4chan

A pretty general explanation is that a number consists of an length and an angle on the number line. Positive numbers have angle = 0. Negative numbers have angle = pi (or 180° if you want to work with degrees instead of radians).

Multiplication is an operation where you add together the angles to retrieve the resulting angle and multiply together lengths to get the resulting length (yes, kinda recursive, but we're only working with purely positive numbers here).

So 3 * (-3) means
Length = 3 * 3 = 9
Angle = 0 + pi = pi (or 0 + 180° = 180°)

Of course this is very pedantic, but it works in more complex scenarios as well (pun intended).

Imaginary numbers have angle pi/2 (or 90°) or 3pi/2 (or 270°). So if you for instance want to find the square root of i, you can solve it by finding the length:

1 = x * x

And angle:

pi/2 = y + y
(can use modulus 2pi to acquire 2 solutions here)

Solving the equations and resolving the real and imaginary part with trigonometry, we get

1/sqrt(2) + 1/sqrt(2)*i

And

-1/sqrt(2) - 1/sqrt(2)*i