This page serves as a supplement to my number list, listing any negative or complex numbers I feel are worth talking about. For further information on the purpose of this list click here.
Among the real numbers, the negative numbers (such as negative one) have no square root. But if you let the (positive) square root of negative one be called "i" (short for "imaginary"), then we open up a whole new world of numbers, called the complex numbers, which are numbers that can be expressed as a+bi, where a and b are both any real number. Complex numbers are a superset of the real numbers. Complex numbers have quite some significance to mathematics, as I discuss in a few later entries.
Here are some facts about i:
i2 = -1, i3 = -i, i4 = 1, i5 = i, i6 = -1, i7 = -i, i8 = 1, and so on in a cycle.
1/i = -i
eiπ + 1 = 0. This bizarre formula was discovered by Euler, and interestingly, it combines five super-important numbers (e, pi, i, 1, 0).
ii is a a real number equal to about 0.207879576...
A lot of these strange facts lead to some interesting complex numbers, as I discuss on the next few entries.
This is just a simple example of a complex number, one that has a real part and an imaginary part. When expressing a complex number as a+bi, a is the real part and b is the imaginary part. For all real numbers the imaginary part is 0; for this number the real part and imaginary part are both 1.
If i is the square root of negative one, then what is the square root of i? I remember being baffled by this question when I first learned about complex numbers. Though you might expect the square root of i to be a whole new kind of complex number, it's really an easy-to-express complex number: √2/2 + √2/2 * i. Simply observe:
(√2/2 + √2/2 * i)2
= (√2/2 + √2/2 * i) * (√2/2 + √2/2 * i)
= √2/2*√2/2 + √2/2*√2/2*i + √2/2*√2/2*i + √2/2*i*√2/2*i
= 1/2 + 1/2*i + 1/2*i + 1/2*i2
= 1/2 + i + 1/2*(-1)
= 1/2 + i - 1/2
= 0 + i
= i
This is a little weird, but it relates to the complex plane, a coordinate plane which maps all complex numbers (that includes the reals) to points where the x-coordinate is the real part and the y-coordinate is the imaginary part.
Remember that technically each number that is not 0 has two square roots (√x and -√x); this is simply i's "negative" square root.
Googologists have suggested that some people might try to cheat in the large number discussion by making some "phenomenally large 'number'", named "unfathomillion" perhaps, defined as:
unfathomillion = 10unfathomillion
so in order to calculate unfathomillion, we first need to calculate unfathomillion. At first glance this appears to be an infinite number since we need to infinitely use the 10x function, but an "unfathomillion" would really be the solution to the equation:
x = 10x
If you graph the functions y=x and y=10x, it's easy to see they don't intersect at all. Therefore the equation x = 10x has no solution, and unfathomillion is not a number at all ... or is it?
That equation has no real solution, but it does have complex solutions. One such solution is 0.5294805... + 3.342716... —this serves as an example of the interesting oddities that can arise from the complex numbers.
Actually, the equation x = 10x has not one, not two, not ten, but infinitely many solutions. Therefore, this "incredibly huge" number is not one gigantic (or not-so-gigantic) real number, but rather infinitely many complex numbers!
Sbiis Saibian's complex number list explores a long series of complex numbers defined as Ei#x in Hyper-E notation, which is to say, 10^i, 10^10^i, and so on, until the values become impossible to know. You should read his exploration of the iplex series; I'd just be repeating his work if I cover it here. The more plexes i gets, the weirder the numbers become.
I hope I provided a good sampling of a few of the more interesting complex numbers. Now on to the negative numbers.
The negative of the very largest possible infinity, by definition larger than any infinity. This entry used to say absolute infinity is "utterly unspeakably mind-blowing, and not worth explaining here", but the explanation is simple: it's just a theoretical infinite number that's by definition larger than all other infinite numbers. An infinity among infinities if you will. I know of no mathematical use for any negative infinite numbers other than regular plain old infinity.
The negative of the very smallest possible infinity. There are lots of infinite numbers defined, and lots of debate going on about them. For more on all those infinities look here. Once again, this probably shouldn’t be on this list, but it shows up when calculating the double logarithm of one, the triple logarithm of 10, the quadruple logarithm of 1010, the quintuple logarithm of 1010^10, and so on. Negative infinity has importance when calculating limits of functions—for example, the limit of the function y = ln(x) as x approaches 0 from the right, as seen in the graph to the right, is negative infinity.
The negative of what the googology community once considered the largest named number, but its definition is now seen as flimsy. The number is called BIG FOOT, and it was defined by LittlePeng9 of Googology Wiki using first-order oodle theory (FOOT for short), an extension to the first-order set theory used to define Rayo's number. It is defined using a function FOOT(x), defined as the largest number definable in FOOT using x symbols or less, and it is thus defined as FOOT10(10100) (the superscript 10 indicates iteration of the FOOT function). Sbiis Saibian suggested the name for this number.
You may wonder: is negative BIG FOOT a really really small number or a really really large negative number? It doesn't make sense to consider, say, negative 0.000001 "larger" of a number than negative one million. For our purposes, we can say that a number, positive or negative, is larger the further it is from 0. Therefore this can be considered not a very small number, but rather a very large negative number. Sbiis Saibian has advocated this usage (see 1 for details).
The negative of what was for a while a the largest number ever coined. It is so big that it can’t be computed at all, and yet it has quite a simple definition. It is defined as the smallest number larger than any number expressible with no more than a googol symbols in the language of first-order set theory. It was considered the largest number in googology until LittlePeng9's BIG FOOT came along.
This is the negative of Jonathan Bowers' largest googolism from BEAF, which could well be among the largest numbers known. Bowers has created tons of montstrous numbers with colorful and often whimsical names. which is famously montrously ginormously incomprehensibly enormous, but not the largest computable number since it's beaten by Loader's number. Seriously, this number is OUTRAGEOUS and doesn’t even have a clear way to be computed; therefore in a way it's more of a construction than a number. After all, Bowers doesn't give us even hints at what "array of L's" would solve to.
Negative of the second largest number coined by Bowers in BEAF. It is also famously large, not as overwhelming as its big brother but still SUPER CRAZY OUTRAGEOUS. As with its brother, this googolism doesn't really have a clear way to be computed.
Now let’s breeze through the negatives of some googolisms. This is the negative of another INCOMPREHENSIBLY GINORMOUS Bowersism which is defined as follows: Use the array representing the gongulus to define an array of 10’s, and solve it. The gongulus is a number whose negative we’ll see in the next entry. You can’t even understand how the array used to define this number is formed, and as of yet nobody has provided a fully working definition of this number!
This is the negative of a MONSTROUS number is a SHIT LOAD smaller than the positives of the previous three numbers, but still INCREDIBLE. You can’t even understand how it can be computed, but it’s so small at least you can understand how the array is formed. Simply put, this is the result of solving a 100-dimensional 10x10x10x10 ... x10 (with 100 10s) cube of 10s in array notation, not like anyone’s EVER gonna be able to do that.
Another fact: Bowers mentions a negative gongulus in an article on his website called “Why Does God Exist?”, making it the lowest non-infinite number mentioned on his site!
The negative of the number equal to 10 10s in a linear array in array notation. Still a CRAZY ENORMOUS number, but it’s so small that once you understand how array notation works, you can sort of understand how it’s computed! But still, there’s a lot between that and the mighty gongulus, a lot between the gongulus and the godly golapulus or beyond, and a lot between this and Graham’s number.
Another negative of a number that I'm only mentioning because Graham's number is so well-known—if I don't mention it, someone else will, and that's true for any googolism's negative. That's about it.
This is the negative of yet another Bowersism, the boogol. Its clearest definition is with up-arrow notation which I just did (if you're not familiar with up-arrows check out my article on them here); that notation is gonna show up a lot when we look at the large numbers. This number is so tiny that it can be expressed with a direct extension of mainstream mathematics.
The negative of this number is SO VERY SMALL that it can be expressed with notations used in everyday mathematics, specifically as 10^10^10^ ...... ^10 with 100 10’s. Another way to imagine this is with stage 1 being 10, stage 2 being 1 followed by 10 0s (or 10 billion), stage 3 being 1 followed by stage 2 zeroes, stage x being 1 followed by stage x-1 zeroes, etc. with giggol being stage 100. Seems staggering compared to 10^^100, even though those are the same thing.
The negative of this number is stage 10 as previously defined. I like to consider this number the boundary between the smaller numbers and the bigger numbers; it also used to be the end of part 1 of this list until that part was getting too long. It’s a staggering number all right, but SO AMAZINGLY SMALL that you can sort of imagine how one would calculate it.
I’m only including the negative of a googolplex because it’s so well known. The more well-known a large number is, the more likely someone is to mention its negative. It’s 1 followed by a googol zeros, where a googol is 1 followed by 100 zeros. It’s so INCREDIBLY TINY that it even has some real-world meaning! It’s less than the number of possible distinct parallel universes if units below Planck units are not considered meaningful. Most people think it’s monstrous, but as far as large numbers go it isn’t that big.
A googol is the younger brother of a googolplex, and I’m only putting its negative in for the same reason as a negative googolplex. It’s so IMPOSSIBLY TINY that if the universe would be filled with the incredibly large particle known as protons, a googol would be much less than that number of protons!
If you had this as a debt, then this is certainly a very real number. If you dug a hole a million feet deep, then this is a real number. I mostly included a negative million to transition to the smaller negative numbers. A million is SO MINISCULE that you could count to it in only a year!
And now for some negative numbers that relate to the real world! Forgive me if this site is inconsistent with its usage of metric and imperial units. Most of the content that I wrote in 2014-16 as a teenager puts imperial units first, but the more recent content from 2022 onwards prioritizes metric units. The Challenger Deep is a valley amidst the Mariana Trench in the western Pacific ocean, close to the island of Guam. It's the deepest point in all of Earth's oceans.
This is the value of absolute zero temperature in degrees Fahrenheit, the standard temperature scale in the United States and a small handful of other countries. In this scale, water freezes at 32 degrees and boils at 212 degrees. Absolute zero is a temperature that's impossible to reach in reality, but scientists have come incredibly close. To convert a temperature from Fahrenheit to the seldom-used Rankine scale, you just need to add 459.67.
The shoreline of the Dead Sea is the lowest land elevation on Earth. It's a dense, extremely salty lake in the Middle East bordered by Jordan, Israel, and the West Bank. Unfortunately this number is gradually getting larger (more negative), because the National Water Carrier of Israel is decreasing the size of the sea.
The exact value of absolute zero temperature in degrees Celsius, the world's most common temperature scale. Most people don't know that the Celsius scale was originally defined so that water freezes at 100 degrees and boils at 0; the values were actually reversed after the inventor of the scale died, so that water freezes at 0 degrees and boils at 100. Today, this scale is officially defined in relation to the Kelvin scale based on thermal energy of gases. You might have heard that to convert from Celsius to Kelvin, you need to add 273, but it's actually 273.15. While Fahrenheit and Celsius are in fierce competition as to which is more logical in daily life, Kelvin is far more logical in scientific terms because it starts at absolute zero.
Most people know of the Celsius, Kelvin, and Fahrenheit temperature scales, but there have been far more competing standards throughout history. The Réaumur temperature scale is similar to Celsius, except water boils at 80 degrees instead of 100, and it was once widely used in Europe. Today, it's occasionally used in manufacturing food and not much else.
The Rømer scale is another obsolete temperature scale, devised not long after the Newton scale. It was originally defined with the freezing point of brine at 0 degrees, and the boiling point of water at 60 degrees; later, the definition was tweaked so that water freezes at exactly 7.5 degrees.
Isaac Newton made the world's first attempt at a temperature scale, defining numeric units based on various real-world temperatures like the melting points of water, wax, and varying alloys of tin and bismuth. While his scale doesn't match with any of the modern temperature scales, it can be reinterpreted based on two points he specified: water freezes at 0 degrees and boils at 33 degrees. His temperature scale was designed not for measuring the weather, but to measure melting points of metals. While the definitions of his temperature scale seem vague and unsound today, it was still an important step in the research of accurately measuring temperatures.
There's one more value of absolute zero on my number list, but it's actually a positive number.
This number is notable as the point when the Celsius and Fahrenheit temperature scales are equal (-40 degrees Celsius = -40 degrees Fahrenheit). This is a good example of a negative number that is significant by itself and not just the negative of a well-known number.
For more temperature related numbers, see 32, 37, 98.6, 212, and 273.
Andre Joyce coined two systems he dubs merology to turn words into numbers by letters; both were intended to turn "zero" into 0, "one" into 1, etc. The first, more complicated system, doesn't even remotely work. The second, simpler one works partially (even though it needs a small correction to properly return 6 for six), but still has many false values, as noted by Joyce himself. Negative nine is the result of plugging "googol" into the second system.
See also meroogol.
In his list of negative numbers, Sbiis Saibian attempted a method to generate large negative numbers, and you can see here that it's not particularly efficient.
The limit of the function log(log(x)), as x approaches one, is negative infinity. This means that the closer a number x > 1 is to 1, the closer its double logarithm is to negative infinity. So what happens when you plug in a number close to 1, like 1.000001, into log(log(x))? Let's find out:
log(log(1.000001))
= log(0.00000043429...)
= -6.36221591...
As you can see, taking the double logarithm of a number close to 1 isn't an efficient way to make large negative numbers (i.e. numbers between negative 1 and negative infinity), as the number you get will be roughly equal to the number of zeros in the 1.000...0001 you plug in.
The formula for the nth triangular number is n * (n + 1) / 2, and its inverse is (-1 ± √(1+8x))/2. This means you can say a number has two "triangular roots" just like how integers have two square roots, but the "negative triangular root" isn't just the negative of the regular one; rather, it's the regular negative minus one. See Δ√2 in PGLN1 for more information.
Negative two is a number that I gave a Wikipedia article when I was 11 years old or so. The article was quickly deleted since there's not much special to say about -2 other than that it's the negative of 2. Negative 1, however, is notable enough to get its own Wikipedia article.
I can think of one googological function that makes use of subtracting 2 (which is to say, adding -2): the double factorial. (TODO: link to 105) That function actually outputs smaller numbers than the factorial and is mainly used in combinatorics, but it's notable to googology because recreational googologists have invented all sorts of factorial-inspired functions.
It's easy to forget that each positive real number technically has two square roots, since if x > 0, then there are two numbers a that satisfy the equation a2 = x (the one is positive, and the other is the one's negative); this is the "other" square root of 2. However the square root of x, by convention, refers to the positive square root unless otherwise specified.
This is negative one, the most important negative number. It is the boundary between small negative numbers and large negative numbers using Sbiis Saibian's idea of what defines small and large numbers (see 1 for details). It also has an important value in mathematics being the negative of the most important number in existence, which we will encounter soon. Some example of -1’s use in mathematics are that x-1 is 1/x and that the square root of -1, or i, opens up a whole new world of complex numbers, some of which are on the beginning of this list.
In almost all googological functions, you'll need to make use of subtracting 1. For example, up-arrow notation is defined with the rules:
Rule 1: x↑y = xy
Rule 2: x↑ay = x↑a-1(x↑a(y-1)) if y isn't 0
Rule 3: x↑a0 = 1
Noticee the uses of -1's in rule 2: that is use of recursion which is essential for googological functions. Subtracting 1 is equivalent to adding -1, and therefore I'd say negative one has some real significance in googology.
This is log(log(2)), an example of a small negative number (i.e. a number between negative 1 and 0) that can be formed through double logarithms. It's also possible to use double logarithms to make large negative numbers if the input number is close to 1, though that's not an efficient way to make them.
Believe it or not, this number, despite being negative, has some use in googology. Here's how:
Consider a number that is a really big power of two, like 210^100. To convert it into base 10 (i.e. using 10 as the base of the exponent), we'll need to make use of the double logarithm of two:
210^100
= (10log(2))10^100 (10log(x) = x by definition)
= 10log(2)*10^100
= 1010^log(log(2)) * 10^100
= 1010^(log(log(2))+100)
= 1010^(100+log(log(2)))
~ 1010^(100+(-0.52139))
= 1010^(100-0.52139)
~ 1010^99.47610
Even though this is a small reducing factor for the hyper-exponent (a hyper-exponent is the number c in an expression of the form abc), we need to make use of double logarithms like this if we want to be precise with estimation, and that kind of thing is necessary with ordering large numbers (especially if there's a close call). Therefore, despite being negative, double logarithms like this do indeed have a use in googology.
If you take the double logarithm of a number close to but less than ten, you can get some very small negative numbers.
Just as you can make kind of large negative numbers with the double logarithm of numbers slightly greater than one, you can use the double logarithm of numbers slightly less than ten to make some really small negative numbers.
This number is an example of a very small negative number, and Sbiis Saibian has suggested an interesting way to get an idea of how small it is: Imagine that had this much as a debt, which would be indistinguishable from breaking even. If that debt compounded by 7% annually (which is faster than you think), then it would take you 195 years to owe the bank a whole penny!
Googolminex is a googolism by John H. Conway and Richard K. Guy defined as the reciprocal of a googolplex. Read the entry for googolminex for more. This is an example of an extremely small negative number (see also next entry).
This is just an example of a negative reciprocal of a very large number, the kind of number that I'll only mention because if I don't mention it, someone else will, since Graham's number is so well known. It's a little scary to think that we will probably never know any digit (after all the zeros) in this or the positive version of this number.
This number is just -0.00000.........000001 with an inconceivably large number of 0s. We’ll talk about it when we reach its positive analog.
If we will never know how Rayo’s number starts or ends, then we will never EVER know how its reciprocal starts or ends. All we do know is that it has a shit load of 0s before we reach the mysterious sequence of digits.
Ditto.
Wait, negative zero?! Isn't that just zero? Well, normally it is, but this list already has a generous definition of "number". There are a few notations of numbers where negative zero is something distinct from zero. For one thing, if you calculate the limit of a function as it approaches zero, it can sometimes have a different value depending on which direction you approach zero from. The limit of 1/x is infinity for positive zero (0+), negative infinity for negative zero (0-). Of course, that doesn't mean 0 and -0 are two different numbers: the limit of 1/(x-1) is infinity when approaching 1 from above (1+), negative infinity when approaching 1 from below (1-).
Negative zero can sometimes informally be used in science, when you're rounding a negative number to zero. It can be helpful when measuring temperatures that are almost negligible but below the freezing point of water. It also shows up in computer science, since some floating point data types have a separate positive and negative zero.
Now that we're done with the negative numbers, click here to continue to positive numbers.