Complex Numbers - Division

Complex Numbers
Division

A few days ago I began talking about complex numbers, and we learned how to raise the complex number i to any power. You can see that post here: Complex Numbers – Examples and Powers of i

We then reviewed how to add, subtract, and multiply complex numbers, and I gave you a problem to try involving multiplication. You can see those posts here: Addition and Subtraction Multiplication

Today I will go over how to divide complex numbers. But first I will provide a solution to yesterday’s multiplication problem. Here is the problem one more time, followed by a solution:

Example: Compute (2 – 3i)(–5 + 6i)

Solution: (2 – 3i)(–5 + 6i) = (–10 + 18) + (12 + 15)i = 8 + 27i

Division

The conjugate of the complex number a + bi is the complex number a – bi.

Examples:

The conjugate of –5 + 6i is –5 – 6i.

The conjugate of 3 – 2i is 3 + 2i.

Note that when we multiply conjugates together we always get a real number. In fact, we have

(a + bi)(a – bi) = a² + b²

We can put the quotient of two complex numbers into standard form by multiplying both the numerator and denominator by the conjugate of the denominator. This is best understood with an example.

Example:

$\frac{1+5i}{2-3i}=$

Hint: Multiply both the numerator and denominator of the fraction by the conjugate of the denominator. The denominator will them become a real number, and the resulting fraction can then be written in the form a + bi.

I will provide a solution to this question tomorrow. Meanwhile, please post your attempted solutions in the comments below.

For those of you that prefer videos…