Addition of Complex Numbers

Addition of Complex Numbers

Let $z_1=a+ib$ and $z_2=c+id$. Their sum is defined by adding real parts together and imaginary parts together:

Equivalently: $\mathrm{Re}(z_1+z_2)=\mathrm{Re}(z_1)+\mathrm{Re}(z_2)$ and $\mathrm{Im}(z_1+z_2)=\mathrm{Im}(z_1)+\mathrm{Im}(z_2)$.

Worked examples.

1. $(2+3i)+(4+3i)=(2+4)+(3+3)i=6+6i$.
2. $(-2+5i)+(7+3i)+(6-4i)=[(-2)+7+6]+[5+3+(-4)]i=11+4i$ — with three or more terms, all the real parts add together and, separately, all the imaginary coefficients add together.

Properties of addition. For any complex numbers $z_1,z_2,z_3$:

• (i) $z_1+z_2=z_2+z_1$ (commutative)
• (ii) $z_1+(z_2+z_3)=(z_1+z_2)+z_3$ (associative)
• (iii) $z_1+0=0+z_1=z_1$ (the complex number $0$ is the additive identity) …