To review, the square root of a negative number is non-real, or an imaginary number. The imaginary unit i is defined as the square root of -1.
If you were to square both sides of this equation, you would have i^2 on the left side, and -1 on the right side, so you also know that i^2 is equal to -1.
A complex number is a value in the form below, in which a and b are real numbers, and i is the imaginary unit.
Suppose you want to multiply an imaginary number by a complex number, as in the example:
You would begin by multiplying these numbers together using distribution, which provides:
Next, you can simplify by remembering that i^2 is equal to -1, so you can rewrite your expression as:
Multiplying 6 times -1 gives you the complex number below, which you would rewrite in standard form with the real part first, -6, and the imaginary part second, 10i, to provide your final answer:
When multiplying complex numbers, use the FOIL method, because of the addition or subtraction that occurs between the real and imaginary parts of complex numbers. Therefore, multiplying complex numbers together is similar to multiplying binomials together.
Suppose you want to multiply the following complex numbers. Using FOIL, you start by multiplying according to the steps for distributing the factors.
Going in order, this provides:
Recalling that i^2 is equal to -1, you can substitute -1 in for i^2, then multiply negative 24 times -1.
To simplify your expression, you know that the real parts are like terms, so you can combine 16 minus -24, which equals 40.
You also know that your imaginary parts are like terms so you can combine 48i minus 8i, which equals 40i, so your final answer is 40 plus 40i. Note that this is in standard form.
Source: This work is adapted from Sophia author Colleen Atakpu.