#imaginary 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).

unfortunately getting tossed after the final so gotta preserve it while i can 💔

lil mistake—if the discriminant is < 0, it has two imaginary solutions! but if it asks for only real solutions, there are zero real solutions (note to self)

@ghostyghost202 i loved your coinpin doodles so much i was like, i gotta take this with me somehow. tysm your art is literally sacred to me

also asymptote reference? (those who know)

missed opportunity to put four and x on there i feel like they would love this. but i need coinpin to get through this final 🙏

Hanako- “isn’t this math why are there letters” ass kid

Nene- ????? So confused, they give her a head ache

Kou- thinks that the one 3D graph he was shown was cool looking, does not get it

Aoi- Gets them, does not like them, I think they light a rage in her soul more than they should

Mitsuba- why would I have an opinion on a math concept???? What??

Akane- Enjoys math and imagery numbers more than one probably should (got sad when the teacher said to ignore them because we don’t go into 3D in highschool(and honestly disappointed in himself that he thinks there so cool and fun))

Teru- can use them, understands them, thinks they’re difficulty is overhyped. Does not think about them after they are one the test

Tsukasa- “WHAt????!?!?? Can I eat it!!?!!?”

Natsuhiko- “How are numbers even imaginary… what does that even mean??”

Sakura- could learn them and use them, but does not care to bother herself with that