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
Complex Number Modulus/Magnitude - dCode
Tag(s) : Arithmetics, Geometry
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Complex Number Modulus/Magnitude
Modulus (Absolute Value) Calculator
Complex from Modulus and Argument Calculator
Answers to Questions (FAQ)
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 $
See also the page about 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 the modulus of its conjugate are equal:
$$ |\overline z|=|z| $$
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Complex Number Modulus/Magnitude on dCode.fr [online website], retrieved on 2023-10-01,
- Modulus (Absolute Value) Calculator
- Complex from Modulus and Argument Calculator
- What is the modulus of a complex number? (Definition)
- How to calculate the modulus of a complex number?
- How to calculate the modulus of a complex number in exponential form?
- How to calculate the modulus of a real number?
- What are the properties of modulus?
