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If z is a complex number for which z² = (conjugate of z)², then which statement is correct?

1. z is purely real
2. z is purely imaginary
3. either z is purely real or purely imaginary
4. none of these

Correct answer: either z is purely real or purely imaginary

Solution

The equation factors into (z - z-bar)(z + z-bar) = 0, giving z = z-bar (real) or z = -z-bar (imaginary).

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