Math: complex numbers

Tutoring university level math, one might run into complex numbers. The tutor begins about them.

The following is by my understanding.

In the Real number system, there is no square root of -1. Yet, in the Complex number system, sqrt(-1) = i. In some contexts, people call it j: sqrt(-1) = j. Therefore, depending on your context, i^2=-1 or j^2=-1. Herein, I will use i^2=-1.

It follows that, while the binomial x^2 + 1 doesn’t factor over the Real numbers, it does over the Complex numbers: x^2 + 1 = (x-i)(x+i), since -i^2 = 1.

Since i^2=-1, (2i)^2=4(-1)=-4 and so on.

I plan to continue about Complex numbers.

Source:

Travers, K.J., Dalton, L.C., Brunner, V.F., Taylor, A.R. (1977). Using Advanced Algebra. Doubleday Canada Limited.