Essential understanding

The complex numbers are based on a number whose square is -1.

The square of say, 5, is 25. And the square of -5 is also 25. So, is it possible to square a number and have the result be negative?

No

The product of two numbers of the same sign will always be positive. This is why you learned in Algebra I that the square root of a negative number is undefined in the set of real numbers.

But at some point, some math guy or math gal wanted to continue to do math with these types of numbers.

The imaginary unit is born

Recall

​The Product Property of Square Roots

Square Root of a Negative Real Number

Imaginary Numbers

Operations with Complex Numbers

We can add, subtract, multiply, and divide complex numbers. The procedures might seem kind of familiar.

Addition/subtraction: combine the real parts and the imaginary parts separately

Multiplication: multiply two complex numbers as you would multiply binomials

Examples:

Examples:

Complex Conjugates

Complex Conjugates