Addition and Subtraction of Complex Numbers

Addition and Subtraction of Complex Numbers

For $z_1=a+ib$ and $z_2=c+id$ ($a,b,c,d\in\mathbb{R}$), addition is defined componentwise:

That is, real parts add with real parts and imaginary parts add with imaginary parts — exactly as if $i$ were an ordinary algebraic symbol being collected. Since $a+c$ and $b+d$ are again real numbers, $z_1+z_2$ is again of the form (real) $+i$(real): $\mathbb{C}$ is closed under addition.

Subtraction is addition of the negative: $-z=-a-ib$ (both parts negated), and

$(5-3i)-(2+4i)=(5-2)+i(-3-4)=3-7i$.

Properties of addition, each following directly from the corresponding property of real-number addition applied separately to the real and imaginary parts:

1. Commutative: $z_1+z_2=z_2+z_1$, since $a+c=c+a$ and $b+d=d+b$ for reals.
2. Associative: $(z_1+z_2)+z_3=z_1+(z_2+z_3)$, by associativity of real addition in each part.
3. Additive identity: $z+0=z$ for every $z$, taking $0=0+0i$.
4. Additive inverse: for every $z=a+ib$, the number $-z=-a-ib$ satisfies $z+(-z)=0$. …