Complex Analysis: Theory and Applications · Problem 1.2.13

Complex Numbers — Howell & Matthews

Mathematics — Complex Analysis · multiplicative-inverse-computation

Problem

From Exercise 12 and basic cancellation laws, it follows that z−1=1z=z‾zz‾z^{-1} = \dfrac{1}{z} = \dfrac{\overline{z}}{z\overline{z}}. The numerator here, z‾\overline{z}, is trivial to calculate and, as the denominator zz‾z\overline{z} is a real number (Exercise 3), computing the quotient z‾zz‾\dfrac{\overline{z}}{z\overline{z}} should be rather straightforward. Use this fact to compute z−1z^{-1} if z=2+3iz = 2 + 3i and again if z=7−5iz = 7 - 5i.

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