Two positive numbers can't multiply together to create a negative number, so it should be impossible.

But... if you 'imagine' that there IS some number that is multiplied by itself to get a negative number, you can actually use that 'imaginary' number to get useful results. All we do is define that number explicitly, so you say i*i= -1.

From this you get:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

A 'Complex number' is the sum of imaginary number and a real number. (for example 3i - 1) The complex plain is a coordinate system that plots complex numbers using the imaginary component for one axis and the real component for the other axis.

If you're using a regular x/y coordinate system, you can easily move the point on the X axis by adding or subtracting from the x value, and you can move the point on the Y axis by adding or subtracting from the y value, but what if you want to do something like rotating the coordinate around the origin? It's not super straight forward to find an operation that will turn 3x-y to -3y-x that will work for all coefficients. Sure, you could just add -4x -2y, but that only works for those specific coordinates.

But if you're using a complex plain, you unlock another operation! You can rotate the coordinate around (0,0) simply by multiplying it by some power of i!

Multiplying by 'i' rotates the coordinate 90 degrees clockwise, multiplying by i^x rotates the coordinate by 90*x degrees.

i^4 = i*i*i*i = (-1)(-1) = 1 because you've completed the 360 degree rotation returning the coordinate to where it started.

So if you're using 3i-1 instead of 3x-y, you can easily rotate it.

(3i-1) * i

3(i*i) - i

3(-1) -i

-3-i

And this operation will work regardless of what your starting coordinates are. Say you want to rotate -9i -5 by 270 degrees, you just multiply by i^3.

(-9i -5) * i^3

(-9i * i^3) + (-5 * i^3)

(-9i^4) + (-5i^3)

(-9 1) + (-5-i)

5i-9