I think it would have been clearer if they weren't called "numbers".

For example, look at matrices [1]: no one asks themselves whether they are "real" or not. We know that we have defined operations on arrays of numbers which are useful to some degree.

If complex numbers were just called "complexes", a lot of confusion would just wither away.

Also, interestingly, quaternions and octonions aren't called "numbers".

[1] Complex numbers may, of course, be seen as special kinds of matrices, but this is beside my point here.

I particularly like this book by Steven Pemberton. https://homepages.cwi.nl/~steven/Numbers.pdf

Imaginary numbers never made sense to me in high school. But this book gradually builds it up in a way that made the conclusion positively intuïtive. Highly recommended!

To me their purpose only came clearly recently. They're just a way to rotate smoothly one dimension into another. And control the speed of the rotation.

x into y. Potential energy to kinetic energy. Charge into current.

Keeping this definition but going higher in the dimensions: x into y, y into z and z into x. Going meta, the x into y rotation defines a plane; And a plane going into another direction is a volume. (x into y) into z.

What made it click, what made it beautiful is Geometric Algebra:

https://youtu.be/60z_hpEAtD8?is=e3GxYU4J1SdpsumW

I've come to understand it as something with both a magnitude and a phase. When you have a unity sine wave, it has two points per period where the magnitude is zero but those points have different phases.

I really was surprised on this when learning about these in Welch labs [1] and 3b1b.

[1] https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...

> I now feel like "complex" numbers etc. are the representation of data through multiple numbers where a single value (scalars) is not enough.

Ok, now find a single term that you think would adequately describe that concept.

'Imaginary' is a misnomer, they should actually be called Orthogonal Numbers instead since they lie on orthogonal axes.

Agreed, the naming system is atrocious. The one that really pisses me off at this point is that "real" numbers get to be called that. Imaginary numbers with rational coefficients are SIGNIFICANTLY more "real" than a converging infinite series, imo

I had a calculus prof who described it thus:

Calculus is a mathematical model of the world, and it is wrong in the sense of the famous quote about "all models [being] wrong, but some are useful", so mathematicians used the imaginary unit almost like a hotfix for their model.

Whether that's an accurate/rigorous statement or not, it helped me as a learner.

You can, of course, express things like oscillation without i, using vectors and matrices. But the ability to use Euler's formula to express complicated sinusoids as compact exponentials with base e simply makes the analysis much easier. In the words of my prof, it's merely a mathematical structure to express things more easily that "regular calculus" isn't good at expressing.

To me, that made the apparent paradoxes of expressing e.g. phase angles and reactive power and stuff using i a lot easier to stomach.

So much of mathematics is just this.

The "visual intro" misses a lot, and wastes half the space by introducing the complex numbers algebraically without a single visual because it's LLM slop.

A "visual intro" that doesn't even mention Mobius transformations[1] or Riemann surfaces [2] or any of the beautiful, colorful ways to visualize complex functions [3], or..

..well, it's just lazy LLM slop.

Is it better than the average high school introduction to the subject?

Sadly, yes. But it's such a low bar.

[1] https://www.youtube.com/watch?v=0z1fIsUNhO4

[2] https://www.youtube.com/watch?v=Ax5wbKsPBqA

[3] https://www.youtube.com/watch?v=NtoIXhUgqSk

> Why do you sum up coordinates?

People use sums to denote vector components, too. It's common to see ai + bj + ck (pretend there are hats on i, j, and k) to mean the vector [a b c].

So with complex numbers, you can think of it as a vector aj + bi where j and i are basis vectors, but we call j "1", because complex numbers have a notion of multiplication, and in that notion, j acts like 1, in that zj = jz = z for any complex number z.

Nothing bad happens if you write all complex numbers a + bi as vectors or tuples [a b]. It's just less convenient.

>what's the meaning of '+' there?

Mathematicians often reuse the same notation for different things where it's clear (for the mathematicians) what the implied meaning is. That means however that non-mathematicians can get confused.

The plus here is the operation that we call "the sum of complex numbers", where the numbers are a and bi. We can work towards this by first defining complex numbers as a tuple of two real numbers with particular sum and multiplication laws for these tuples. So (a, b) + (c, d) = (a + b, c + d) where the left side plus is for complex numbers and the right side plus is for real numbers. Similarly for multiplication, (a, b) * (c, d) = (ac - bd, bc + ad).

We associate (1,0) with the real number 1, and similarly for tuples of the form (a,0). The association is what allows us to see the real numbers as a subset of the complex numbers rather than being some unrelated set. It makes sense because the set of tuples of the form (a,0) with the multiplication laws of complex numbers behaves the same way as the real numbers and is in 1:1 correspondence with them. This association is used every time we write (a,0) as a(1,0) or just "a". We also write (0,b) as b(0,1) since (b,0)*(0,1) = (0,b), from the multiplication law. A general tuple (a, b) then can be written as (a,0) + (0,b) or, as what we've said is the same thing, a + b(0,1). Now we have simplified notation for all of these things except (0,1). It follows from the multiplication law that (0,1)*(0,1) = -(1,0) = -1. The complex number whose square is -1 is denoted as i. All these simplifications allow you to write (a,b) as a + bi.

> More fundamentally what's addition and multiplication? Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?

I'm not personally aware of any concept of continuity between these operations. In abstract algebra, operations are fundamental to an algebraic structure. The field of real numbers or the field of complex numbers are algebraic structures defined by their elements, the binary operation called addition, the binary operation called multiplication, and the laws that these operations satisfy. For example, both are commutative and associative, but the distribution law shows that they're different: a*(b+c) = a*b + a*c, but not a+b*c = (a+b)*(a+c). Also, a+0 = a, but a*1 = a, so the operations have different special elements that "fix" the other element when an operation is applied. It may be interesting to think what a "half-way operator" would have to be like with respect to these laws.

>What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line?

The idea of complex numbers and the more general idea of field extensions, which describes the process of obtaining complex numbers from reals, came from the study of polynomials. Polynomials can only have terms of integer degree. They satisfy many nice properties and quite rich mathematical structures arise from them which would not be possible with general non-integer degrees. Second degree polynomials with real coefficients, like x^2 + 1, are the lowest-degree polynomials such that they may not have real roots, and x^2 + 1 is the simplest example, but any other polynomial can be used to produce the complex numbers as well, if has a factor like x^2 + a. This is where the 2 comes from. It's not significant from a real line perspective (2 is just a number among the infinite reals) but it's significant from an algebraic one.

My, what excellent questions! Thank you for asking. Let's get to it!

>But when you represent a complex number as a+bi, what's the meaning of '+' there?

Here, treat a as the notation for "move a steps to the right", and bi as "move b steps up". Think about it like moving a pawn on a chess board.

These operations are translations: taking something and moving it. You can execute them as instructions.

"+" means composition of these two operations; that is doing them one after another.

In which order? Doesn't matter, the result will be the same.

Since we talk about "+", let's talk about "-".

The operations we are invertible, you can undo each of them (or any combination).

If "3" stands for the command "Move a pawn 3 steps right", you can introduce a notation "-3" meaning "move it back" (i.e., move it 3 steps left).

You have then: a + -a = 0, and to save yourself some ink, you just write a - b instead of a + (-b).

Same with bi; -bi means move b steps down.

>In Argand diagram, the numbers are coordinates. Why do you sum up coordinates?

Say, you move a pawn 3 steps right and 2 cells up.

Then you move it again by 1 cell right and 4 cells up.

That's the same as moving it (3 + 1) = 4 cells right and (2 + 4) = 6 cells up.

In our previous notation: (3 + 2i) + (1 + 4i) = (3 + 1) + (2 + 4)i = 4 + 6i.

If you know linear algebra: it works this way because the operations 1 (moving one step right) and i (moving one step right) form a basis[1] for the vector space of translations in a plane.

In simpler terms, it works this way because you can combine the move operations in any order (we can understand that easily), and so we can:

1. Write each move operation as "move ___ left/right and __ up/down"

2. Add up left/right and up/down components separately

>More fundamentally what's addition and multiplication?

We are treating numbers here as translation operators, i.e. commands: "move by this much".

Addition is composition, i.e. doing commands in order (which, turns out, doesn't matter).

Multiplication - well, that's where the salt of the article is. Or rather, should have been.

First, let's look at multiplication of "real" numbers.

Geometrically, the operation is proportional scaling. You can think of it as redefining the unit.

Say, we agreed that "3" means "make 3 steps to the right". But for that to make sense, we need to know what a step is.

We can define, by decree, that a step on a chess board is one cell (1). So "3" by itself really means "3 * 1" (3 of 1 step to the right).

We can scale a number by changing the step size; defined geometrically by proportion:

(a * b) is a distance such that (a * b) : b = a : 1.

This might seem weird, but that's how Euclid would've done it, because it allows us to think of a and b as quantities of equal kind (geometric lengths), avoiding distinguishing geometric quantities, like lengths and translation operators (aka vectors), and scalars, i.e. quantities you can scale by.

Side note: vector literally means "carriers", because the vector operation (translation) moves something from one place to another. A mosquito is a malaria vector.

scaling (multiplication) and adding (composition) interact in a familiar way:

c(a + b) = ca + cb

This is fine and dandy as long as we have only one dimension to deal with.

When we have two, we can first restrict ourselves to multiplying by "real" numbers, because we can say that multiplying by a is simply making everything a times bigger or smaller, and it doesn't break anything.

You still get the same interplay of addition and multiplication.

If we want to keep all the other niceties that we had from real numbers, we will be forced to have the following meaning of multiplication of couples of them (i.e. 2D vectors, translations in a plane):

To get a product of two 2D vectors, u and v, you ADD the angles that u and v make with the x-axis, and you MULTIPLY their lengths.

If you denote this operation as u * v, then you have all the rules for * and + that worked for real numbers still working.

It turns out, essentially, that there's no other way to cook up a product of 2D vectors that has these properties.

And that for 3D vectors, there is no such way at all, a consequence of the Hairy Ball Theorem[5].

And that for 4D vectors, you can define a product rule of that sort only if you give up on the order not mattering. For real and complex numbers, a * b = b * a, but in 4D, the order matters.

And for other dimensions, you really don't get anything nice.

Dimensions 1, 2, and 4 are special; but 2 more so than others.

> Is there a continuity between them, like an operator that represents both addition and multiplication meeting them half-way?

There is a continuity between addition and multiplication.

You can define multiplication of integers by treating one as "number of times to do a thing" and the other as "distance to translate by":

3 * 5 = (move by 3 steps to the right 5 times)

      = (move by 5 steps to the right 3 times)

So real number multiplication flows from real number addition by continuity.

Another connection is that multiplication of powers is addition of exponents:

a^n * a^m = a^(n + m).

a^n = a * a * a ... * a (n times)

This is how you can define exponentiation. There are other ways. But that's to say that addition and multiplication are inherently inseparable when it comes to real numbers because an integer can be thought of as either a length or a count (that's to say, in 1D vector are scalars), and keeping that puts a lot of structure on what you do.

I think the true spirit of your question deserves this answer though:

Complex multiplication exactly "meets multiplication and addition half-way", because you add the angles and multiply lengths.

>What's the significance of square and square root? Aren't they just some exponent values (2, 1/2) which have no particular significance on the real line? Why should imaginary numbers have a special place if they are tied to square roots (exponent of 0.5)?

This, my friend, is known as the Fundamental Theorem of Algebra, and boy was it not easy to figure out. There was about a handful of failed attempts to prove it (i.e. proofs that people universally agreed were correct, but later realized they weren't).

In short: there is no significance, on the surface at least. You could have picked a root of any polynomial equation that has no real solution as your "imaginary unit", and got the same set of complex numbers in the end - and nothing more.

But you'll end up with sqrt(-something) anyway.

This comes from the following (non-obvious) fact. Any real polynomial in x can be written as a product of the form c(x - r_1)(x - r_2) ... (x - r_n), where c is a real number and r_i are the roots. For the coefficients to end up being real numbers, the roots must come in conjugate pairs (if (a + bi) is a root, so is (a - bi)), so when you multiply (x - (a - bi))(x - (a + bi)) out, you get a quadratic with real coefficients: x² - 2ax + a² + b²

So you can decompose any real polynomial into a product of real polynomials of degree 2 (you might have one term (x - [real root]) left over if the degree is odd - which is another reason why odd-degree polynomials always have a real root).

If you extend your real number field[4] by adjoining a real number, you get nothing new.

And if you get something new, you have adjoined a root of a quadratic equation.

Why sqrt(-1) instead of the root of x^2 + 5 = 0? There is convenience.

(1, sqrt(-1)) is convenient pair to define the complex plane because they form an orthonormal basis [3], i.e. a neat coordinate grid, with easy multiplication rule.

Nobody is forcing you to use i=sqrt(-1). You can define j as a non-real root of x^3 - 7 = 0. Throwing it into the bucket will get you the same "complex" numbers.

But now they'll be really annoying to work with. E.g.: if j = sqrt(-3), then

(a + bj)(c + dj) = (ac - 3bd) + (ad + bc)j

This extra 3 is annoying. In the end, you can still write your j in the form a + b * sqrt(-1), which is a nicer form, and you no longer have "extra" constants popping up in your formulas.

So you do that.

[1] https://en.wikipedia.org/wiki/Basis_(linear_algebra)

[2] https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

[3] https://en.wikipedia.org/wiki/Orthonormal_basis

[4] https://en.wikipedia.org/wiki/Field_extension

[5] https://en.wikipedia.org/wiki/Hairy_ball_theorem

[6] https://en.wikipedia.org/wiki/Quaternion

I've been teaching math for 35 years, which I admit, doesn't say much, but I assure you the difficulties students have with complex numbers by and large have nothing to do with the name. It's the same with everything else: lack of preparation in arithmetic and basic algebra.

You are very right.

But, might be more complete to say that complex numbers are two-dimensional numbers used to model a lot of different things. So simply saying they are two dimensional numbers only part of the explanation. To fully understand them, you need to consider the notion of a 2-d number fits the application area, and what it means in context.

For example, complex numbers can provide an elegant way to represent rotations, oscillations, waves, etc. making certain calculations much easier.

But complex numbers are also used to extend the real number system in a way that ensures every non-constant polynomial has a root. In that context, they aren't merely a convenient tool for representing two-dimensional phenomena; they help complete the number system itself in a mathematically precise sense.

And then there is quantum mechanics, where complex numbers play a fundamental role in the mathematical description of reality itself.

Gauss agrees with you. He said "If the numbers we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary (or impossible) units, such an obscurity would have been out of the question."

Hear, hear. "Complex" makes it sound difficult, when the original idea was that they were a complex--as in compound--of two numbers. Complex analysis turns out to be almost trivial, thus taking an orthogonal journey, compared to the gnarliness of real analysis.

There is proof of this theorem in it that blew my mind some years ago [1]. Someday I want to finish reading the rest of the book.

> Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another.

https://en.wikipedia.org/wiki/Van_Aubel%27s_theorem

Wow. Thank you for sharing this relatively ancient online treasure.

Aside: as a commentary on how AI-brainrotted I've let myself become, I am awestruck by the animations and writing on that page predating the LLM coding era by that much. Already, the idea of someone acquiring the theoretical knowledge and practical software skills and time management and executive-function skills to create that page feels like it will only get more rare as more of us work at ever higher levels of abstraction.

My feelings exactly.

It doesn't just appear to be substantially written by an LLM; it was written by an LLM.

It lacks the punch. Anyone who didn't know the geometric definition would just be bored to hell by going over the history of how they were discovered which has nothing to do with geometry.

The article says "Visual Intro", but the introduction is literally not visual. The complex numbers are introduced through historical/algebraic lens, and then reintroduced visually, by which time the audience has been lost.

Of course this article will rise to the top here.

Not a tangent really. This explanation was very clearly LLM-produced.

For starters, this "visual intro" introduces complex numbers algebraically.

The text has a visual explanation, but the intro had zero pictures.

> Why does it work with sqrt(-1), and not with dividing by 0

This is pretty straightforward to explain.

With dividing by zero, any number you multiply by zero becomes zero. There's no other way to get zero through multiplication. So, the only sensible thing to divide by zero is zero (since "What times zero equals six?" doesn't have an answer).

But because anything multiplied by zero is zero, even if you divide zero by zero you have an undefined answer. Zero divided by zero could be anything! So that's also not meaningful.

The square root of negative one, however, only has to be defined once to get meaning out of all negative square roots.

It might help to point out to students that not all things called “number” in math are for counting/measuring.

Even before complex numbers, I was confused by negative numbers. Why negative times negative suddenly positive?

I spent time figuring that out. When complex numbers came I was ready. I knew they were not really numbers that would count something. And that they are just called numbers.

I think the best introduction is by taking the geometric perspective: Addition is translation, multiplication with a positive number is scaling, multiplication with -1 is reflecting across the origin, multiplication with any negative number is scaling and reflecting at once. multiplying with a square root of a positive number does only part(half) of the scaling. multiplying twice with the square root does the full scaling.

Now the only question left is: what is the half of a reflection? a 90 degree rotation into a new dimension. Doing it twice returns to the original dimension but on the opposite side of the origin.

This also explains why division of 0 is a problem: multiplying by 0 scales everything to the origin (ie deletes the original size). Division would need to restore the original size. But as a sibling comment has already said. Division by zero could just scale everything to infinity. But then everything can only flip between 0 and infinity.

>Why does it work with sqrt(-1), and not with dividing by 0 ?

Good question!

It does work with dividing by zero :D

See: https://en.wikipedia.org/wiki/Riemann_sphere