August 14, 2023
Complex numbers are often noted as the "link between algebra and geometry". They provide intuitive explanations for complicated ideas, and give rise to some of the most beautiful mathematical structures.
The key to complex numbers is to visualize numbers as vectors, and operations, such as addition or multiplication, as movement.
Addition and subtraction work as movement in a direction. Multiplication works as rotating the number around by the angle of that number. If you were to multiply by a positive number, there is no rotation, because positive numbers have an angle of 0. When you multiply by a negative, you rotate by 180 degrees. This is the reason that a negative times a negative makes a positive.
But its quite restrictive to rotate only by 180 degrees. Now we know to interpret multiplication as a rotation, how can you rotate by some other ammount? What we need is a number that we can multiply by that rotates us only 90 degrees, half of a negative.
We can figure out what such a number needs to be by noticing a certain fact about rotation. If you rotate by 90 degrees, and then do it again, the effect is the same as rotating 180 degrees.
Using this idea we can write that any number multiplied by our special number (lets call it \(\color{#F09B24}{i}\)) twice, rotates it 180 degrees, wich is the same as multiplying by -1. So we can work out:
$$ \color{#5492f0}{n} \cdot \color{#F09B24}{i} \cdot \color{#F09B24}{i} = \color{#5492f0}{n} \cdot (-1) $$ $$ \color{#F09B24}{i}^2 = -1 $$ $$ \color{#F09B24}{i} = \sqrt{-1} $$
The normal question now is usually "but what does it mean to take the square root of a negative?". But we've already answered this question. It is the answer to our question of rotation.