Simplifying Complex Numbers: Conjugate of the Denominator

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Calculations with complex numbers sometimes produce a complex number in the denominator. In this lesson, we show how to simplify these results by using the complex conjugate.

Simplifying Complex Numbers

Sometimes, we can take things too literally. Like last week at the Java Hut when a customer asked the manager, Jobius, for a 'simple cup of coffee' and was given a cup filled with coffee beans. Simple, yet not quite what we had in mind.

A simple cup of coffee
A simple cup of coffee

This lesson is also about simplifying. What we have in mind is to show how to take a complex number and simplify it. Are coffee beans even chewable?

A Complex Number in the Denominator

Looking at our first example, we see the number 1 divided by a complex number. The complex number has the form of 'a + bi' where 'a' is the real part and 'b' is the imaginary part. The complex number in the denominator has a real part equal 'a' equal to 3 and an imaginary part 'b' equal to -4.

1/(3-4i) example

To simplify this fraction we multiply the numerator and the denominator by the complex conjugate of the denominator. When we reverse the sign of the imaginary part, we have the complex conjugate. Another way to think of this is to replace all the 'i' with '-i'. The complex conjugate of 3 - 4i is 3 + 4i.

multiplying num and den by complex conjugate 3 + 4i

When multiplying the numerator by 3 + 4i and the denominator by the same thing, 3 + 4i, we are not changing the value of the fraction. (3+4i)/(3+4i) is 1, and multiplying by 1 does not change the quantity being multiplied.

The next steps are to simplify. Multiplying the numerators together, we get 1 times 3+4i which is just 3+4i. When multiplying the denominators, we distribute the multiplication giving (3-4i)(3+4i) equal to 9 + 12i -12i - 16i^2.

multiplying nums and dens

A couple of things to notice: the 12i and the -12i cancel; the i^2 part is equal to -1 because i is the square root of -1. After substituting for i^2, the denominator becomes 9 - 16(-1).

simplifying further

Of course, -16(-1) is +16:


And 9 + 16 is 25:


The 25 in the denominator is dividing both the real part and the imaginary part of the numerator.

3/25 + 4i/25

We could stop here, because our result is in the form of 'a + bi'. In this example, the fractions can be written nicely as decimals: 3/25 is 0.12 and 4/25 is 0.15. Going one step further:

0.12 + 0.16i

We have used the complex conjugate of the denominator of 1/(3 - 4i), to simplify this fraction as 0.12 + 0.16i. The key step is multiplying the numerator and the denominator by the complex conjugate of the denominator. We then simplify the expression. Ready for something else? I wonder what we would get if we asked Jobius for a muffin to go. Maybe a muffin with wheels attached?

A Complex Number in Both Numerator and Denominator

What happens if the fraction we want to simplify is more general? In this example, instead of a 1 in the numerator, we have a complex number.


Just a comment: yes, we are simplifying a fraction, but we can also think of this as dividing a complex number by a complex number. As in the earlier example, first, multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 2 + 4i is 2 - 4i.

multiplying by complex conjugate

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