Imaginary Numbers

Note: I’ve got no idea what I’m doing. I just needed a quick intro to imaginary and complex numbers. Learning in a hurry.

The number i, sometimes called “the imaginary unit”. i is defined as the number whose square is equal to negative 1.

i can also be defined as the principal square root of -1. The principal square root is the positive square root. E.g. the principal square root of 4 is 2.

The powers of i

• $i^0 = 1$ follows from any number to the power of 0 is 1
• $i^1 = i$ follows from any number to the power to 1 equals that number
• $i^2 = -1$ follows from the definition of i
• $i^3 = i^2 \cdot i = -1 \cdot i = -i$ don’t overthink this, -1 times i equals negative i
• $i^4 = i \cdot i^3 = i \cdot -i = -i^2 = +1$ follows from i x i by definition equals -1, -1 x -1 = 1

So $i^4$ is the same as $i^0$.

What’s $i^5$? $i^5 = i^4 \cdot i = 1 \cdot i = i$.

What’s $i^6$?

The cycle keeps going: 1, i, -1, -i, 1, i, -1, -i,...

References

Khan Academy. “Introduction to i and imaginary numbers | Imaginary and complex numbers | Precalculus | Khan Academy”. July 11, 2011. https://www.youtube.com/watch?v=ysVcAYo7UPI.