Algebraic Properties of Complex Numbers

Algebraic Properties of Complex Numbers

Definition. A complex number is an expression $z=a+ib$, where $a,b\in\mathbb{R}$ and $i$ satisfies $i^2=-1$. The real number $a$ is called the real part of $z$, written $\mathrm{Re}(z)=a$, and $b$ is called the imaginary part, written $\mathrm{Im}(z)=b$. Note that $\mathrm{Im}(z)$ is itself a real number — it is the real coefficient of $i$, not $ib$ itself.

• If $a=0$ and $b\neq0$, $z=ib$ is called purely imaginary.
• If $b=0$, $z=a$ is an ordinary real number; this is how $\mathbb{R}\subset\mathbb{C}$.
• The complex number $0+0i$ is simply the number $0$, and it is the only complex number that is simultaneously real and purely imaginary.

Equality of two complex numbers. Two complex numbers $z_1=a+ib$ and $z_2=c+id$ are defined to be equal exactly when their real parts match and their imaginary parts match:

This is a genuinely new rule — it says a single complex equation is equivalent to two real equations, obtained by equating real parts and imaginary parts separately. It is used constantly to solve for unknowns hidden inside a complex equation.

If $(x+2)+i(y-3)=5+4i$ for real $x,y$, equate parts: $x+2=5\Rightarrow x=3$, and $y-3=4\Rightarrow y=7$.