Rectangular Form

Definition (rectangular form). A complex number is written $x+iy$ (or, equivalently, $x+yi$), where $x$ and $y$ are real numbers. Here $x$ is called the real part of the complex number and $y$ is called the imaginary part.

• If $x=0$, the complex number is purely imaginary.
• If $y=0$, the complex number is (identified with) a real number.
• Zero ($0=0+0i$) is the only number that is at once real and purely imaginary.

The standard rectangular form $x+iy$ is usually denoted $z$, and we write $x=\operatorname{Re}(z)$, $y=\operatorname{Im}(z)$. For instance, for $z=5+7i$: $\operatorname{Re}(z)=5$ and $\operatorname{Im}(z)=7$; for $z=5-7i$: $\operatorname{Re}(z)=5$ and $\operatorname{Im}(z)=-7$.

Numbers of the form $\beta i,\ \beta\ne0$ are called imaginary (non-real complex) numbers. …