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 \( z \).
Complex Number Modulus - dCode
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
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: \( z = 1+i \) (of abscissa 1 and of ordinate 1 on the complex plane) then the modulus equals \( |z| = \sqrt{1^2+1^2} = \sqrt{2} \)
The calculation also applies with the exponential form of the complex number.
How to calculate the modulus of a real number?
The module 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 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:
$$ |\overline z|=|z| $$
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