# How to Rationalize Denominators with Complex Numbers

Complex fractions contain complex numbers in the numerator and the denominator, for example, (a + bi)/(c + di). However, every complex fraction can be transformed into an equivalent complex number of the form x + yi, where x and y are real. This process is called rationalizing the denominator of a complex fraction.

To rationalize the denominator, you must multiply the top and bottom of the fraction by the *complex conjugate* of the denominator. Conjugate pairs differ only in the +/- sign between the real part and the imaginary part. Some complex conjugate pairs are

3 + i and 3 - i

-5 + i√2 and -5 - i√2

13 - 4i and 13 + 4i

10 + πi and 10 - πi

You can follow the example below to simplify complex fractions, or use the calculator on the left.

## Example

Rationalize the denominator of the complex fraction60 - 25i

————————

7 - 4i

The first step is to multiply the top and bottom by the complex conjugate of 7 - 4i, which is 7 + 4i.

60 - 25i 7 + 4i

———————— x ——————

7 - 4i 7 + 4i

Now multiply the two numerators, keeping in mind that i² = -1. The new numerator is

60x7 + 60x4i - 7x25i + 4x25

= 420 + 240i - 175i + 100

= 520 + 65i

And when you multiply the two denominators, the new denominator is

7x7 - 7x4i + 7x4i + 4x4

= 49 + 16

= 65

Notice how the imaginary part disappears when you multiply two complex conjugates. Now the equivalent fraction is

520 + 65i

—————————

65

Which simplifies to 8 + i.

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