 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ia/paper-1/5c/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For

z \neq = 0

, rationalizing the denominators gives

1/ z = (x - i y) / (x^{2} + y^{2})

and

1/ \overline{z} = (x + i y) / (x^{2} + y^{2})

. Their imaginary parts cancel, so

Re (\frac{1}{z} + \frac{1}{z}) = \frac{2 x}{x ^{2} + y ^{2}}, Im (\frac{1}{z} + \frac{1}{z}) = 0.

This also follows because the summands are complex conjugates.

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