Exercise 423Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Properties of the conjugate

#conjugate#properties#modulus

Statement

Prove that for all $z,w\in\mathbb{C}$ :
1. $\overline{z+w}=\bar z+\bar w$ ;
2. $\overline{zw}=\bar z\bar w$ ;
3. $z\bar z=|z|^2$ ;
4. $z+\bar z=2\operatorname{Re}(z)$ and $z-\bar z=2i\operatorname{Im}(z)$ .

Hint

Write $z=a+bi$ , $w=c+di$ .

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Solution

1. $\overline{(a+c)+(b+d)i}=(a+c)-(b+d)i=(a-bi)+(c-di)=\bar z+\bar w$ .

2. $zw=(ac-bd)+(ad+bc)i$ , so $\overline{zw}=(ac-bd)-(ad+bc)i$ . And $\bar z\bar w=(a-bi)(c-di)=ac-ad i-bc i-bd=ac-bd-(ad+bc)i$ . Equal.

3. $z\bar z=(a+bi)(a-bi)=a^2+b^2=|z|^2$ .

4. $z+\bar z=2a=2\operatorname{Re}(z)$ ; $z-\bar z=2bi=2i\operatorname{Im}(z)$ .

Conclusion. All four properties hold.

Properties of the conjugate

Statement

Solution

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