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)

You get rational numbers from this, and real numbers by completion.

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

A Visual Intro to Complex Numbers