Pointless Gigantic List of Non-Positive Numbers

Introduction
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.

Imaginary and Complex Numbers
Entries: 5
i
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, etc.
- 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...
- many others

A lot of these strange facts lead to some interesting complex numbers, as I discuss on the next few entries.

1+i

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.

The square root of i
0.7071... + 0.7071...i

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.

The negative square root of i
-0.7071... - 0.7071...i

Remember that technically each number that is not 0 has two square roots (√x and -√x); this is simply i's "negative" square root.

The "unfathomillion"
0.5294805... + 3.342716...i

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 things 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!

I hope I provided a good sampling of a few of the more interesting complex numbers. Now on to the negative numbers.

Negative Numbers

-infinity ~ 0

Entries: 43


Negative Absolute Infinity


The negative of the very largest possible infinity, by definition larger than any infinity. It is utterly unspeakably mind-blowing, and not worth explaining here. All we can say is that it really shouldn’t be on this list, but I’m putting it here for the sake of having an absolute end to this large number stuff.

I should point out that absolute infinity is not a cardinal, not an ordinal, not any sort of infinite number. It's really just a theory, making a negative version of it arguably meaningless. Who ever heard of a negative theory?


Negative 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 101010, etc. 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.

Negative BIG FOOT -FOOT10(10100)

The negative of what currently holds the honor of being considered the largest named number. 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).


Negative Rayo’s number

-Rayo(10100)


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.

Negative meameamealokkapoowa oompa

-{LLLL ...... LLLL, 10}10,10 with a meameamealokkapoowa array of L’s


This is the negative of Jonathan Bowers' largest googolism, 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 meameamealokkapoowa

-{L100,10}10,10


Negative of the second largest number coined by Bowers. 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.


Negative golapulus

-10100&10&10


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!


Negative gongulus

-10100&10 or -{10,10(100)2}


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!


Negative iteral

-{10,10,10,10,10,10,10,10,10,10}


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.


Negative Graham’s number

-G(64)


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.


Negative boogol

-10^^^...^^^10 with 100 ^s


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.


Negative giggol

-10^^100


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.


Negative decker

-10^^10


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.


Negative googolplex

-1010100


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.


Negative googol

-10100


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!


Negative million

-1,000,000


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!


Negative forty

-40


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.


Negative nine

-9

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.

Double logarithm of one point oh oh oh oh oh one

-6.36221591...

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.


Negative two

-2

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.

Negative square root of two

-1.41421356...

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.


Negative one

-1


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 a small number vs a large number is (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: xay = xa-1(xa(y-1)) if y isn't 0
Rule 3: xa0 = 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.

Double logarithm of two

-0.521390228...

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 210100. 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:

210100

= (10log(2))10100 (10log(x) = x by definition)

= 10log(2)*10100

= 1010log(log(2))*10100

= 1010(log(log(2))+100)

= 1010(100+log(log(2)))

~ 1010(100+(-0.52139))

= 1010(100-0.52139)

~ 101099.47610

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.

Double logarithm of nine

-0.0203412405...

If you take the double logarithm of a number close to but less than ten, you can get some very small negative numbers.

Double logarithm of nine point nine nine nine nine nine nine

-0.0000000188611...

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!

Negative googolminex

-1/1010100

Googolminex is a googolism by John H. Conway and Richard K. Guy defined as the reciprocal of a googolplex. Rread the entry for googolminex for more. This is an example of an extremely small negative number (see also next entry).


Negative reciprocal of Graham’s number

-1/G(64)


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.


Negative reciprocal of meameamealokkapoowa oompa

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


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.


Negative reciprocal of Rayo’s number

-1/Rayo(10100)


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.

Negative reciprocal of BIG FOOT

-1/FOOT10(10100)

Ditto.

Now that we're done with the negative numbers, click here to continue to positive numbers.

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