Complex Numbers

## Description

The field of complex numbers in Sage is denoted by CC. The square root of $-1$ is denoted by I or i. Complex arithmetic works as expected.

## Sage Cell

#### Code

z = 2 + 3*I
w = -2 + 5*I
z*w


## Options

#### Option

We can use CC to create complex numbers from ordered pairs. CC is an approximation to the field of complex numbers using floating point numbers.

#### Code

z = CC(2,3)
w = CC(-2,5)
z*w


#### Option

real or real_part returns the real part of a complex number. Similarly, imag or imag_part returns the imaginary part of a complex number.

#### Code

z = CC(2,3)
z.real()


#### Option

If $z = a + bi$, then $|z| = \sqrt{a^2 + b^2}$ is the absolute value or modulus of $z$. abs returns the absolute value or modulus of a complex number.

#### Code

z = CC(2,3)
z.abs()


#### Option

The argument (angle) of the complex number, normalized so that $-\pi < \theta \leq \pi$ can be calculated using arg() or argument().

#### Code

z = CC(0,1)
z.arg()


#### Option

The conjugate of the complex number $z = a + bi$ is $\overline{z} = a - bi$ and can be computed using the command conjugate().

#### Code

z = CC(1,1)
z.conjugate()


## Tags

Primary Tags: Complex Analysis

Secondary Tags: Arithmetic

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## Attribute

Author: T. W. Judson

Date: 02 Aug 2018 19:46

Submitted by: Tom Judson