Exercise 220Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Basic operations on complex numbers

#complex numbers#addition#multiplication#conjugate

Statement

Let $z_1=2+3i$ and $z_2=1-2i$ . Compute $z_1+z_2$ , $z_1-z_2$ , $z_1z_2$ , $\bar{z_1}$ , and $z_1/z_2$ .

Hint

Use $i^2=-1$ . For the quotient, multiply numerator and denominator by $\bar{z_2}$ .

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Sum. $z_1+z_2=3+i$ .

Difference. $z_1-z_2=1+5i$ .

Product. $z_1z_2=(2+3i)(1-2i)=2-4i+3i-6i^2=2-i+6=8-i$ .

Conjugate. $\bar{z_1}=2-3i$ .

Quotient.
$\frac{z_1}{z_2}=\frac{(2+3i)(1+2i)}{(1-2i)(1+2i)}=\frac{2+4i+3i+6i^2}{1+4}=\frac{-4+7i}{5}=-\frac45+\frac75 i.$

Conclusion. $z_1+z_2=3+i$ , $z_1-z_2=1+5i$ , $z_1z_2=8-i$ , $\bar{z_1}=2-3i$ , $z_1/z_2=-4/5+7i/5$ .