Complex Number

Complex Number

Definition. A number of the form $z=a+ib$, where $a,b\in\mathbb{R}$ and $i=\sqrt{-1}$ (so $i^2=-1$), is called a complex number. Here $a$ is called the real part of $z$, written $\mathrm{Re}(z)$ or $R(z)$, and $b$ is called the imaginary part of $z$, written $\mathrm{Im}(z)$ or $I(z)$.

The set of all complex numbers is denoted $\mathbb{C}$:

Worked table of examples. To read off the real and imaginary parts of a complex number, first rewrite it in the standard $a+ib$ form (converting any negative-radicand surds to multiples of $i$ along the way), and then simply read off the coefficient of $1$ as the real part and the coefficient of $i$ as the imaginary part. For instance $5i$ is really $0+5i$, so its real part is $0$ and imaginary part is $5$; and $5+\sqrt{-16}$ becomes $5+4i$ once the surd is converted, giving real part $5$ and imaginary part $4$.

Five standing notes, referred to throughout the chapter:

1. A complex number whose real part is zero is called a purely imaginary number. Such a number has the form $z=0+ib=ib$.
2. A complex number whose imaginary part is zero is simply a real number: $z=a+0i=a$.
3. A complex number whose real part and imaginary part are both zero is the zero complex number: $0=0+0i$. …