1.2 Complex Numbers

Up until this point, we have only looked at numbers which are subsets of the real numbers (\(\mathbb{R}\)) However, to steal an archetypal example, how do we solve an equation like \(x^2 + 1 = 0\)? We can’t do this in the reals, so we ‘invent’ a solution by defining \(i:=\sqrt{-1}.\) This gives us a new set of numbers, namely \[\mathbb{C}:= \lbrace a + ib \;\lvert\; a,b\in\mathbb{R}\rbrace,\] and call members of this set complex numbers. There is a ton of theory on these numbers, which is left for the lecture notes.1 Instead, we just repeat a list of important definitions, and one important result.

Definition 1.1: (Complex Numbers)

Given a complex number \(z = a + ib,\) we define the following quantities:

  1. The real part of \(z\) is \(\mathcal{R}(z) := a.\)
  2. The imaginary part of \(z\) is \(\mathcal{I}(z) := b.\)
  3. The complex conjugate of \(z\) is \(\bar{z}:= a - ib.\)
  4. The modulus of \(z\) is defined to be \(\lvert z \rvert := \sqrt{a^2 + b^2}.\)

Just like the absolute value on the real numbers, the modulus defines a distance on the complex numbers.2 In particular for complex numbers \(z_1\) and \(z_2\), the quantity \(\lvert z_1 - z_2 \rvert\) tells us how ‘far apart’ the two numbers are. Plotting these numbers on an Argand diagram, we see that \(\lvert z_1 - z_2 \rvert\) gives us the distance of the straight line joining \(z_1\) and \(z_2.\)

Since the modulus defines a distance on \(\mathbb{C},\) it had better obey the triangle inequality. Luckily for us, it does!

Proposition 1.3: (Triangle Inequality)

For \(z_1,z_2 \in \mathbb{C}\), \[\begin{align*} \lvert z_1 + z_2 \rvert \leq \lvert z_1 \rvert + \lvert z_2 \rvert. \end{align*}\]

  1. We could even spend the entire semester studying complex analysis instead of real analysis, but that adds a whole other layer of — for want of a better word — complexity. Having two dimensions to deal with instead of one causes a whole raft of issues!↩︎

  2. The more technical term is that the modulus defines a metric on \(\mathbb{C}.\) You’ll see more of this next year, but if you’re interested, see the section on metrics in this document!↩︎