# Argument of a complex number

Allowed: constants, operators and i. To multiply use a*b not ab

This online tool calculates the 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