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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 $.

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Complex Number Modulus -

Tag(s) : Arithmetics, Geometry

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Complex Number Modulus

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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 $.

Answers to Questions

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

The modulus 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 module 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} $

The calculation also applies with the exponential form of the complex number.

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 its conjugate are equal:

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

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