Complex number

The set of complex numbers is denoted by $\mathbb{C}$ . The set of complex numbers contains the set $\mathbb{R}$ of the real numbers but is much wider. Every complex numbers has a real part, denoted by $\Re$ or simply $\mathrm{Re}$ , and a imaginary part, denoted by $\Im$ or simply $\mathrm{Im}$ . So if $z\in \mathbb C$ , we can write $z=\mathrm{Re}(z)+i\mathrm{Im}(z)$ where $i$ is the imaginary unit.

As you can see, complex numbers enable us to remove the restriction of $x\ge 0$ for the domain of $f(x)=\sqrt{x}$ .

The letters $z$ and $\omega$ are usually used to denote complex numbers.

Operations

Addition
Subtraction
Multiplication
Division
Absolute value/Modulus/Magnitude (denoted by $|z|$ ). This is the distance from the origin to the complex number when graphed.

Simple Example

If $z=a+bi$ and $w=c+di$ ,

$\mathrm{Re}(z)=a$ , $\mathrm{Im}(z)=b$
$|z|=\sqrt{a^2+b^2}$
$\mathrm{Re}(w)=c$ , $\mathrm{Im}(w)=d$
$|w|=\sqrt{c^2+d^2}$
$z+w=(a+c)+(b+d)i$
$z-w=(a-c)+(b-d)i$

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Complex number

Contents

Operations

Simple Example

Topics

See also