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