When a complex number is in the denominator, why is it necessary to multiply by the conjugate?

Here is the explanation.

For the general complex number (a + bi), its conjugate is (a - bi).

By definition, i² = - 1.

Evaluate (a + bi) * (a - bi) to obtain

(a + bi) * (a - bi) = a² - abi + abi - b²i²

= a² - b² * (-1)

= a² + b²

This means that multiplying a complex number by its conjugate yields a real number.

For this reason, it is customary to make the denominator of a complex rational expression into a real number, by multiplying the denominator by its conjugate.

Of course, the numerator should also be multiplied by the same conjugate.

Example:

Simplify (2 - 3i) / (1 + 4i) into the form a + bi.

The denominator is (1 + 4i) and its conjugate is (1 - 4i).

Multiply the denominator by its conjugate to obtain

(1 + 4i) * (1 - 4i) = 1² + 4² = 17.

Also, multiply the numerator by the same conjugate to obtain

(2 - 3i) * (1 - 4i) = 2 - 8i - 3i + (3i) * (4i)

= 2 - 11i + 12*i²

= 2 - 11i - 12

= - 10 - 11i

Therefore

(2 - 3i) / (1 + 4i) = - (10 + 11i) / 17