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Homemodulus of complex number

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# modulus of complex number

Definition
Let $z$ be a complex number, and let
$\overline{z}$ be the complex conjugate of $z$.
Then the *modulus*, or *absolute value*, of $z$ is defined as

$|z|:=\sqrt{z\overline{z}}.$ |

There is also the notation

$\mod{z}$ |

for the modulus of $z$.

If we write $z$ in polar form as $z=re^{{i\phi}}$ with $r\geq 0,\;\phi\in[0,\,2\pi)$, then $|z|=r$. It follows that the modulus is a positive real number or zero. Alternatively, if $a$ is the real part of $z$, and $b$ the imaginary part, then

$\displaystyle|z|$ | $\displaystyle=$ | $\displaystyle\sqrt{a^{2}+b^{2}},$ | (1) |

which is simply the Euclidean norm of the point $(a,\,b)\in\mathbb{R}^{2}$. It follows that the modulus satisfies the triangle inequality

$|z_{1}+z_{2}|\leq|z_{1}|+|z_{2}|,$ |

also

$|\Re{z}|\leq|z|,\quad|\Im{z}|\leq|z|,\quad|z|\leq|\Re{z}|+|\Im{z}|.$ |

Modulus is multiplicative:

$|z_{1}z_{2}|=|z_{1}|\cdot|z_{2}|,\quad\left|\frac{z_{1}}{z_{2}}\right|=\frac{|% z_{1}|}{|z_{2}|}$ |

Since $\mathbb{R}\subset\mathbb{C}$, the definition of modulus includes the real numbers. Explicitly, if we write $x\in\mathbb{R}$ in polar form, $x=re^{{i\phi}}$, $r>0$, $\phi\in[0,2\pi)$, then $\phi=0$ or $\phi=\pi$, so $e^{{i\phi}}=\pm 1$. Thus,

$|x|=\sqrt{x^{2}}=\begin{cases}x&x>0\\ 0&x=0\\ -x&x<0\end{cases}.$ |

## Mathematics Subject Classification

32-00*no label found*30-00

*no label found*12D99

*no label found*

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