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This calculator finds the square root, modulus, inverse, conjugate and converts complex numbers to polar form. The solver explains each operation in detail, step by step.

**solution**

The inverse of $ z $ is:

$$ z^{-1} = \frac{ 4 }{ 25 }-\frac{ 3 }{ 25 }i $$**explanation**

We will denote the inverse of $ z $ as $ z_1 $. The inverse can be found in three steps.

**Step 1: ** Rewrite complex number as its reciprocal

**Step 2: ** Multiply top and bottom by complex conjugate of $ z $

**Step 3: ** Simplify

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Examples

ex 1:

Calculate the complex conjugate of z = 2/3 – 3i.

ex 2:

Find the modulus of z = 1/2 + 3/4.

ex 3:

Find the inverse of 3 - 3i.

ex 4:

Find the polar form of z = 1/2 + 4i.

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Find more worked-out examples in our database of solved problems..

This calculator performs five operations on a single complex number: modulus, conjugate, inverse, roots and polar form. We will demonstrate how to solve each of these problems using five basic examples.

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:

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

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.

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:

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

1. How to find conjugate and modulus ?

2. Polar form of a complex number — video tutorial with solved examples.

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