3.1 Complex numbers  (Page 3/8)

Multiplying complex numbers together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

Because $i^2=-1,$ we have

To simplify, we combine the real parts, and we combine the imaginary parts.

Dividing complex numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate    of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of $a+bi$ is $a-bi.$

Note that complex conjugates have a reciprocal relationship: The complex conjugate of $a+bi$ is $a-bi,$ and the complex conjugate of $a-bi$ is $a+bi.$ Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

Suppose we want to divide $c+di$ by $a+bi,$ where neither $a$ nor $b$ equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

Multiply the numerator and denominator by the complex conjugate of the denominator.

Apply the distributive property.

Simplify, remembering that $i^2=\mathrm{-1.}$