Absolute Value of a Complex Number
Online calculator and formulas for calculating the magnitude of a complex number
Magnitude Calculator
Absolute Value of a Complex Number
The absolute value of a complex number is the length of its vector in the complex plane, calculated using the Pythagorean theorem.
Graphical Representation
Vector in the Complex Plane
The complex number is represented as a vector. The magnitude corresponds to the length of this vector.
Formulas for the Absolute Value of a Complex Number
The absolute value (also called magnitude or modulus) of a complex number \(z = a + bi\) is the length of the corresponding position vector in the complex plane.
Standard Definition
Pythagorean formula for the distance from the origin
Conjugate Representation
Multiplication with the complex conjugate
Properties of the Absolute Value
Basic Properties
- \(|z| \geq 0\) for all \(z \in \mathbb{C}\)
- \(|z| = 0 \Leftrightarrow z = 0\)
- \(|z| = |\overline{z}|\) (magnitude of conjugate)
- \(|-z| = |z|\) (magnitude of negation)
Calculation Rules
- \(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\) (product rule)
- \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\) for \(z_2 \neq 0\)
- \(|z^n| = |z|^n\) (power rule)
- \(|z_1 + z_2| \leq |z_1| + |z_2|\) (triangle inequality)
Calculation Examples
Example 1: z = 3 + 4i
\(|z| = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
This is a classic example with the 3-4-5 Pythagorean triple.
Example 2: z = 3 - 4i
\(|z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
The magnitude is identical to Example 1, since \(|z| = |\overline{z}|\).
Example 3: Conjugate Representation
\(|z| = \sqrt{z \cdot \overline{z}} = \sqrt{(3-4i) \cdot (3+4i)} = \sqrt{9 - 16i^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
The conjugate representation uses the property \(z \cdot \overline{z} = a^2 + b^2\).
Absolute Value of a Complex Number - Detailed Description
Geometric Interpretation
In the complex plane, each complex number \(z = a + bi\) is represented by a point with coordinates \((a, b)\).
• Real part \(a\) = x-coordinate (horizontal axis)
• Imaginary part \(b\) = y-coordinate (vertical axis)
• Magnitude \(|z|\) = distance from the origin (0,0)
• The vector from the origin to the point forms a right triangle
Pythagorean Theorem
The magnitude results from the Pythagorean theorem: In a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have: \(c = \sqrt{a^2 + b^2}\)
Important Note
The magnitude is always positive (or zero), regardless of the signs of the real and imaginary parts. It holds: \(|z| = |-z| = |\overline{z}| = |-\overline{z}|\)
Practical Applications
The absolute value of a complex number is used in many areas:
• Electrical engineering: Impedance and AC circuit analysis
• Signal processing: Amplitude of frequency components
• Mathematics: Convergence of sequences and series
• Physics: Amplitudes of oscillations and waves
Programming
The magnitude is often referred to as Abs() (from "Absolute value")
or Magnitude() in programming languages.
Code Examples
abs(3+4j) → 5.0MATLAB:
abs(3+4i) → 5C#:
Complex.Abs(new Complex(3,4)) → 5JavaScript:
Math.hypot(3,4) → 5
Special Cases
- Real numbers: \(|a + 0i| = |a|\) (classical absolute value)
- Imaginary numbers: \(|0 + bi| = |b|\)
- Zero: \(|0 + 0i| = 0\)
- Unit circle: All numbers with \(|z| = 1\)
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More complex functions
Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •