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 .

We ask: What would happen if had a solution?

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

Definition. The complex numbers are all numbers of the form , where and are real numbers and is a constant satisfying .

• Addition: If and are complex numbers, then (we add like terms).
• Multiplication: If and are complex numbers, then (distribute, use the fact that , and collect like terms).

Example: Find the product and sum of the complex numbers and .

### The geometric perspective

#### “Show me how to multiply points in the plane”

We start with the coordinate plane , consisting of ordered pairs of real numbers . When we look at a point, we will also consider its size or modulus and its angle (formed with the positive -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 and , we add their corresponding coordinates To multiply two points, we will multiply lengths and add angles.

• Multiplication. To multiply two points and , we find the point such that the length of is the product of the lengths of and , and the angle of is the sum of the angles of and • Shocking fact: Example: Find the product and sum of the points and .

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