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Further complex numbers — Question 259

Original QCE Vault practice · 5 marks

Q259 · Practice questionTechnology-freeComplex familiar5 marks

QUESTION 259 (5 marks)

Let z,w∈Cz,w\in\mathbb C, with w≠0w\ne0. You may use uu‾=∣u∣2u\overline u=|u|^2 and the modulus product identity.
a)
Prove ∣z/w∣=∣z∣/∣w∣|z/w|=|z|/|w| without assuming the quotient identity.
[3 marks]
b)
Hence prove ∣1/z∣=∣z−1∣=1/∣z∣|1/z|=|z^{-1}|=1/|z| for z≠0z\ne0, and state why the restriction matters.
[2 marks]
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