. The only other number with this property is 8191.

Thirty-two / Binary-eyelash mite
32

This number, due to being a power of two (second penteract), has sort of a digital feel to it. This is because computers run on binary digits, and therefore basing things on powers of two is often most convenient. A direct example of 32 in computing is the system32 folder and 32-bit.

32 is the third member of the sequence 11+22+33 ... +nn, expressible as 11+22+33. 32 seems to be the largest perfect power in the sequence, but honestly that's just a guess.

32 is also notable in science as the freezing point of water in degrees Fahrenheit. For more temperature related numbers, see -40, 37, 98.6, 212, and 273.

32 is the exact value of a very small googolism by Sbiis Saibian - it's called binary-eyelash mite. Since an eyelash mite is defined as 2*10^4, a binary-eyelash mite can be thought of as being 2*2^4, which evaluates to 32. See binary-small fry, another small googolism by Saibian, for more on that.

Thirty-three
33

33 is the smallest number in the first semiprime triplet (as I call it). A semiprime triplet is a set of three neighboring integers that are all semiprimes: 33 = 3*11, 34 = 2*17, 35 = 5*7. The next semiprime triplet is (85, 86, 87), followed by (93, 94, 95) and (121, 122, 123). Semiprime quadruplets are impossible because one of the four consecutive numbers must be divisible by 4, much like how triplets of twin primes are impossible except for 3, 5, and 7. For similar groups but with prime numbers, see 47, 211, and 251.

As a consequence of Fermat's Little Theorem, if p is a prime other than 2 and 5, (10^(p-1)-1)/p is a whole number. 33 is the smallest number that can be generated with that formula: here, p = 3, so plugging 3 in gives us (10^(3-1)-1)/3 = (10^2-1)/3 = (100-1)/3 = 99/3 = 33. This formula sometimes forms cyclic numbers (33 is not cyclic) - for more on that, see 142,857.

Thirty-four
34

34 is the eighth Fibonacci number. It's the middle number in the first semiprime triplet (see 33), and as such it's the smallest number with the property that it and its neighbors have the same number of divisors. It is also known as part of an Internet meme, Rule 34, which states that "if it is on the Internet, there is porn of it, no exceptions". Similarly, Rule 34 of googology (jokingly stated by Sbiis Saibian) states that if it is a googolism, there is a salad number for it. A few salad numbers are on later parts of this list.

Thirty-five
35

35 is the the largest number in the first semiprime triplet (see 33). It's also an example of a 5-rough number. N-rough numbers are the antithesis of n-smooth numbers, because while a n-smooth number only has prime factors n or smaller (e.g. 48 and 81 are 3-smooth numbers), a n-rough number only has prime factors n or larger. 3-rough numbers are just a synonym for odd numbers, and the first few 5-rough numbers are 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43... - notice how a lot of them are prime numbers.

Also, 35 is the number of hexominoes (polyominoes made fron six squares, picture below).

Thirty-six
36

36 is 6^2, the sixth square number. It's also the 8th triangular number and the smallest number (aside from the trivial case of 1) to be both a square number and a triangular number. Such numbers are called square triangular numbers, and they are quite rare: the first few are 1, 36, 1225, 41,616, 1,413,721...

36 is also an example of a number which is highly divisible, as it can be divided by two twice and by three twice - its nontrivial factors are 2, 3, 4, 6, 9, 12, 18. It's a record-setter in the number of factors, being the first one to reach seven non-trivial factors. These record-setting numbers are called highly composite numbers. The first few such numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 360, 720, 840, 1260, 1680, 2520, 5040... and they tend to be very popular numbers for dividing things in (e.g. 24 hours in a day, 60 seconds in a minute and minutes in an hour, 360 degrees in a circle) - see also 12, which a famously large number of things are grouped in.

36 is the largest highly composite number to be a perfect power - the only other one (other than the trivial case of 1) is 4.

Thirty-seven
37

37 is the twelfth prime number. It's also equal to hydra(3), the last trivially sized value of the Kirby-Paris hydra function. The hydra function is defined using a single-player game which has a few simple rules:

• Start with a finite sequence of matched parentheses such as (()(()(())((())))).

• Pick an empty pair () and a natural number n.

  1. Delete it.

  2. If its parent is not the outermost pair, take its parent and append n copies of it.

The parent of a parentheses pair is the innermost pair of parentheses that encloses the pair and fully encloses all parentheses within the pair, and all that is within that pair. For example the parent of the bold pair in (()(()(())((())))) is the the red parentheses in (()(()(())((())))).

Then hydra(n) is defined as the number of steps it takes to reduce a hydra (((...((()))...))) with n pairs of (), always picking the rightmost pair of () and using 1 as n on the first move, 2 on the second move, etc. Hydra(0) = 0, Hydra(1) = 1, Hydra(2) = 3, Hydra(3) = 37, and Hydra(4) and beyond are very very large.

Unlike the busy beaver function, the hydra function is computable, with a growth rate of epsilon-zero in the fast-growing hierarchy. See also 51.

1/37 is 0.027027027.... and 1/27 = 0.037037037.... - this nice relationship is because 27*37 = 999.

Thirty-seven was another entry in my "Very Important Numbers" list. This one was because 37 multiplied by 3, 6, 9, 12, etc. produced 111, 222, 333, 444, etc. Mathematicians would find this property trivial, but I found it cool at the time that such a random (37 is psychologically random) number could produce repdigits (and that's also related to the correspondence between 1/27 and 1/37). 37 remains one of my favorite two-digit numbers, though I don't seem to pay attention to it as much as I do for 17, 43, and 79.

Because 37 is a factor of 111, it has an interesting divisibility test:

1. If the number is longer than 3 digits, divide it in groups of 3 digits starting from the right, e.g. 47,356,892 -> 47, 356, 892

2. Add the groups together, e.g. 47+356+892 = 1295

3. If the result has more than 3 digits, repeat the process - doing that gives you 296

When the number no longer has more than 3 digits:

1. If it's 1-2 digits, it's divisible by 37 if it's 37 or 74

2. If it's a 3-digit repdigit, then it's divisible by 37.

3. Otherwise, subtract the number from the nearest multiple of 111, and if the result is 37 or -37 then the number is divisible by 37. Here, we find that 333 is the closest multiple of 111 to 296, and 333-296 = 37. This means that 47,356,892 is divisible by 37.

In science, 37 is the normal body temperature in degrees Celsius. For other temperature related numbers, see -40, 32, 98.6, 212, and 273.

I consider 37 to be the quintessential psychologically random number because 37 is said to be the number people are most likely to choose when asked to name a number from 1 to 100. It's one of the survivors when we filter out the non-random numbers with this process:

First we can remove all the single-digit numbers, leaving 91 numbers left. Then, we can delete the multiples of 10, leaving 81 left. After that, we can get rid of the even numbers, removing 36 numbers and leaving 45. While we're at it we can remove the repdigits as well, leaving us with 40 numbers, Even the numbers that end in 5 can be removed, leaving left 32 numbers (13, 17, 19, 21, 23, 27, 29, 31, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 87, 89, 91, 93, 97).

To narrow down further, we'll need to make further observations: the remaining numbers all end in 1, 3, 7, and 9. Of these, the most random ending digit seems to be 7, leaving us with 17, 27, 37, 47, 57, 67, 87, and 97. Of those, we'll now want to focus on size. We use and think about the smaller ones in our daily life more often than the larger ones. The larger numbers are less likely for us to think of, and the smaller ones are so common that they don't seem random. 37 is just the right combination of "large enough to be random" and "small enough to be random". And that, is how you get 37 as the "most random" number (though some give the honor to 47).

As it turns out, 37 is a cult number as well, like on this website. The property I noted in my Very Important Numbers list happens to be one of the many many properties listed on the website.

Thirty-eight
38

38 is an example of a psychologically random even number, for reasons similar to 37. It's also part of a twin semiprime pair with 39. 38 is also the largest even number that can't be written as a sum of odd composite numbers.

When writing each number that can be expressed in Roman numerals (i.e. numbers 1 to 3999) and alphabetizing them, the last Roman numeral would be 38 (XXXVIII).

Thirty-nine
39

39 is infamous as part of a claim that 39 is uninteresting: it was claimed to be the "smallest uninteresting positive integer, which makes it especially interesting" in the first edition of David G. Wells's book The Penguin Dictionary of Curious and Interesting Numbers (a list of numbers with notable properties much like Robert Munafo's). In response, some people gave connections between 39 and the famed number 666, but that isn't all that special since so many other numbers have been connected to 666 (like 1998). But Wells wasn't aware that 39 did indeed have an interesting property.

What is that property? 39 is the sum of a sequence of consecutive primes (3+5+7+11+13) and the product of the first and last numbers in the sequence (3*13). Very few numbers have this property - the only other numbers known to have that property are 10, 155, 371, and 4,545,393,575,304,421. That property was noted as early as 1999, in an old version of Robert Munafo's number list which was a lot shorter than his modern list.

In the second edition of his book, Wells added that property to 39, and his first uninteresting number is instead 51. As a side note, the largest number in his book is Graham's number. (What were you expecting?)

Forty
40

o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o

This number is close to the limit of our secondary number sense. It’s getting a bit tough to accurately visualize, say, forty people in a bus. This number is spelled differently from what we might expect, “fourty”. This number’s prefix is tetraconta-, but by now you won’t see these prefixes outside of lists. Take a guess what the prefixes for fifty, sixty, seventy, eighty, and ninety are.

Forty appears many times in religion, particularly with periods of forty days. For example, Lent is noted as the forty says before Easter. It also appears in the phrase "forty acres in a mule" which refers to what freed slaves hoped for after the American Civil War, a promise that was unfulfilled.

Forty is also the only number whose English name has the letters in alphabetical order.

Forty-one
41

41 part of an interesting formula discovered by Leonhard Euler (x^2+x+41) that will always output a prime number when x is between -40 and 39, inclusive, and for some other values as well. Another way to think of the formula is:

41 is prime

Add 2, 43 is prime

Add 4, 47 is prime

Add 6, 53 is prime

Add 8, 61 is prime

Add 10, 71 is prime

Add 12, 83 is prime

You can continue the pattern 39 times, till you reach 1601.

This is interesting because the formula, unlike others which are a bit more wide ranged, is easy to remember. In fact, it's the most widely ranged prime-generating polynomial that is easy to remember. After this formula was discovered, people wondered if there were polynomials that would always generate primes for any integer input - this was proven impossible.

41 is also the next prime after 19 to be part of a twin, cousin, and sexy prime pair - not a lot of prime numbers have this property.

Personal: 41 is part of the product 41*27

Phiplex
10^((1+√5)/2)
~ 41.4986...

Since x-plex can be thought of as 10^x because a googolplex is 10 to the power of a googol, we can call 10^phi (phi is the golden ratio) phiplex - it's a relatively small number (less than 100), but can be considered a googolism nonetheless.

Forty-two
42

Forty-two is famous for being calculated as the Answer to the Ultimate Question of Life, the Universe, and Everything in The Hitchhiker’s Guide to the Galaxy—a book I must sadly admit I haven't read—which many people interpreted as the meaning of life or something similar. Therefore, 42 is a very culturally significant number. It's an example of a number whose reputation has been swamped into one strong association due to pop culture. As a result, 42 is a famous cult number - as with numbers like 666, many people have tried to use real-world connections to 42 to deduce why 42 is the "meaning of life".

Despite all those theories about 42, Douglas Adams, author of the book that made 42 famous, said that all those theories are nonsense, and that he only chose 42 because it is an "ordinary, smallish number". It's a random-seeming number, but not too random like the psychologically random numbers 17 or 23 or 47, which already were large cult numbers - psychologically random numbers and cult numbers are strongly related. If the "meaning of life" was something like 17, then the cult of whatever number it would've been in the book would likely have a very different story.

Before Adams, Lewis Carroll also referenced 42 several times in his writing, but he did not manage to get 42 to be a cult phenomenon like Adams did.

For another number that appears in The Hitchhiker's Guide to the Galaxy, see the Hitchhiker's Number.

Forty-three
43

43 is a prime number that appears in an interesting sequence called Sylvester's sequence, a sequence that starts with 2 (or alternately 1, doesn't make a difference to the sequence), and each member of the sequence is the product of all the previous terms plus 1. It begins:

1, 2, 3, 7, 43, 1807, 3,263,443, 10,650,056,950,807...

so the sequence grows very quickly, faster than even the factorial. In fact, it achieves hyper-exponential growth rates, which basically means that the number of digits grows exponentially - in this function's case, the number of digits in any term of the sequence is about twice the number of digits in the previous term. Another function that achieves hyper-exponential growth is the fuga- function.

It is fairly easy to see that all the terms have no prime factors in common, and the sequence has been used in one of the many many proofs that there are infinitely many prime numbers. Another such proof involves showing that all the Fermat numbers have no prime factors in common.

Another interesting thing about the sequence is that the reciprocals of its members (not counting 1) add up to one: 1/2+1/3+1/7+1/43+1/1807.... = 1. It's the fastest growing sequence whose reciprocals converge to a rational number.

43 can also start a sequence of primes by adding digits either from the right (43, 439, 4391, 43,913, 439,133, and 4,391,339) or from the left (43, 443, 8443, etc, up to 6,933,427,653,918,443), and is a record setter for length of said sequence when growing from the left (for two-digit numbers that is).

43 was also one of my childhood favorite numbers, and one of my most loved two-digit numbers as a kid (see also 79). This one's story is kind of weird: I believe it came from a VHS (see also 1107) to teach kids about math I watched several times in first grade. In one scene, the narrator listed the names of some prime numbers, and as he narrated each one, the number appeared in a different part of the screen, and it ended up being filled with about twelve primes. 43, for some reason, struck me as a funny number that I should pay attention to, probably because it was the last one they listed and/or because it's psychologically random, which always makes a number appealing. From then on I have often compulsively checked lists just to see what the 43rd entry is (ok I admit it, I still sometimes do that), among other things with the number 43.

In chemistry, 43 is the atomic number of technetium, the first radioactive element. Its name comes from the ancient Greek word "technetos" (artificial, manmade), because scientists thought it did not occur in nature. But it turned out that technetium does occur in nature, due to radioactive decay of uranium. However, the best way to obtain technetium is to synthesize it from smaller elements.

xkcd

Booga-pi with Nayuta Ito's system
~43.825904

Remember booga-e, equal to about 10.30495? Well, with the same generalization of up-arrows you can make values like booga-pi, equal to pi{pi-2}pi, with {pi-2} indicating pi-2 (or about 1.14) up-arrows, which can be imagined as a hyper-operator just beyond exponents.

Calculating booga-pi with that system is a lot harder than calculating booga-e, so I ran a program to calculate the value and got 43.825904 as the result.

Forty-four
44

44 is an example of a subfactorial number, and the fifth one. The subfactorial (sometimes denoted !n) is a function similar to the factorial but not as quickly growing - it represents the number of ways n objects can be ordered such that none of them are in their original place. The sequence of subfactorials begins 1, 2, 3, 9, 44, 265, 1854....

For example, four subfactorial is equal to nine: the nine arrangements for four objects (A, B, C, D) such that none are in their original place are:

BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA

A way to imagine the subfactorial in real life is like so: Imagine that there are five people, each with one hat, who meet together to trade hats. Each person wants to end up with a different hat after the trading is over. Then, there are 44 ways this could happen, i.e. 44 ways such that in the end nobody has their original hat.

Curiously, you can calculate n subfactorial easy with the surprisingly elegant formula [n!/e] where [n] is the nearest integer to n.

For another weaker version of the factorial see 420.

Forty-five
45

45 shows up sometimes when working with degree measures of angles in geometry, because a 45 degree angle is halfway between a right angle and a fully closed angle. That angle is equivalent to slopes halfway between horizontal and vertical, and an example of such a slope is the slope of the function y = x.

More notably, 45 is what Robert Munafo calls "the quintessential Kaprekar number": Kaprekar numbers are a special type of number best explained with an example: 45^2 = 2025, 20+25 = 45.

Kaprekar numbers are numbers that have this property for squares. 999 is another Kaprekar number: 999^2 = 998,001, 998+001 = 999.

45 is also a Kaprekar number for cubes (45^3 = 91,125, 9+11+25 = 45) and fourth powers (45^4 = 4,100,625, 4+10+6+25 = 45). In fact it is the only known number that is a Kaprekar number for squares, cubes, AND fourth powers.

The first few Kaprekar numbers are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, and 9999 - notice how they tend to be repdigits.

45 is also one of the numbers that appears in the ending loop when performing Kaprekar's routine (see 6174) on two-digit numbers - the loop is 45, 9, 81, 27, 63, back to 45, 9, 81, etc.

Forty-six
46

The largest even number not expressible as a sum of two abundant numbers. Also the number of chromosomes in a typical human cell - 22 pairs of normal chromosomes and two sex chromosomes, with 23 pairs.

Forty-seven
47

Forty-seven is part of an equally spaced group of 3 consecutive primes (47, 53, 59). That triplet is the smallest one after 3, 5, and 7; the next is 151, 157, 163; and a little after, such triplets become more common. See also 33, 211, and 251.

47 is also a strictly non-palindromic number. A strictly non-palindromic number is a number n that is not a palindrome (reads the same forwards and backwards, like 14,641) in any numeral base b, where 2 ≤ b ≤ n-2 (that is, in any base from binary to base n-2). That means that in any base from binary to base 45 and all bases in between, 47 won't be a palindrome. The first few strictly non-palindromic numbers are 1, 2, 3, 4, 6, 11, 19, 47, 53, 79, 103, 137, 139....

47 is yet another cult number (for being psychologically random), one large enough to be an in-joke in some places. A few of its cult websites are this one and this one - one calls it the "quintessential random number", but I think 37 deserves that title more. The second website I listed seems to be for a university that has connections to 47 and lists other places where 47 is an in-joke.

Forty-eight
48

48 is the number of known Mersenne primes, as of October 2014. It's also notable for having a lot of factors (2, 3, 4, 6, 8, 12, 16, 24), and it's another highly composite number.

48 is an example of a 3-smooth number, a number that has no prime factors greater than three. The first 3-smooth numbers are 1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216...... In fact, it's a member of a sequence of 3-smooth numbers of the form 3*2^n that starts 3, 6, 12, 24, 48, 96, 192, 384, 768...... these numbers all have a pretty large amount of divisors, and of these, 6, 12, 24, and 48 are also highly composite numbers. Therefore they occur in fields like display resolutions, e.g. 1024x768. However no 3-smooth numbers larger than 48 are highly composite numbers.

See also log(3).

Forty-nine
49

Forty-nine is the seventh square number. It's also the smallest number with the property that it and its neighbors are squareful (divisible by a square number other than 1): 48 = 12*22, 49 = 72, 50 = 2*52.

49 is sometimes given a connotation of luck since it's the square of a number (7) well-known for being considered lucky. For example, in Chinese culture the number 7 symbolizes togetherness, and therefore 49 is sometimes associated with good luck as well and used in some Chinese rituals.

The Post-Secondary Range
50 ~ 99
Entries: 40

Fifty
50

o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o = fifty o’s

Now this number is about the ending of our secondary sense and the beginning of our tertiary sense. Tertiary number sense allows us to roughly visualize numbers and somewhat approximate them, but it’s significantly less accurate than our secondary sense.

Fifty shows up a lot in real life in general, being half of one hundred (see also 25). For example, it's the number of states in the United States, and it's been that way since 1959. That value is unlikely to change soon, as it would be odd to deviate from that round number, and we'd need to update the flag that we Americans have such a silly obsession with. Additionally there is the fifty-move rule in chess which states that if the game goes fifty moves (a move is a turn by white followed by a turn by black) without a pawn move or capture, the game ends with a draw. See 14,296.

50 is also the smallest number that can be expressed as a sum of two squares in two different ways: 52+52 = 12+72 = 50. See also 17, 65, and 1729.

Fifty-one
51

The book where 39 was claimed to be uninteresting, in its second edition, instead has 51 as the first uninteresting number. For more on that see 39.

51 is notable in googology for being the value of Worm(2). Worm(n) is a fast growing function in googology (growth rate epsilon-zero in the fast-growing hierarchy) invented by Lev D. Belkemishev. It is closely related to the Kirby-Paris hydra (see 37), but a little more complicated. The worm function is defined as how many steps it takes to reduce a chain of integers [n] down to an empty chain using a specific set of rules. Those instructions are made in such a way that it will keep on expanding unless conditions allow you to remove a value - therefore it can take very long to reduce a chain to an empty chain, but it will always take a finite number of steps. A chain [1] takes 3 steps to reduce to an empty chain, so Worm(1) is 1, and a chain [2] takes 51 steps to reduce to an empty chain. Not much is known about Worm(3) and higher values, except that they're very very huge.

Fifty-two
52

Fifty-two is the number of playing cards in a traditional deck: four suits (diamonds, hearts, club, spade) multiplied by thirteen faces for each suit (ace, 2 through 10, jack, king, and queen). Therefore it has a bit of numerical significance, and shows up sometimes when working with combinatorics in card games (see 649,740).

52 is also usually given as the number of weeks in a year. That's an estimate, since dividing a year into 7 day weeks gives 52.14 weeks, 52.28 if it's a leap year.

Fifty-three
53

53 is notable for being the first prime not to be part of a twin or cousin prime pair. Its two closest primes are 47 and 59, both of which are sexy primes with 53. The first prime not to be part of a twin, cousin, or sexy prime pair is my favorte 3-digit number 211.

53 is said to be a magic number relating to birthdays - the chance that no two of any fifty-three people in a room share their birthday is approximately 1 in 53.

And 53 is also the number of bits used to represent the part before the exponent in the commonly-used double floating point format in computing - for more on that see the entry on the maximum value of that format, 1.7976*10^308.

Fifty-four
54

54 is the number of stickers on a traditional 3x3x3 Rubik's Cube - therefore, it shows up sometimes in combinatorics with Rubik's cubes. This leads to some pretty big numbers, such as the number of ways to arrange the stickers of a Rubik's cube.

Fifty-four is also the number of words in the English language used to name integers. They are:

negative                                hundred
zero          ten                        thousand      decillion                 vigintillion
one           eleven                   million          undecillion             googol
two           twelve      twenty    billion          duodecillion            centillion
three         thirteen    thirty      trillion         tredecillion             googolplex
four          fourteen    forty       quadrillion   quattuordecillion
five          fifteen        fifty       quintillion     quindecillion
six           sixteen      sixty       sextillion      sexdecillion
seven      seventeen  seventy   septillion      septendecillion
eight       eighteen    eighty      octillion       octodecillion
nine       nineteen    ninety      nonillion      novemdecillion

(click on any number above to go to its entry, except for 80 which doesn't have an entry so it links to 81 instead)

Up to 10^66-1, the naming is unambiguous, but after that it gets a little more uncertain - for a more on that, see the entry for a vigintillion.

Fifty-five
55

Fifty-five is the ninth Fibonacci number. The next one is 89. It's also the tenth triangular number (sum of the integers 1 through ten), and it's the largest number to be both a Fibonacci number and a triangular number.

Fifty-six
56

A Latin square is a x-by-x grid where each slot is filled with one of x numbers (or letters or colors or symbols or anything, doesn't matter) such that each row and has each number/symbol exactly once. For example,

4 3 1 2
2 1 4 3
1 2 3 4
3 4 2 1

is a Latin square.

A reduced Latin square is a Latin square such that the first row and column have the numbers ordered. By rearranging the rows and columns, any Latin square can become reduced. For example, the square above can become a reduced Latin square if we rearrange the rows and swap the two righmost columns:

1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1

56 is the number of reduced 5x5 Latin squares - after that term, the number of reduced x*x Latin squares grows quickly, with 9408 reduced 6x6 Latin squares and 16,942,080 reduced 7x7 Latin squares. The number of non-reduced x*x Latin squares grow even faster.

Latin squares are notable because of the well-known game Sudoku: all rows and columns in a Sudoku grid must have the each number 1 to 9 exactly once (with the added 3x3 box restriction). There are 6.67 sextillion possible Sudoku grids, and 5.524 octillion possible 9x9 Latin squares (Sudoku grids without the 3x3 box restriction). Also these squares are sometimes used in artistic designs, with different colors instead of numbers.

56 is also the largest number you can raise to its own power on most scientific calculators which overflow when reaching a googol. See also 69 and 449.

~56.961

This is the number that, when raised to its own power, is equal to a googol.

Fifty-seven
57

57 is the first integer that, when raised to its own power, makes a number larger than a googol.

The number 57, for being a random-sounding number, has sometimes been jokingly referred to as a prime. For example, it's been called the Grothendieck prime because in a certain story, mathematician Adam Grothendieck gave 57 as an example of a prime number. It isn't acually a prime, as its factorization is 3*19. It is a semiprime though.

57 is also a number associated with Heinz ketchup (and sort of a cult number among its fans): H. Heinz, the founder, promoted his ketchup by lying that there were 57 varieties, and the number 57 is still printed on the ketchup bottles to this day. Heinz seems to still be attached to the number 57: for example, its 1934 cookbook had 57 recipes, and its headquarters are P.O. Box 57. Heinz 57 is even sometimes used as slang for a dog with mixed breeds, or a person from mixed ethnicities.

Why was 57 chosen? Heinz said that he chose 57 because 5 was his wife's lucky number and 7 was his, and also because the number ends in 7, and numbers that end in 7 tend to be more appealing than other numbers. In his own words, Heinz said that seven was chosen largely because of "the psychological influence of that figure and of its enduring significance to people of all ages". I'm telling you, numbers ending in 7 really do appear disproportionately often.

Number of degrees in a radian
180/π
~57.296

Radians are an alternate way to measure angles, used in higher mathematics. Unlike degrees, which ultimately were arbitrary in their choice of measurement, radians are very non-arbitrary - an arc of one radian has the same arc length as the radius of the circle. Therefore this number can be thought of as relating an arbitrary measure (degrees) to a non-arbitrary measure (radians).

This number is equal to 180/pi, since pi radians is 180 degrees - it's easy to tell that the measure of radians relates directly to pi.

Fifty-eight
58

58 = 2+3+5+7+11+13+17, the sum of the first 7 primes, and one less than the 17th prime number, 59. This means that 58 has connections with my favorite numbers 7 and 17. The first few numbers that are the sum of the first x primes are 2, 5, 10, 17, 28, 41, 58, 77, 100, 129, 160...

Sums of primes seem to often have special properties - the case of 58 isn't too interesting, but there are some cool examples, like 39, 77, and 100.

Fifty-nine
59

The seventeenth prime number (sorry I had to)

Also, 59 is an example of a weak factorial prime, a type of prime number I coined in analogy to factorial primes and primorial primes. A weak factorial prime is a prime number one more or less than a weak factorial - the sequence starts with 2, 3, 5, 7, 11, 13, 59, 61, 419, 421, 839, 2521, but the next isn't until 232,792,559. It's interesting to note that all numbers neighboring a weak factorial (other than 1) are prime up to a certain point, but soon after, weak factorial primes become pretty sparse. See also 841.

59 is also the number of stellations of an icosahedron, counting the icosahedron itself as a stellation.

Sixty
60

Sixty is a highly composite number, a number with more divisors than any smaller number.

60 is the "weak factorial of five", meaning it's the smallest number divisible by 1 through 5. It's also the weak factorial of 6 since 60, the smallest number divisible by 1 through 5, is already divisible by 6. See 420 for more.

60 was used as a base by the Babylonians, with a fairly intricate numeral system with 3, 10, and 30 all used as sub-bases. See my large number timeline for more.

Because of that usage, there are 60 seconds in a minute, and 60 minutes in an hour, making 60 important in everyday life. Sixty was a good choice because it is divisible by many numbers: 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 can all be divided into 60.

Sixty-one / gagthree
61

Sixty-one is a weak factorial prime (see 59) and twin primes with 59, centered around the important number 60.

More notable to googology, 61 is expressible as A(3, 3) in the best known version of the Ackermann function. The Ackermann function is defined as follows:

A(a, b) =

• b+1 if a = 0

• A(a-1, 1) if b = 0

• A(a-1, A(a, b-1)) otherwise

So A(3,3) begins solving as follows:

A(3,3)
= A(2,A(3,2))
= A(2,A(2,A(3,1)))
= A(2,A(2,A(2,A(3,0))))
= A(2,A(2,A(2,A(2,1))))
= A(2,A(2,A(2,A(1,A(2,0)))))
= A(2,A(2,A(2,A(1,A(1,1)))))
= A(2,A(2,A(2,A(1,A(0,A(1,0))))))
= A(2,A(2,A(2,A(1,A(0,A(0,1))))))
= A(2,A(2,A(2,A(1,A(0,2)))))
= A(2,A(2,A(2,A(1,3))))
= A(2,A(2,A(2,A(0,A(1,2)))))
= A(2,A(2,A(2,A(0,A(0,A(1,1))))))
= A(2,A(2,A(2,A(0,A(0,A(0,A(1,0)))))))
= A(2,A(2,A(2,A(0,A(0,A(0,2))))))
= A(2,A(2,A(2,A(0,A(0,3)))))
= A(2,A(2,A(2,A(0,4))))
= A(2,A(2,A(2,5))))
= A(2,A(2,A(1,A(2,4)))))
...

and with continued iteration and iteration and iteration we eventually get to ... 61.

If it takes that long just to get 61, then how are we supposed to even estimate higher values? Fortunately, there are actually easy mathematical formulas to make computing the Ackermann function much faster:

A(0,n) = n+1
A(1,n) = n+2
A(2,n) = 2n+3
A(3,n) = 2^(n+3)-3
A(4,n) = 2^^(n+3)-3
A(5,n) = 2^^^(n+3)-3
etc.

This makes computing A(3,3) a lot less of a hassle.

The Ackermann function has a surprisingly innocent definition, and A(3,3) is not too bad of a number, but starting with A(4,x) it QUICKLY blasts into the stars - see my article on the function for more.

61 is a number expressible with the gag- prefix: gag(x) = A(x,x). Gagzero = 1, gagone = 3, gagtwo = 7, gagthree = 61, but gagfour, the next member of the sequence, is a FAR bigger number.

Sixty-two
62

62 is a number whose square root has an interesting pattern of digits: it starts 7.874007874011811019685034448... - 11811 is 7874*1.5 and 19685 = 7874*2.5. Amazingly, this is not a coincidence, as it can be derived from some less surprising patterns of reciprocals - for an explanation of this read Robert Munafo's entry on the square root of 62.

62 is also the sum of the number of vertices, edges, and faces of both a dodecahedron and an icosahedron - that is, the number of vertices, edges, and faces of a dodecahedron adds up to 62, and same for an icosahedron.

Sixty-three
63

63 is used in the defintion of each of the seven Fish numbers. The Fish numbers are a group of unusual googolisms coined by Kyodaisuu ("very large number" in Japanese) of Googology Wiki, who sometimes calls himself "Fish". He says he used 63 because he wanted his googologisms to be like Graham's number, but he said he doesn't remember why he used 63 and not 64.

Here are links to the entries for each of the Fish numbers (except numbers 4 and 7, which will have entries when part 7 is released):

Fish number 1, Fish number 2, Fish number 3, Fish number 4, Fish number 5, Fish number 6, Fish number 7

Sixty-four
64

64 is a power of two (26), and therefore it's another number associated with computing, for example in the term 64-bit. It’s the 8th square, the 4th cube, and the second hexeract (6 dimensions). Therefore, it's the first number expressible as a power in three different ways. It can be shown that if a number is the first number expressible as a power in x ways, it's equal to 2 to the power of the first number to have x factors - in other words, these kinds of numbers are always equal to 2 to the power of a highly composite number. For numbers with similar properties, see 16, 81, and 4096.

64 may be best known in googology for showing up in the definition of Graham's number - Graham's number is the 64th member of Graham's sequence. That designation was not arbitrary - it had to do with the context of the problem where Graham's number arose, since there are 64 ways to color all the lines in a K4 red or blue (see here for what all that means).

Sixty-five
65

65 is a number with interesting properties related to squares. It is the smallest number expressible as the sum of two different squares in two different ways: 65 = 42+72 = 82+12. See also 50.

65 is also the smallest number whose square is expressible as the sum of two different squares in more than 2 different ways: 652 = 162 + 632 = 332 + 562 = 392 + 522 = 252 + 602. This also means that 65 is the smallest number to be the hypoteneuse of more than different Pythagorean triples, i.e. an integer c that satisfies the equation a2+b2 = c2 where a and b are also integers.

Another cool property of 65 is that it's the smallest number that becomes square if its reverse is added or subtracted to it: 65-56 = 9 = 32 and 65+56 = 121 = 112.

Sixty-six
66

The number 66 currently holds the honor of being this list's first "uninteresting" number (that is, a number without properties I consider interesting enough to talk about). This is an arbitrary designation, since some people may find that I should talk about something interesting about 66. It's also a little paradoxical: is 66 made interesting by the fact that it's uninteresting? In any case, I won't go to other numbers just to label them as uninteresting. Robert Munafo's first uninteresting number is 74.

66 likely won't be this list's first uninteresting number forever, as I'm constantly finding more interesting numbers to talk about, or interesting things to say about numbers that are already listed here. For example, 44 was uninteresting until I decided to take note of subfactorials, 56 until I decided to discuss Latin squares, 58 until I added the idea of sums of primes, and 65 until I added some cool properties relating to squares.

Sixty-eight
68

68 is the largest known number that can be written as a sum of two prime numbers in exactly two ways: 68 = 7+61 = 31+37. It's conjectured to be the largest. This closely relates to Goldbach's conjecture that all even integers above 2 can be expressed as the sum of two primes, which is one of the most famous unproven conjectures in mathematics.

68 is also the 2-digit decimal string that takes the longest to appear in the digits of pi.

68 is the smallest composite number that becomes prime when you turn it upside-down, but that property is a coincidence that relies entirely on the glyphs used (see also 69).

Sixty-nine
69

69 is the largest number whose factorial you can take on a typical scientific calculator which overflows when reaching a googol - see also 70! in part 2, and 449, the equivalent number for more expensive calculators.

69 is also a number that has a notable popularity partially due to having a sexual meaning. This makes 69 another number which is almost entirely associated with a certain meaning (akin to 42).

The digits 69 are the same when you turn them upside-down, making 69 a strobogrammatic number. That last property is why 69 (along with 96) were on my "Very Important Numbers" list as a pair of numbers. However, nowadays I don't consider this property all that special - it relies not only on the base but also on the glyphs used - therefore, really, a number being strobogrammatic isn't much of a numerical property at all. Nonetheless if an interesting number does happen to be strobogrammatic then that makes for a nice coincidence - see 99,066 for an example of this.

69 being itself written upside-down and its sexual connotation are the very properties that make it a cult number - see also 105 and 69,105.

Seventy
70

Seventy is one of the numbers that appears down the the middle of Pascal's Triangle, a famous triangle in mathematics formed by continually adding numbers as the rows go down, starting with a 1:

                        1
                     1    1
                  1    2     1
              1    3     3    1
            1    4    6     4    1
        1    5   10   10    5   1
     1    6   15   20   15   6   1
  1   7   21   35   35   21   7   1
1   8   28  56   70   56   28   8   1
etc.                etc.                  etc.

Each number is the sum of the two numbers above it except for the 1's.

Pascal's triangle has many properties in mathematics. For example:

• to find the number of combinations (not permutations) of x objects from y objects to choose from, look at the xth entry on the yth row.

• the rows add up to powers of 2

• the (n+1)th row is the coefficients of (x+y)n when expanded as a polynomial - for example, (x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

• the 3rd numbers on each row are the triangular numbers, the 4th are tetrahedral numbers, the 5th are pentatope numbers (sums of the first x triangular numbers), and so on

• when converting each number in the triangle to black if it's odd and white if it's even, you get a fractal known as the Sierpinski triangle

• the first few rows coincide with the digits of the powers of 11

• diagonals taking the last or second last number in a row and going down to the left such as the one colored red add up to the Fibonacci numbers

Unrelated to that, the rule of 70 is sometimes used as an alternative to the rule of 72 because 70, unlike 72, is divisible by 5 and 7, and also because 70 works better as an approximation with lower percent increase rates - see 72 for details.

Seventy-one
71

The number of ways two gliders can collide in Conway's game of life, a famous "game" in mathematics which I discuss in detail in another site of mine. Those 71 collisions are very diverse - some of them produce nothing, some of them turn into a block, a blinker, a traffic light, or whatever else, and some of them become methuselahs that take over 100 generations to stabilize. The collision that takes the longest to stabilize is the so-called two-glider mess, which takes 530 generations to stabilize. To find out what all this means look at my site on Conway's game of life.

Seventy-two
72

Seventy-two is the first number that is a product of perfect powers but isn't itself a perfect power: 2^3 * 3^2. Numbers that can be expressed as products of perfect powers, whether they are or aren't themselves perfect powers, are called powerful numbers. Not all perfect powers are powerful numbers.

The number 72 is also significant in the rule of 72 in economics. The rule of 72 states that to approximate how long it takes for an amount of money to double, divide 72 by the annual percent increase. For example, if an amount of money increases by 2% each year, it'll take about 36 years to double, and if it increases by 12% each year, it'll take about 6 years to double.

It probably isn't very surprising that 72 approximates a value here. However, it may be surprising that 72 approximates a value that varies - when the percent rates are low, it's about 69.3, or 100*ln(2). The value 72 approximates increases with the percent rate, and is closest value to 72 at around 8%.

The real formula involves logarithms, and is a hassle to work with - that's why the rule of 72, which is so much simpler, is convenient in such scenarios.

Seventy-three
73

You may have heard about the four fours problem, which is the question of how many integers can be expressed with mathematical operations on four 4’s and no other digits. The problem with this puzzle is, to keep it interesting you need to restrict which operators are and aren’t allowed. The American computer scientist David A. Wheeler, in his four fours answer key, made his own set of rules organized by impurity levels, and his goal is to express as many integers as possible from 0 to 40,000 with the lowest impurity level. In impurity level 0, he allows concatenation, addition, subtraction, multiplication, division, decimal points, square roots, exponents, and parentheses. This means that writing out 4444 is a valid solution, as is 88 as 44 + 44, or 26 as 4! + (4 + 4) / 4.

With these rules, he can express every integer from 0 to 72 with four fours, but at 73 he hits a problem. He must move to impurity level 2,* which allows one new operator: the overline for repeating decimals, which I’ll substitute with an underline here. For example, 0.4 = 4/9. Since 4/9 is the square of two thirds, it’s a very handy tool in four fours. It even lets you express 3 with two fours:

√(4/.4)
= √(4/(4/9))
= √9
= 3

David A. Wheeler expresses 73 with a more complex combination of square roots:

√(√(√4)4!)) + 4/.4
= √(√(224)) + 9
= √(212) + 9
= 26 + 9
= 64 + 9
= 73

Noah Franske suggested an alternative expression which I think is more elegant:

(4! + 4! + √.4) / √.4
= (24 + 24 + 2/3) * 3/2
= (48 + 2/3) * 3/2
= 48 + 48/2 + 1
= 72 + 1
= 73

The original reason I added 73 to this list isn't quite as interesting. 73 and 37 are an unusual pair of prime numbers: 73 and its reversal 37 are prime, and 73 is the 21st prime while 37 is the 12th prime.

This number had an appeal to me as a kid, for the same reason as 79, but for some reason 79 ended up being the "golden number" instead of 73.

* Exponents and square roots were originally introduced in impurity level 1, but later he moved them to level 0, so now level 1 is unused.

Seventy-five
75

75 is an example of a 5-smooth number, a number that has no prime factors larger than 5. The first few are starts 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 50, 54, 60, 64, 72, 75 ... and the 3-smooth numbers are a subset of the 5-smooth numbers. I have considered both 3-smooth and 5-smooth numbers since I was a kid (and to some extent 7-smooth numbers and higher), but I was surprised when I learned from Robert Munafo's number list that such numbers have their own name.

This is also the number of uniform polyhedra (see also 1849 and 8190), which Jonathan Bowers categorizes into the regulars (this includes the 5 Platonic solids and four others), truncates, quasiregulars, trapeziverts, omnitruncates, and snubs. Bowers discusses those and gives a short name for each on the polyhedron section of his website.

75 has also gained some cultural significance for being exactly 3/4 of the way to 100, and it is also a Keith number.

Seventy-six
76

76 is an automorphic number, a number n such that n2 ends with the digits of n. In 76's case, 76^2 = 5776. The first few such numbers are 1, 5, 6, 25, 76, 376, 625, (0625), 9376, (09376), 90625 ...

In base ten, automorphic numbers (besides 1) follow an interesting pattern: for any number of digits n, there are exactly two automorphic numbers with n digits, one ending in a 5 and the other ending in a 6, and all automorphic numbers will themselves end in the digits of automorphic numbers. Observe the pattern:

• 6 is automorphic

• 76 is automorphic

• 376 is automorphic

• 9376 is automorphic

• (09376 is automorphic)

• 109,376 is automorphic

If a number is automorphic, that means that not only its square ends in the original number's digits, but so do all its other powers:

76^2 = 5776
76^3 = 439,876
76^4 = 33,362,176
etc.

Seventy-seven
77

77 = 2+3+5+7+11+13+17+19, sum of primes up to 19, and also 7*11, a product of consecutive primes. This isn't as interesting as the case of 39 though.

77 is allso, according to this webpage, the largest positive integer that can't be written as a sum of numbers whose reciprocals add up to 1, although it gives no explanation for this claim.

Personal: 77 was another number I found appealing as a kid because its digits were all 7's, but not as much as numbers like 43 and 79.

Seventy-nine
79

79 was yet another of my childhood favorite numbers, along with 7, 17, and 43. I think I liked this one because as a kid, I liked to associate each 2-digit number with a certain thought. Most were hard to put in words, but 79, for some reason, just felt cool and awesome (largely due to containing a seven). I have occasionally used the number 79 in Internet usernames and passwords as a kid.

This is also the smallest whole number not to appear on Robert Munafo's list at all, which would have been horrifying for my childhood self.

I don't think this entry should solely include an arbitrary personal connection to 79, so what else is notable about it? Ooh, I know: it's the atomic number of gold, one of the few elements known since antiquity. Clearly, nature loves the number 79 as well. :)

Eighty-one
81

81 is the ninth square and the third tesseract. It's also the second number expressible as a power in two ways (3^4 and 9^2, see also 16, 64, and 4096).

It's notable for being the square of the sum of its own digits, since (8+1)^2 = 81. That property applies to the largest 2-digit square number in any base (in base 10 that's 81).

For more completely indisputably true facts about the number 81, consult this article.

Eighty-three
83

83 is the number of right-truncatable primes in base 10 - a right-truncatable prime is prime number that, when you remove the rightmost digit, is still prime, AND remains prime each time you repeat the process. For more on such prime numbers, see 73,939,133, the largest such prime.

In the real world, 83 is the number of chemical elements that naturally occur on Earth in significant quantities: almost everything from hydrogen to bismuth (except the radioactive oddballs, technetium and promethium), plus the radioactive but long-lived thorium and uranium (yes, I know bisumth is also technically radioactive). According to Wikipedia, the rarest in Earth's crust (besides noble gases, which are more common in the atmosphere) is an obscure metal named rhenium.

See also 94 and 118, for more numbers related to the periodic table.