Complex number calculator

The calculator does the following: extracts the square root, calculates the modulus, finds the inverse, finds conjugate and transforms complex numbers into polar form. For each operation, the solver provides a detailed step-by-step explanation.

problem

Find the inverse of

$$z = 4+3i$$

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

$$ z_1 = \frac{1}{ 4+3i } $$

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

$$ z_1 = \frac{1}{ 4+3i } \cdot \frac{ 4-3i }{ 4-3i } $$

Step 3: Simplify

$$ z_1 = \frac{ 4-3i }{ 25 } $$$$ z_1 = \frac{ 4 }{ 25 } - \frac{ 3 }{ 25 } \cdot i$$$$ z_1 = \frac{ 4 }{ 25 }-\frac{ 3 }{ 25 }i $$

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Operations with complex number
five operations with a single complex number
show help ↓↓ examples ↓↓ tutorial ↓↓
$ z = 4 + 6i $
$ z = 2 - \frac{3}{2}i $
$ z = \sqrt{2} - \sqrt{5}i $
Modulus (Magnitude) Conjugate
Inverse Roots
Polar form
Find approximate solution
Hide steps
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EXAMPLES
example 1:ex 1:
Find the complex conjugate of $z = \frac{2}{3} - 3i$.
example 2:ex 2:
Find the modulus of $z = \frac{1}{2} + \frac{3}{4}i$.
example 3:ex 3:
Find the inverse of complex number $3 - 3i$.
example 4:ex 4:
Find the polar form of complex number $z = \frac{1}{2} + 4i$.
TUTORIAL

Operations on complex numbers

This calculator performs five operations on a single complex number.
It computes module, conjugate, inverse, roots and polar form.

1 : Modulus ( Magnitude )

The modulus or magnitude of a complex number ( denoted by $ \color{blue}{ | z | }$ ), is the distance between the origin and that number.

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

$$ |z| = \sqrt{a^2 + b^2} $$ complex number modulus

Example 01: Find the modulus of $ z = \color{blue}{6} + \color{purple}3{} i $.

In this example $ \color{blue}{a = 6} $ and $ \color{purple}{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 need to change the sign of the imaginary part. The conjugate of $ z = a \color{red}{ + b}\,i $ is:

$$ \overline{z} = a \color{red}{ - b}\,i $$ complex number conjugate

Example 02: The complex conjugate of $~ z = 3 \color{blue}{+} 4i ~$ is $~ \overline{z} = 3 \color{red}{-} 4i $.

Example 03: The conjugate of $~ z = - 4i ~$ is $~ \overline{z} = 4i $.

Example 04: The conjugate of $~ z = 15 ~$ is $~ \overline{z} = 15 ~$, too.

4 : Inverse

The inverse or reciprocal of a complex number $ a + b\,i $ 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 + i\,b$ is given as $ z = |z| ( \cos \alpha + i \sin \alpha) $.

ploar form of complex number

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

To find a polar form, we need to calculate $|z|$ and $ \alpha $ 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:

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