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

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