Search for a tool
Complex Number Modulus

Tool for calculating the value of the modulus of a complex number. The modulus of a complex number $$z$$ is written $$| z |$$ (absolute value) and consists of the length of the segment between the point of origin of the complex plane and the point $$z$$.

Results

Complex Number Modulus -

Tag(s) : Mathematics

dCode and you

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!
You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? Contact-me!

Team dCode read all messages and answer them if you leave an email (not published). It is thanks to you that dCode has the best Complex Number Modulus tool. Thank you.

# Complex Number Modulus

## Modulus (Absolute Value) Calculator

Tool for calculating the value of the modulus of a complex number. The modulus of a complex number $$z$$ is written $$| z |$$ (absolute value) and consists of the length of the segment between the point of origin of the complex plane and the point $$z$$.

### How to calculate the modulus of a complex number?

The module is the length (absolute value) in the complex plane, qualifying the complex number $$z = a + ib$$ (with $$a$$ the real part and $$b$$ the imaginary part), it is denoted $$| z |$$ and is equal to $$| z | = \sqrt{a ^ 2 + b ^ 2}$$.

Example: Consider $$z = 1+i$$ (of abscissa 1 and of ordinate 1 on the complex plane) then the modulus equals $$|z| = \sqrt{1^2+1^2} = \sqrt{2}$$

The module of a real number is equivalent to its absolute value.

Example: $$|-3| = 3$$

### What are the properties of modulus?

Consider the complex numbers $$z, z_1, z_2$$, the complex module has the following properties:

$$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$$

$$\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \iff z_2 \ne 0$$

$$|z_1+z_2| \le |z_1|+|z_2|$$

A modulus is an absolute value, therefore necessarily positive (or null):

$$|z| \ge 0$$

The modulus of a complex number and its conjugate are equal:

$$|\overline z|=|z|$$