## 0.1 Complex numbers

One of the fundamental ingredients of quantum information science (and, indeed, of quantum physics in general) is the notion of **complex numbers**.
It would be disingenuous to expect that a few paragraphs would suffice to make the reader sufficiently familiar with subject, but we try our best here to give a speedy overview of the core principles, and end with some exercises that can be a helpful indicator of which things you might want to read up on elsewhere.

The “classical” way of arriving at complex numbers is as follows: start with the **natural numbers** **integers** **rationals** **monoid**, to a **group**, to a **field**.
Algebraically, then, we seem to be done: we can do all the addition and multiplication that we like, and we can invert it whenever it makes sense to do so (e.g. we can divide, as long as it’s not by

But there are lots of numbers that turn up in geometry that are not rational, such as **real analysis** — something which we won’t touch upon here — to end up with the **real numbers**

Well the reals have one big problem: they are not **algebraically closed**.
That is, there exist polynomials with no roots, i.e. equations of the form ^{2}
Somehow the most fundamental such example is the equation

It turns out that if we just throw in this one extra number *any* polynomial — a theorem so important that it’s known as the **fundamental theorem of algebra**.
We call the result of doing this the **complex numbers**, and denote them by

This gives us an algebraic way of understanding what a complex number is: it is a real number **imaginary** number

But what about multiplication and division?
Following the rules of the game, we can figure out what the product of two complex numbers is by treating the imaginary number

Division works similarly — the most simple example of inverting a complex number

This other complex number **complex conjugate**^{3} of a complex number **modulus** (or magnitude, norm, or absolute value).
Note then that we can simply write

Now things are looking somewhat nice, but the story isn’t complete.
We have a good geometric intuition for what a complex number is (a vector in

To understand these we need to switch from our **rectangular coordinates** **polar coordinates** — instead of describing a point **radians**”.
We already know, given

It would be nice to know how to go in the other direction though, but this can also be solved with some trigonometry:

Great!
… but what’s the point of polar coordinates?
Well, it turns out that they give us a geometric way of understanding multiplication: you can show^{4} that

There is one last ingredient that we should mention, which is the thing that really solidifies the relation between rectangular and polar coordinates.
We know that rectangular coordinates

Given polar coordinates **Taylor series**^{5} of **exponential function**

We have just “proved”^{6} one of the most remarkable formulas in mathematics: **Euler’s formula**

Addition and subtraction are most neatly expressed in the planar form

We know how to perform addition, multiplication, inversion (which is a special case of division), and complex conjugation on complex numbers in planar form, but we’ve only described how to do the last *three* of these in polar form: we haven’t said how to write

You do not need to know everything about this whole story of algebraically closed fields and so on, but it helps to know the basics, so here are some exercises that should help you to become more familiar.^{7}

- The set
\mathbb{Q} of rational numbers and the set\mathbb{R} of real numbers are both fields, but the set\mathbb{Z} of integers is not. Why not? - Look up the formal statement of the fundamental theorem of algebra.
- Evaluate each of the following quantities:
1+e^{-i\pi}, \quad |1+i|, \quad (1+i)^{42}, \quad \sqrt{i}, \quad 2^i, \quad i^i. - Here is a simple “proof” that
+1=-1 :1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i^2=-1. What is wrong with it? - Prove that, for any two complex numbers
w,z\in\mathbb{C} , we always have the inequality^{8}|z-w| \geqslant|z|-|w|. - Using the fact that
e^{3i\theta}=(e^{i\theta})^3 , derive a formula for\cos3\theta in terms of\cos\theta and\sin\theta .

To explain why we care so much about polynomials would be the subject of a whole nother book, but one important reason (of the

*many*!), for both analysts and geometers alike, is the**Weierstrass Approximation Theorem**.↩︎The more common notation in mathematics is

\bar{z} instead ofz^\star , but physicists tend to like the latter.↩︎**Exercise.**Prove this!↩︎If you don’t know about Taylor series, then feel free to just skim this part, but make sure to read the punchline!↩︎

It is very important to point out that this “proof” is not rigorous or formal — you need to be very

*very*careful when rearranging infinite sums! However, this proof*can be made rigorous*by using some real analysis.↩︎Note that we have not really given you enough information in this section to be able to solve all these exercises, but that is intentional! Sometimes we like to ask questions and not answer them, with the hope that you will enjoy getting to do some research by yourself.↩︎

*Hint: use polar form, draw a diagram, and appeal to the*↩︎**triangle inequality**.