small_numbers

Sbiis Saibian's

ULTIMATE

Small Numbers List

Introduction

Welcome to my Ultimate Small Numbers List. Why is there a small numbers list on a large numbers site?! Because certain smallish numbers such as zero, log2, log3, and even a few negatives such as loglog2 and -1 have some application in googology. More importantly, the purpose of this site is to celebrate numbers for their own sake, not for any practical purpose they might serve. Lastly there are certain obscure numbers like negative gongulus that if it weren't for googologist's would never get any mention. Besides, I just can't resist the temptation to include as many kinds of numbers as possible!

However a distinction does need to be made between something which is merely a "number" and something which can be deemed a "large number". Louis Epstein originally made the criticism that numbers under a million don't really belong to a large numbers list. I would beg to differ: does not a thousand belong to a large numbers list, seeing as a million simply means great thousand. However Louis Epstein does make a valid criticism that one should make a distinction between a "number" and a "large number". Is zero a large number by any stretch of the imagination? It's the largest non-positive number, sure, but does that make it "large"?

When my "Large Numbers List" was originally released I played loose with my own rules and ultimately decided to begin the list at zero, only forbidding negative numbers. This was because I didn't want to exclude zero on the account that it is the default value for some googological functions, such as the Ackermann function for example.

However, I have recently come to a new policy in regards my large numbers list. For the purposes of this site a Large Number is any number between one and infinity, possibly including one, a proper small number is any number between zero and one, possibly including zero, and a small number is any real number less than one, including negative numbers. In order to make this distinction explicit I have split my original "Large Numbers List" into a "Small Number List" for reals numbers less than one, and a "Large Numbers List" for real numbers equal to or greater than one. Technically one is a borderline case, being neither small nor large, but I'd rather not give one it's own category, and so I have opted to include it amongst the large numbers, thereby giving the Large Number List a definite starting point.

The purpose of the present list is to discuss some real numbers that can't justifiably be called "large numbers". These are "small numbers" in the second sense, so negatives are included. If you want to get right to the action, go to the Large Numbers List Part I, and you'll start right off with one and then the sky is the limit ... literally ...

Total Entries: 67

Negative Numbers

-infinity ~ -0.00000001

Entries: 34

-∞

negative infinity

Technically speaking, this entry shouldn't even be here! It's debatable whether infinities of any kind should be considered as numbers. However so that we have a definite starting point, for the Small Numbers List, just as we have a definite starting point for the Large Numbers List, I have allowed it. Besides, I don't want to have to make a category just for negative infinity.

Note that there is really no equation that results in a negative infinity. Even -1/0 can't really be called negative infinity, because the expression in itself is undefined. It can only have meaning as the limit of some function, say -1/x. However the value of that limit depends on whether x is approaching zero from the left or the right. So the expression is still technically ambiguous. The best I can do in terms of a mathematical expression representing this number is a negative sign followed by the infamous lemniscate symbol. This symbol simply means a changing quantity whose value eventually drops below any given negative number, at least in calculus. In a technical sense this is not a number, but rather a "concept" for "falling without bound". However it is not difficult to treat negative infinity as a "definite quantity" defined as:

-∞ < x : Ax=R

In other words, by definition, negative infinity is a quantity such that negative infinity is less than x, for all x in the set of real numbers. It's smaller than any and every real number! The problem is ... by how much? If we add one to it, do we get a real number? No, because if we did then negative infinity would have to be a real number, which contradicts the above definition. But this would be true for any finite positive number, therefore negative infinity must be infinitely less than any real number. But then negative infinity is not unique in being less than the real numbers which means the original definition is flawed because the above property CAN NOT be used to define a definite quantity. So on some fundamental level the idea of treating infinities as any kind of ordinary numbers is flawed. I would go one step further and suggest that perhaps we should take the properties of ordinary numbers as axiomatic and therefore infinite number should be treated as impossibilities by definition. One of the most basic properties that googology is based on is that n < n+1 for all n. However -∞ = -∞+1, because -∞+1 must still be less than all real numbers, and therefore by the definition is the same as -∞. But this contradicts the fundamental axiom that n<n+1 for all n. Therefore -∞ is not an element of googology.

The only time that negative infinity shows up in googology is when we try to compute loglog1. Since log1=0, and log0 can be thought of as negative infinity it follows that loglog1 = log0 = -infinity.

-(10^^^100 & 10)

negative kungulus

Currently the largest number on my large number list is a kungulus. So why not also have the negative version as well! No one but a googologist would ever think up such a number! We can think of this as a VERY VERY VERY "Large" Negative number, though normally it would be called a "very very very small number". As I've argued before however, small should refer to numbers between 0 and 1. We can then break up the negative numbers into "large negative numbers" (numbers between -infinity and -1), and "small negative numbers" (numbers between -1 and 0).

-(3^^^3 & 3)

negative triakulus

The second largest number on my large number list is a triakulus. Here is the negative version. Negative triakulus is inconceivably less than the next entry :)

-E100#^^#100

negative tethrathoth

The tethrathoth is one of the largest numbers in my system. The largest number in my system even borrows the name as a base. So here is negative tethrathoth. When googologist's invent large numbers they also make it possible to define a whole family of related numbers. For every large number a googologist defines, a reciprocal can be defined, a negative, and a negative reciprocal. So googologist's really get four terms for the price of one! Although I only consider real numbers as relevant to googology because they can be "ordered", one can also use large numbers to create large imaginary numbers, the sums large numbers and large imaginary numbers, or even add reciprocals and negatives into the mix. The number of possible derivative terms quickly multiplies as we include even more unorthodox things such as quaternion and octonion units. For our purposes however, none of this stuff has much baring on googology since i, the imaginary unit, can not be put anywhere along the real axis.

-<10,10 (100)2>

negative gongulus

In an article called "Why Does God Exist?" written by Jonathan Bowers, famed googologist and inventor of array notation, he makes mention of "minus gongulus" in passing to make a point that every number has "trueness". Technically this makes a minus gongulus the least real number explicitly mentioned on Jonathan Bowers' entire site! That's got to count for something!

-10^10^100

negative googolplex

Because if I don't mention it, someone else will. In googology if there is a googolism, there is probably a negative version of it somewhere. The more notorious the number, the more likely that's to be so.

-10^100

negative googol

Ditto.

-1,000,000

negative million

Imagine having that as a debt; Tell me that isn't a "real number" then!

loglog1.000001

-6.36221590585...

double logarithm of one point zero zero zero zero zero one

As the argument of the double logarithm approaches 1 from above, the output approaches negative infinity . Even when we use 1.000001 as the argument however, it doesn't result in a very large negative number. In fact the number of zeroes after the decimal point is roughly the negative number that will result. Thus this is an inefficient method for generating very large negative numbers.

Curiously, if the argument of the double logarithm is larger than about 1.25 then the result is a small negative number, and if it's less than 1.25 but greater than 1 then the result is a large negative number. (See -1)

loglog1.1

-1.38307639985...

double logarithm of one point one

log(1/12)

-1.07918124605...

logarithm of one twelfth

log(1/11)

-1.04139268516...

logarithm of one eleventh

-1

negative one

Negative one is kind of special among the negative numbers. If I was only going to mention a single negative number, this one would be it. It is the square of the imaginary unit: i^2= -1. It also pops up in this very strange equation:

e^(i*pi) = -1

This equation can be used to develop a system of complex exponentiation! For googology, it's purpose is simply to define the predecessor of any integer. It is used explicitly in the definition of the predecessor function:

P(n) = n - 1

Beyond that negative numbers don't really have much use in googology. After all, we aren't interested in making numbers smaller, but larger! However one of the catches to this is that you need to take a step back now and then when defining googological functions or else the function does not terminate. Every googological function must have a base case, and every googological function must make use of the predecessor function so that the evaluation of any expression is eventually forced back to the base case. Hence "minus one" is being used implicitly all the time in googology, even though we usually never think of it as a number in it's own right. Yet without those implicit "-1"s, googology wouldn't even function the way it does. So I'd say some credit is due to negative one.

Just a small note of passing interest: -1 = loglog1.25892541179... = log0.1

log(1/9)

-0.954242509439...

Since 1/10 < 1/9, it follows from the fact that the logarithm is a strictly increasing function that log(1/10) < log(1/9). Thus log(1/9) must be greater than log0.1 which is -1. The absolute value of this number is log9.

Although ordering negative numbers seems confusing at first, just remember that in this case a "larger negative" is smaller than a "smaller negative". In other words the order is reversed. In the case of negative numbers, the number closer to zero is always larger. This is in contrast to positive numbers where the number further from zero is always larger.

log(1/8)

-0.903089986992...

logarithm of one eighth

log(1/7)

-0.845098040014...

logarithm of one seventh

log(1/6)

-0.778151250384...

logarithm of one sixth

loglog1.5

-0.754262201319...

double logarithm of one point five

log0.2

-0.698970004336...

logarithm of one fifth

log0.25

-0.602059991328...

logarithm of one quarter

loglog2

-0.521390227654...

double logarithm of two

This number has some importance in googology believe it or not. When attempting to compute 2^^6, we find that 2^^6 = 2^2^65,536. Naturally we want to convert this into base 10 form. Roughly speaking we could change the 2's into 10's but that isn't very accurate, especially for a number this small (tetrationally speaking). So instead we use logarithms:

2^2^65,536 = (10^log2)^(10^log2)^65,536 = (10^log2)^10^(65,536log2) =

10^(log2*10^(65,536log2)

log2 is approximately 0.301, so we can simplify 65,536log2 to about 19,728. Since in log2*10^19728 the log2 won't have much effect on 10^19278 it is sometimes ignored. However it has a small reducing factor, that can be accounted for by 10^(19,728+loglog2). Since loglog2 is negative, it means that it slightly reduces the top most exponent to about 19,727.7804056. Interestingly loglog2 is just enough to decrease the top most exponents integer part from 19,728 to 19,727, so it's effect is not completely negligible, especially considering it's a second exponent. In fact:

(10^10^19,727.7804)^3.32 ~ 10^10^19,728.3017

In otherwords, the corrected reduced estimate has to be cubed to get the rough estimate. Considering how large 10^10^19,727.7804 is, you have to imagine that shrinking to an unimaginably small dot amongst exactly that many dots, then imagine that as an unimaginably small dot amongst that many dots to get close to the rough estimate. So you can imagine, there is a big difference from factoring it in, from an ordinary perspective. Although we routinely ignore huge differences like this in googology (numbers are often so far apart that such differences are insignificant), such accuracy for smaller numbers is sometimes necessary to settle a close call. So the number loglog2, although negative, does in fact serve a practical purpose in googology.

log(1/3)

-0.47712125472...

logarithm of one third

loglog3

-0.321371236131...

double logarithm of three

log0.5

-0.301029995664...

logarithm of a half

This number is also the additive inverse of log2. It turns out that |logx| = |log(1/x)|.

loglog4

-0.22036023199...

double logarithm of four

log(2/3)

-0.176091259056...

logarithm of two thirds

Two thirds is the simplest non-unit fraction. It's logarithm is -0.176091259056...

loglog5

-0.155541461208...

double logarithm of five

loglog6

-0.108935980359...

double logarithm of six

loglog7

-0.073092905527...

double logarithm of seven

loglog8

-0.044268972935...

double logarithm of eight

loglog9

-0.020341240467...

double logarithm of nine

loglog9.9

-0.001899759965

double logarithm of nine point nine

loglog9.999999

-0.000000018861...

double logarithm of nine point nine nine nine nine nine nine

As the argument of the double logarithm approaches 10 from below the result gets arbitrarily close to zero from below. This would be an example of a very small negative number. If this was your bank account balance it would be a debt so small that it would be virtually indistinguishable from breaking even. If the debt were to compounded at 7% interest annually, it would take roughly 195 years for you to owe the bank a whole penny!

-1/E100#^^#100

negative reciprocal of a tethrathoth

This is the smallest negative number on this list. It is one of the four possible flavors of a "tethrathoth" using negatives and reciprocals. Since it is both negative and reciprocal it is probably the strangest out of the four possible combinations.

Properly Small Numbers

0 ~ 0.99999999

Entries: 33

0

zero

"Zero" can be thought of as the smallest quantity possible. After all you can't have less than nothing, or can you? Sometimes "negatives" are thought of being "smaller" than zero, but this seems to defy logic since you can't have something smaller than something which is infinitely small to begin with! It is better to think of negatives as "less than zero" rather than "smaller than zero".

Negative numbers do not relate so much to the concept of quantity as to "position". An axis can have a "central point" at zero, with negatives to one side and positives to another. Quantity however can only travel in one direction from zero, namely, towards positive infinity.

The exclusion of negatives from this list much easier to justify than the exclusion of zero (although I've decided to include them none the less). This is because the negatives really serve almost no purpose in the large numbers field. In order for algorithms to terminate it is necessary to have a minimum value for every argument. This means we have to choose a least number allowable as an argument. Common choices for the minimum value are 1 and 0. If however we allow any integer value, including negatives, we kind of drop the bottom out so to speak and the result is either a function which does not terminate for all values, some values, or requires at least 3 rules (a base case, a less-than-base case, and a more-than-base case). None of these options provides any advantages over simply deciding on a minimum integer value for the function. This is why zero has some importance in the large number field, because it serves as a beginning value. Some of the uses of zero in googology are as the minimum value of an argument in the Ackermann function, and the minimum order of a separator in array notation. It is also the minimum arity of an array. In cantor's system of ordinals, it is the smallest possible ordinal.

Zero crops up when attempting to extend the hyper-operators to all integers. By definition a^^1=a. Since log_aa^^n = a^^(n-1), for n>2, we can define this as a law for all integer values. Thus we obtain that a^^0 = log_aa^^1 = log_aa=1. Thus a^^0=1. Next let a^^(-1) = log_aa^^0 = log_a1 = 0. Thus we find that any positive integer>1, a, that a^^(-1)=0.

Zero is equal to the double logarithm of 10: loglog10=0. It is also equal to log1.

Zero sometimes leads to undefines as in 1/0 and log0. For this reason I have occasionally been wary of zero. However we rarely have such problems in googology since the functions we work with are usually integral and not continuous.

In some respects zero serves as a natural starting point for large numbers. We can think of it as the starting line, and any positive real becomes part of the race coarse. The end goal can be thought to be at infinity, although it is probably better to think of this as a race with no end goal! We can begin this race, but we can never finish it! Better get going then ...

1/E100#^^#100

reciprocal of a tethrathoth

1/G₆₄

reciprocal of Graham's Number

Let's begin our journey very slowly. We will have plenty of time to accelerate towards the infinite! Our first non-zero entry is a number so incredibly small that you'd have to multiply it by Graham's Number (seen later on this list) to get 1. By necessity this number must begin as 0.0000000000000000000000000000000000000000000000000000000000000000............ but we have no way of knowing exactly what the first non-zero digit is, or where exactly it would occur! It's that small! As far as the race towards infinity, it's as if we haven't even left the starting line yet, but in fact we have.

2/G₆₄

two divided by Graham's Number

This number is just as far from our last entry as our last entry was from 0. If we want to get somewhere however we're going to have to pick up the pace because we'd have to have a Graham and one entries just to reach one!

1/(10^10^100)

googolminex

Conway and Guy have suggested the name "googolminex" for the reciprocol of a googolplex (seen later). It's an example of an extremely small number with an actual name! One of the many consequences of being able to define very large numbers, is that we can also define very small ones. We simply have to take the reciprocal of some large number, and we get it's inverse: a number that is just as small as the original number was large! You can imagine this number as 0.0000000000000000000000000000000000000..................................000000001 where there a googol-1 zeroes after the decimal point. This number is tremendous when compared to the reciprocal of Graham's Number, and yet it is still mind-bogglingly "googol-scopic". If we were to continue with the multiples of a googolminex, such as two googolminex, three googolminex, etc. We would never even have a hope of reaching 1, let alone actually large numbers. So once again we must pick up the pace...

1E-100

0.00000000000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000000001

reciprocal of a googol

Here is an extremely small number so large that I can actually write out it's decimal expansion in full. This is the reciprocal of a googol (seen later). It can be most compactly defined as 10^-100. This number is smaller than some of the very smallest numbers in physics. 10^-100 meters would be a distance so small that it couldn't even theoretically be measured, no matter how powerful our particle accelerators got. The reason for this is because quantum effects would distort space and time so much that no meaningful measurements could be made! This number, though vastly larger than the previous entries, is still uselessly small. Even if we were to continue with the multiples of this number we still wouldn't have any hope of reaching 1! Let's now explore some really small numbers in science...

1E-43

Planck Time (in seconds)

This is the smallest time scale that can theoretically be measured. You can think of this as the length of time for a single "frame" of the universe. Another way of looking at it is that the universe has a "frame rate" of about 10^43. See 1.2x10^-17 for comparison.

1E-35

0.00000000000000000000000000000000001

Planck Length (in meters)

This is the smallest distance that can theoretically be measured. This distance is so small that even the diameter of an electron would be about 10^17 Planck Lengths! In string theory this is said to be the size of a typical "string particle". It would also be roughly the size of the curled up dimensions. You can think about it this way. A string is as small relative to an electron as an electron is to us, making a string a "particles particle".

1E-18

0.000000000000000001

electron diameter (in meters)

This is the theoretical diameter of an electron. In truth however the meaningfulness of this is doubtful. Physicists have long since ceased thinking of sub-atomic particles as little billiard balls, and prefer to think of them as mathematical points with an associated "field" surrounding them. The idea that an electron has a size however can be given some meaning based on the proximity another particle has to be in order to interact with it. This distance of interaction should be around this value. This distance is also known as an "attometer".

1.2E-17

0.000000000000000012

Smallest Measured Time (in seconds)

According to a wikipedia article_[1], this is the smallest amount of time scientists have ever actually measured. Interestingly this is equivalent to about 10^26 Planck Times. As you can see we are still a long way from being able to measure changes by the Planck Time!

1/100

0.01

one hundredth

One hundredth of something is considered to be a pretty small precentage of anything. This number represents the same concept as 1%. However, this number is still huge compare some of the really small numbers in science. (check out the previous entries)

1/12

0.08333333333333333333333333333...

one twelfth

1/11

0.090909090909090909090909090909...

one eleventh

1/10

0.1

one tenth

1/9

0.1111111111111111111111...

one ninth

1/8

0.125

one eighth

1/7

0.142857142857...

one seventh

1/6

0.1666666666666666666666666666...

one sixth

1/5

0.2

one fifth

1/4

0.25

one quarter

log2

0.301029995663981195213738894724...

logarithm of two

Here is a small number with an important connection to googology. The common logarithm of two, or log2 is a number such that 10^log2 = 2. The upshot of this is that it allows us to change the base of a power tower of base 2 to base 10. For example, we can estimate 2^65,536 to a high degree of accuracy, obtaining the correct order of magnitude, simply by using this number. Observe:

2^65,536 = (10^log2)^65,536 = 10^(65,536log2) ~ 10^(65,536*0.3) ~ 10^19,661

This is pretty accurate despite the very rough rounding. The number 2^65,536 actually has exactly 19,729 digits.

The common logarithms of the primes are also useful because we can compute the logarithms of other positive integers by decomposing them into their prime factors and then using the laws of logarithms to figure out the value. As an example, we can compute the logarithm of 8 easily if we know the logarithm of 2:

log8 = log(2^3) = 3log2 ~ 3*0.301 = 0.903

Logarithms are indispensable to the study of tetrational class numbers and allow us to make estimates and bounds on numbers like Skewe's Number, or Ballium's Number. A lot can be learned about numbers of this size, without an impractical amount of computation. Thus even small numbers play an important role in googology.

1/3

0.333333333333333333333333333333333...

one third

This is the "decimal expression" for the fractional value, 1/3. This value is of some importance in my early exploration of mathematics and large numbers. This number has the somewhat irksome property that it can not be expressed as a "finite decimal expression". That is, it is not expressible in the form A/B where B is a whole power of 10. The decimal expression for 1/3 can be thought of as an infinite number of 3s following the decimal point. The discovery that the long division of the fraction would not terminate was something of a revelation to me. I was suspicious of infinity to begin with, but up until that point I figured infinity was something "out there". Now I saw that infinity could also be invoked even with the very close regions between 0 and 1.

There is also a curious feature of the decimal expansion of 1/3. It follows from the definition of 1/3 that 3(1/3) = 1. What happens if we multiply the decimal expansion by 3:

3(0.333333...) = 0.999999...

It turns out that 0.999999... is the same as 1. This may seem incredible. However consider what happens when we had 0.333333... to 0.999999...:

0.333333...+0.999999... = 1.333333...

We get 1+1/3. This might seem like a contradiction since 0.3+0.9=1.2, 0.33+0.99=1.32, 0.333+0.999=1.332, etc. However since neither decimal expression is finite, the "2" never "shows up" and the result is an infinite sequence of 3s. This just illustrates the rather counter-intuitive properties of the infinite.

log3

0.477121254719662437295027903255115...

logarithm of three

Normally written as log3, it is the unique real number such that 10^log3 = 3. With this number, and log2, we can approximate many common logarithms without a calculator. For example:

log6 = log(2*3) = log2+log3 ~ 0.301+0.477 = 0.778

log9 = log(3*3) = log3+log3 = 2log3 ~ 2(0.477) = 0.954

log27 = log(3^3) = 3log3 ~ 3(0.477) = 1.431

We can compute any common logarithm in this manner as long as the integer decomposes into 2s and 3s. We would not be able to compute log5, log7, log10, log11, etc. with only log2 and log3.

0.5

one half

0.5 is the decimal expression of the unit fraction, 1/2. This fraction is the simplest possible that can not be simplified as a whole number. It lies exactly half way between zero and one. It is important enough that it even gets the special name "one half".

log4

0.602059991328...

logarithm of four

2/3

0.6666666666666666666...

two thirds

This is the simplest non-unit fraction. It was important enough that there was a special symbol for it in eygptian mathematics.

log5

0.698970004336...

logarithm of five

log6

0.778151250384...

logarithm of six

log7

0.84598040014...

logarithm of seven

log8

0.903089986992...

logarithm of eight

log9

0.954242509439...

logarithm of nine

log9.9

0.995635194598...

logarithm of nine point nine

log9.999999

0.999999956571...

logarithm of nine point nine nine nine nine nine nine

As the argument of the logarithm approaches 10 from below, the output approaches the value of 1.

Next up, the Large Numbers ...

Ultimate Large Number List Part I