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

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Prepare for the A Level Further Mathematics Core Pure Exam. Practice with flashcards and multiple-choice questions, each accompanied by hints and explanations. Ace your exam!

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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