Exercise 221Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

The equation $z=\bar z$

#conjugate#real axis#equation

Statement

Solve in $\mathbb{C}$ the equation $z=\bar z$ .

Hint

Write $z=a+bi$ and identify real and imaginary parts.

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Setup. $z=\bar z$ means $a+bi=a-bi$ , so $2bi=0$ , i.e. $b=0$ .

Conclusion. The solutions are the real numbers: $z\in\mathbb{R}$ .

Interpretation. $z=\bar z$ iff $z$ is real; geometrically, the fixed points of complex conjugation are the real axis.