#complex numbers

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0 (zero; naught)

seed: 4 (60 nominations)

previous opponent: negative one

class: additive identity

biography: one of the most revolutionary numbers in all of mathematics, and not just because of its rotund symbol.

the notion of "nothingness" as a number in and of itself rather than merely as a placeholder was discovered independently in many parts of the world at many different times, particularly in cultures that used forms of positional notation (ie. writing numbers in a way where every symbol has some numerical value, and its position within a number indicates that the value is multiplied by some power of a "base", often ten). due to its strange nature as a number with no value, many properties that are often taken for granted with other numbers do not apply to zero.

in the Number Tournament, zero has beaten some incredibly iconic numbers seemingly effortlessly. fifteen, thirty-six, sixty-four, the golden ratio, and negative one were all no match for naught. it is truly a force to be reckoned with.

zero is a number associated with emptiness, with the void, as well as with new beginnings. it is one of the foundations of all of mathematics, and it is certainly one of the best numbers.

[Wikipedia article]

i

seed: 11 (46 nominations)

previous opponent: NaN

class: imaginary

biography: another groundbreaking number, discovered much more recently than zero. much like zero, in its earliest uses i (the imaginary unit) was considered more of placeholder than a number in its own right, as the name "imaginary number" might suggest.

the imaginary numbers (and the complex numbers they are a part of) were born as an elegant solution to a practical problem, and they've persisted as a tool for modeling things in the physical world, no less real than the "real numbers". complex numbers are useful for "translating" statements about shapes into statements about numbers, and vice versa. they are crucial to the Fourier transform, which itself is a vital part of signal processing and many areas of physics.

in the Number Tournament, i faced off against a series of increasingly tougher challengers: forty-seven, twenty-seven, e, two, and Not a Number, each race closer than the last. i fought hard to get here, and we're all very proud of it for making it this far.

i is associated with the mathematical tradition of taking "you can't do that" as a challenge, and with thinking outside of the box. it is a fundamental component of our modern understanding of the world, and it is certainly one of the best numbers.

[Wikipedia article]

Oh ye! So, you know how when you take any real number, and you square it, even if the real number was negative, you wind up with a nonnegative number? Like, if I want to square two (2^2), I get 2*2, which is 4. If I want to square negative two ((-2)^2), I get (-2)*(-2), which is again 4, because the negatives cancel each other out.

I can do this with any number! 0^2 is 0*0 is 0 (nonnegative). (-1)^2 is (-1)*(-1) is 1 (positive, and therefore nonnegative). pi^2 is just pi^2, because pi is transcendental (a special form of irrational that I could go on about, but that takes a bit of a tangent - but basically, it means that if I plug pi into any polynomial with rational coefficients, then there's no way to get 0).

So, we have a well-defined function that can take any real number, and map it to another real number: f(x) = x^2. That's squaring!

Now, what if we wanted to do the reverse?

Well, we have a function for that, too! The square root function: f(x) = sqrt(x).

This takes in a number x, and it outputs the "primary square root" of x. When x is a nonnegative real number, it outputs the nonnegative real number y such that y^2 = x. So, we have sqrt(0) = 0, because 0*0 = 0. We have sqrt(4) = 2, because 2*2 = 4.

We have sqrt(2) as its own irrational number (somewhere close to 1.41...), with sqrt(2) defined as the nonnegative number x such that x^2 = 2. Similarly, sqrt(5) is the nonnegative number x such that x^2 = 5.

This doesn't address what happens when we want to do sqrt(x) where x is negative. What if I want to do sqrt(-4)? In other words, how do I find an x such that x^2 = -4?

x can't be -2, because, as we said earlier, (-2)*(-2) = +4, not -4.

That's where the idea for i comes in! i is defined as the primary square root of -1: that is, i is defined such that i^2 = -1.

When we first encounter i in classes that talk about imaginary numbers, we normally see this written as i = sqrt(-1). But that leads to the question: which root are we picking? sqrt(-1)^2 = -1, but also, (-sqrt(-1))^2 = -1.

As it turns out, if we base our number system on -i rather than on i, we get basically the same number system. So, rather than force a choice for i, which could be sqrt(-1) or could equivalently be -sqrt(-1), we define i such that i^2 = -1.

This lets us play with imaginary numbers! sqrt(-4) is now 2i. sqrt(-pi) is now i*sqrt(pi).

We can even take the square root of imaginary numbers! That's just asking the question, what is x such that x^2 = i, for example?

As it turns out, we need a number with both a real part and an imaginary part to answer that question: in other words, we need a complex number! A number a + bi, where a and b are real numbers!

And once we have complex numbers, we essentially have everything we need to build functions made out of polynomials and roots, without having to restrict the input and output!

So! TLDR: the idea of i comes from the fact that we can't take the square root of a negative number and come out with a real number. So, we need i to be one of the solutions to the equation x^2 = -1. Rather than write "i = sqrt(-1)" (no good, forces us to choose between sqrt(-1) and -sqrt(-1), why are we pitting our children against each other, -sqrt(-1) deserves to shine), we write "i^2 = -1" (good, does not force us to choose between these siblings, we love our children equally and we show it by not forcing sqrt(-1) into the spotlight and -sqrt(-1) into the shadows, they can share i).

argand diagrams

Rennison’s number

Alright so let me create a method called the “Tropical Forest Method” that we will use to make the biggest number ever… something that operates through various symbols and the Kruskal’s Tree Theorem… we’ll use emojis to represent it:

Repeatable commands:

🍃 - +1

☘️ - Increased to the power of 3

🍀 - increased to the power of 4

🌲 - Adds 1 node colour to tree structure

🌳 - Doubles the node colours of tree structure

🌴 - Makes every individual node worth what the entire structure was (includes +1 bits)

How many times over (applied to all actions)

🪾 - X1

🍂 - X2

🍁 - X10

🌱 - X100

🪴 - X1000

🌹 - X1 Million

🌷 - X1 Billion

💐 - X1 Decillion

⭐️ - XGoogol

🌺 - X1 Centillion

🌟 - XGoogolplex

💫 - XGoogolplexian

✨ - XGrahams Number

🌌 - XRayo’s Number

If anyone is unaware, basically a Googol is (10^100) in size, 1 followed by 100 0s, now a Googolplex is (10^Googol), it has a Googol 0s, and a Googolplexian is (10^Googolplex), you get the point they are large… however for calculating Rennison’s number they are completely worthless and add nothing. Same goes for Graham’s number, which while large, becomes obsolete fast as early as the third three colour node.

The real bearing ultimately comes from the trees, which I will explain. Basically Kruskal’s Tree Theorem operates as a game of making trees, you place down coloured nodes, starting with 1 node and going up 1 each time, however you cannot create the same pattern twice. So like when you place a colour at your first tree you can’t use that colour again since that colour is now recognised as a used pattern, then using two of the same colour at the second tree when you have 2 means you can use that colour but can’t connect 2 together… however if you were to say use 4 of the same colour at the fourth tree, then you can still use 3 or 2 of that colour connected now infinitely, and if you connected 2 of the same together and one of those 2 to a different colour at the third tree then you can’t connect 2 of that same to that other but you can’t still connect the different colours together and the same colours together… if that makes sense.

You start with 1 colour, this is TREE (1), and in it you can only do 1 node since you’ve already wasted it, TREE (2) adds a colour however still stops at the second tree as you expend 1 colour the first tree and connecting the second colour together means you can’t do it again… TREE (3) however, with one added colour opens so much variety that completing every variation would take longer than we can calculate… so, TREE (3) is also known to be one of the longest numbers ever conceived… however not quite the longest.

Rayo’s number is defined as being “The smallest number bigger than any finite number named by an expression in any language of first-order set theory in which the language uses only a googol symbols or less.”, which… in non mathematical terms is basically saying “One above everything we can calculate”… everything else we’ve been looking at has nothing on this, and so this is the modifier that will be used for Rennison’s number.

Anyways… enough serious math, ready to see the largest number ever? Okay:

🌌🌲🌳🌴

That’s it, that’s Rennison’s number… want to know why this is so large? Well let me put it this way… each action has XRayo’s number added to it… so in reality the game immediately jumps to TREE (Rayo’s Number)… doubled by 2, but WAIT, no the 2 is X by Rayo’s number so it’s actually X by Rayo’s number X2, then after all this you add the palm layer which makes every individual node worth what the entire structure was previously… X by Rayo’s number so you actually now already have Rayo’s number layers of this whole process by round 2… of all Rayo’s number of rounds of this rapidly multiplying again and again… the result, my friends, is Rennison’s number.

Understood any of that? No, well fair enough… but yeah this is big. I’ll be honest considering I conceptualised this post as a joke I’m surprised how sophisticated I ended up getting… either way say hello to… Rennison’s Number, it’s really large… and stuff.