How to Find Modulus (Absolute Value) and Argument (Angle) of Complex Numbers?

The formula for complex numbers argumentation

A complex number can be expressed in polar form as \(r (cos\ θ + i sin\ θ)\), where is the \(θ \) argument. The argument function \(arg(z)\) where \(z\) denotes the complex number, \(z=(x+iy)\). The formula for calculating the complex argument is as follows:

\(\color{blue}{ arg (z) =arg (x+iy)= tan^{–1}(\frac {y}{x})}\)

Thus, the argument is expressed as follows:

\(\color{blue}{ θ =tan^{–1}(\frac {y}{x})}\)

Complex number argument properties

Now, let us look at some of the characteristics that complex numbers and their arguments share.

• If \(n\) is an integer and \(z\) is a complex number, then:

\(\color{blue}{arg\ (z^n)=n\ arg (z)}\)

• \(Z_1\) and \(Z_2\) are two complex numbers that we can suppose to have the following properties:

\(\color{blue}{ arg (\frac {z_1}{z_2})=arg (z_1)-arg(z_2)}\)

\(\color{blue}{arg\ (z_1 z_2)= arg (z_1) + arg (z_2)}\)

What is the best way to find the argument of a complex number?

1. From the given complex number, determine the real and imaginary components. \(x\) and \(y\) should be referred to as such.
2. In the formula \(\color{blue}{ θ =tan^{–1}(\frac {y}{x})}\), substitute the appropriate values.
3. If the formula returns a standard value, find the value of \(tan^{–1}\) and write it in the form of \(tan^{–1}\) itself if there is no standard value.
4. The needed value of the complex argument for the specified complex number is this value followed by the unit “radian”.

The Modulus and Argument of Complex Numbers – Example 1:

Find the argument of the complex number. \(z= 3+3\sqrt{3} i\)

In \(z=3+3\sqrt{3} i\): the real part is \(x=3\) and imaginary part \(y= 3\sqrt{3}\).

Now, to find the argument of a complex number use this formula: \(\color{blue}{ θ =tan^{–1}(\frac {y}{x})}\).

\( θ =tan^{–1}(\frac { 3\sqrt{3} }{3})\) \(=tan^{-1}\sqrt{3}\)

\(θ\) \(=tan^{-1}\sqrt{3}=arctan \sqrt {3}=\frac{ π}{3} \)

Therefore, the argument of the complex number is \(\frac {π}{3}\) radian.

The Modulus and Argument of Complex Numbers – Example 2:

Find the modulus of a complex number. \(z=-5+3i\)

Use this formula to find the modulus of a complex number: \(\color{blue}{ |z| =\sqrt {x^2+y^2}}\)

\(|z| =\sqrt {(-5)^2+(3)^2}\)\(=\sqrt {25+9}\)\(=\sqrt {34}\)

The Modulus and Argument of Complex Numbers – Example 3:

If \(arg ({z_1})=\frac{7π }{6}\) and \(arg ({z_2})=\frac{-π }{2}\), find the \(arg ( z_1 z_2)\).

To find the \(arg ( z_1 z_2)\) use this formula: \(\color{blue}{arg\ (z_1 z_2)= arg (z_1) + arg (z_2)}\)

\(arg ( z_1 z_2)\) \(= \frac{7π }{6} + \frac{-π }{2} = \frac{7π }{6} – \frac{π }{2} =\frac{ 7π -3 π }{6}=\frac {4π}{6}=\frac{2π}{3}\)

Exercises for the Modulus and Argument of Complex Numbers

Find the modulus of a complex number.

• \(\color{blue}{ z= {2+i}}\)
• \(\color{blue}{ z= {10-6i}}\)

Find the argument of a complex number.

• \(\color{blue}{ z= {\sqrt{3}-i}}\)
• \(\color{blue}{ z= {-4-4i}}\)

• \(\color{blue}{ |z| =\sqrt {5}}\)
• \(\color{blue}{ |z| =8}\)
• \(\color{blue}{ θ=\frac{-π}{6}}\)
• \(\color{blue}{ θ=\frac{π}{4}}\)