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
