What can be said about z z* for any complex number z?

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Boost your skills in A Level Further Mathematics Core Pure. Study with flashcards and multiple choice questions, each question is followed by hints and explanations. Get prepared for your exam with confidence!

Multiple Choice

What can be said about z z* for any complex number z?

Explanation:
When you multiply a complex number by its complex conjugate, the imaginary parts cancel and you’re left with a real quantity. Write z = a + bi, then z* = a − bi. Their product is (a + bi)(a − bi) = a^2 + b^2, which is |z|^2. This is a real number and is always greater than or equal to zero, being zero only when z = 0. So z z* is real for any complex z.

When you multiply a complex number by its complex conjugate, the imaginary parts cancel and you’re left with a real quantity. Write z = a + bi, then z* = a − bi. Their product is (a + bi)(a − bi) = a^2 + b^2, which is |z|^2. This is a real number and is always greater than or equal to zero, being zero only when z = 0. So z z* is real for any complex z.

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