Complex number calculator

1 : Modulus ( Magnitude )

The modulus or magnitude of a complex number (denoted by |z|), is the distance between the origin and that number.

If the z = a + bi is a complex number than the modulus is

$$ \color{blue}{|z| = \sqrt{a^2 + b^2}} $$

Example 01: Find the modulus of z = 6 + 3i.

In this example a = 6 and b = 3, so the modulus is:

$$ \begin{aligned} | z | &= \sqrt{ a^2 + b^2} = \sqrt{6^2 + 3^2} = \\[1 em] &= \sqrt{36 + 9} = \sqrt{45} = \\[1 em] &= \sqrt{9 \cdot 5} = 3 \sqrt{5} \end{aligned} $$

2 : Conjugate

To find the complex conjugate of a complex number, we just need to change the sign of the imaginary part.

The conjugate of z = a + bi, is z=a – bi

Example 02: The complex conjugate of z=3+4i is z=3-4i.

Example 03: The conjugate of z=-4i is z=4i.

Example 04: The conjugate of z=15 is z=15, too.

4 : Inverse

The inverse or reciprocal of a + bi is

$$ \color{blue}{ \frac{1}{a + b\,i}} $$

Here is an example of how to find the inverse.

Example 06: Find the inverse of the number z=4+3i.

$$ \begin{aligned} \frac{1}{z} &= \frac{1}{4+3i} = \frac{1}{4+3i} \cdot \frac{4-3i}{4-3i} = \\[1 em] &= \frac{4-3i}{(4+3i)(4-3i)} = \frac{4-3i}{4^2 - (3i)^2} = \\[1 em] &= \frac{4-3i}{16+9} = \frac{4-3i}{25} = \frac{4}{25} - \frac{3}{25} i \end{aligned} $$

3 : Polar Form

The polar form of a complex number z = a + ib is given as:

$$ z = |z| ( \cos \alpha + i \sin \alpha) $$

Example 05: Express the complex number z = 2 + i in polar form.

To find a polar form, we need to calculate |z| and α using formulas in the above image.

$$ |z| = \sqrt{2^2 + 1^2} = \sqrt{5}$$ $$ \tan \alpha = \dfrac{b}{a} = \dfrac{1}{2} \implies \alpha = \tan^{-1}\left(\dfrac{1}{2}\right) \approx 27^{o} $$

So, the polar form is:

$$ z = \sqrt{5} \left( \cos 27^o + i \sin27^o \right) $$