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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

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 $$z$$.

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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