Understanding division of complex numbers requires us to understand the concept of a 'conjugate.' In order to create the conjugate of a complex number, we simply alter the sign of the imaginary part of a complex number. We denote conjugation with a "bar" over the expression in question. 5 + 7i with a bar over it is the conjugate of the expression. If we see 5 + 7i with a bar over it, we know to change the + sign to a - minus. Therefore, 5 + 7i with a bar over it, indicating the conjugate, is equal to 5 - 7i. Multiplying a complex number by its own conjugate provides with a complex number whose imaginary component is equal to 0. Therefore, for example, 5 + 6i multiplied by its conjugate it provides us with (5 + 6i)(5 - 6i). The solution is 25 - 30i + 30i - 36i^2 = 25 - 36i^2. Notice that since -30i + 30i equals 0, the imaginary part of the complex number = 0.
This is important to understand with respect to the division of complex numbers. When we are asked to do an equation in which one complex number is divided by another, we multiply the numerator and denominator of the equation by the conjugate of the bottom. We then simplify the answer, and we have our answer.
For example, suppose we have 4 + 6i/8 + 7i. We multiply the top and bottom of the equation by the conjugate of 8 + 7i, which is 8 - 7i.
(4 + 6i)(8 - 7i)
(8 + 7i)(8 - 7i)
We simply solve this equation and then simplify.