A conjugate changes a complex denominator into a real one because (a + bi)(a - bi) = a2 + b2. The sign of i2 is what makes the cancellation work.
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Multiply numerator and denominator
Consider (3 + 4i)/(1 + 2i). Multiply both by 1 - 2i. The numerator becomes 3 - 6i + 4i - 8i2 = 11 - 2i.
The denominator is 1 + 4 = 5, so the quotient is 11/5 - (2/5)i. Multiplying only the denominator would change the value of the fraction.
Check the modulus independently
The numerator's modulus is 5, and the denominator's modulus is √5. The quotient should therefore have modulus √5.
Our result has squared modulus (11/5)2 + (-2/5)2 = 125/25 = 5, agreeing with that check.
Avoid importing a real-number rule
A complex number with a positive real part and negative imaginary part is not simply positive or negative in the ordinary real-number ordering. Interpret its components or modulus according to the question.
Keep i2 = -1 visible until the expansion is secure, then collect real and imaginary parts. This original exercise develops algebra foundations for AMC 12 preparation; current competition information belongs to MAA's official resources.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
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Official information and further reading: Source 1
