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.
ExampleRationalize the denominator of the complex fraction
60 - 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
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
Notice how the imaginary part disappears when you multiply two complex conjugates. Now the equivalent fraction is
520 + 65i
Which simplifies to 8 + i.
© Had2Know 2010