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Complex Number Modulus/Magnitude

Tool for calculating the value of the modulus/magnitude of a complex number |z| (absolute value): the length of the segment between the point of origin of the complex plane and the point z

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Complex Number Modulus/Magnitude -

Tag(s) : Arithmetics, Geometry

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# Complex Number Modulus/Magnitude

## Complex from Modulus and Argument Calculator

### What is the modulus of a complex number? (Definition)

The modulus (or magnitude) 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 $|z| = \sqrt{a^2 + b^2}$.

The module can be interpreted as the distance separating the point (representing the complex number) from the origin of the reference of the complex plane.

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

To find the modulus of a complex number $z = a + ib$ carry out the computation $|z| = \sqrt {a^2 + b^2}$

Example: $z = 1+2i$ (of abscissa 1 and of ordinate 2 on the complex plane) then the modulus equals $|z| = \sqrt{1^2+2^2} = \sqrt{5}$

### How to calculate the modulus of a complex number in exponential form?

A complex number in exponential notation has the form $re^{i \theta}$, the modulus is the value of $r$.

Example: $2e^{i\pi}$ has for modulus $2$

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

The modulus (or magnitude) of a real number is equivalent to its absolute value.

Example: $|-3| = 3$

### What are the properties of modulus?

For the complex numbers $z, z_1, z_2$ the complex modulus has the following properties:

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

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

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

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

$$|z| \ge 0$$

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

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

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