# Argument of a complex number

## Argument of a complex number

Let z be a complex number written in its algebraic form,

`z = x + i * y`

x is the real part of z

y is the imaginary part of z

z has the following graphical representation,

We define the argument of a complex number as follows,

An argument of a non-zero complex number z, denoted by arg (z), is a radian measure `\varphi` of the angle formed by the x-axis and the vector \(\overrightarrow{OM}\), M is the point that represents z in the complex plane (M is said to be the affix of z).

z can be written in polar form,

`z = r *( cos(\varphi) + i * sin(\varphi))`

or in exponential form,

`z = r * e^(i*\varphi)`

`\text{arg}(z) = \varphi`

`|z| = r` where |z| is the modulus from z

**Principal argument**

There are an infinite arguments of z: `\varphi\, \varphi+2pi\, \varphi+4pi, \varphi+2kpi`, with k a relative integer but, there is only one argument that belongs to `(-pi,pi]`, this argument is called the **principal argument** of z.

**How to calculate the argument of a complex number ?**

By writting z in its algebraic and polar forms we get,

`z = |z| *( cos(\varphi) + i * sin(\varphi)) = x + i*y`

To calculate z principal argument, simply find `\varphi` between `-pi` and `pi` that satisfies,

`{(cos(\varphi) = \frac{x}{|z|}),(sin(\varphi) = \frac{y}{|z|}):}`

## Examples of argument calculations

**Example 1**

`z = i`

`|z| = 1, x = 0, y = 1`

`cos(\varphi) = \frac{0}{1} = 0`

`sin(\varphi) = \frac{1}{1} = 1`

So, `\text{arg}(z) = \varphi = pi/2`

**Example 2**

`z = \sqrt(3)+i`

`|z| = \sqrt(\sqrt(3)^2+1^2) = \sqrt(3+1)=2, x = \sqrt(3), y = 1`

`cos(\varphi) = \frac{\sqrt(3)}{2}`

`sin(\varphi) = \frac{1}{2}`

Then, `\text{arg}(z) = \varphi = pi/6`

## See also

Modulus of a complex number

Conjugate of a complex number

Operations on complex numbers