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”.

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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.

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

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

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