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) : Mathematics

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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 \( z \).

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

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.

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:

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

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