Complex numbers are an area where algebraic and geometric perspectives provide very different strengths and intuitions. Having a firm grasp of both perspectives and being able to switch from one perspective to the other on-the-fly gives a much richer “big picture” of how the complex numbers work.

The algebraic perspective

“Give me a square root of -1”

We start with the real numbers \mathbb{R}.

We ask: What would happen if x^2=-1 had a solution?

We call that solution i. By closing under the operations of plus and times, we obtain the complex numbers.

Definition. The complex numbers are all numbers of the form a+bi, where a and b are real numbers and i is a constant satisfying i^2=-1.

  • Addition: If a+bi and c+di are complex numbers, then (a+bi)+(c+di) = (a+c) + (b+d)i (we add like terms).
  • Multiplication: If a+bi and c+di are complex numbers, then (a+bi)\cdot (c+di) = (ac-bd)+(ad+bc)i (distribute, use the fact that i^2=-1, and collect like terms).

Example: Find the product and sum of the complex numbers 4+3i and 2+2i.

The geometric perspective

“Show me how to multiply points in the plane”

We start with the coordinate plane \mathbb{R}^2, consisting of ordered pairs of real numbers (x,y). When we look at a point, we will also consider its size or modulus and its angle (formed with the positive x-axis).

If we want to treat points in the plane like numbers on the number line, we need to find a way to add them and multiply them. To define addition of two points, we can think of adding vectors (as in the image below). This exactly corresponds to “adding coordinate-by-coordinate”.

  • Addition. To add two points (a,b) and (c,d), we add their corresponding coordinates (a,b) + (c,d) = (a+c, b+d)

To multiply two points, we will multiply lengths and add angles.

  • Multiplication. To multiply two points (a,b) and (c,d), we find the point (p,q) such that the length of (p,q) is the product of the lengths of (a,b) and (c,d), and the angle of (p,q) is the sum of the angles of (a,b) and (c,d)
  • Shocking fact: (a,b)\cdot (c,d) = (ac-bd, ad+bc)

Example: Find the product and sum of the points R(3,4) and S(5,12).

Here is a Desmos project where you can experiment with the sum and product of two points.