Precalculus · Problem 5

Rational Functions — Stitz & Zeager

Mathematics — Precalculus · complex-number-arithmetic

Problem

For z=3−5iz = 3 - 5i, w=2+7iw = 2 + 7i, find and simplify the following. Write your answers in the form a+bia + bi: z+w,zw,z2,1z,zw,wz,zˉ,zzˉ,(zˉ)2z + w, \quad zw, \quad z^2, \quad \frac{1}{z}, \quad \frac{z}{w}, \quad \frac{w}{z}, \quad \bar{z}, \quad z\bar{z}, \quad (\bar{z})^2
Intuitive explanation

Complex number arithmetic follows the same rules as polynomial algebra—treat ii as a variable, but remember that i2=−1i^2 = -1. For addition and multiplication, just combine like terms; for division, multiply numerator and denominator by the conjugate to eliminate ii from the denominator. The conjugate zˉ\bar{z} flips the sign of the imaginary part and is your key tool for simplifying fractions and finding magnitudes.

Intuitive explanation

Complex number arithmetic follows the same rules as polynomial algebra—treat ii as a variable, but remember that i2=−1i^2 = -1. For addition and multiplication, just combine like terms; for division, multiply numerator and denominator by the conjugate to eliminate ii from the denominator. The conjugate zˉ\bar{z} flips the sign of the imaginary part and is your key tool for simplifying fractions and finding magnitudes.

Related theory

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