Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
This calculator is also known as Complex Number Modulus Calculator.
Read the complete guideModulus Formula
For z = a + bi, the modulus is |z| = √(a² + b²). This is the Euclidean distance from the origin (0, 0) to the point (a, b) in the complex plane. Key properties: |z| ≥ 0, |z| = 0 iff z = 0, |z₁·z₂| = |z₁|·|z₂|, and |z·z̅| = |z|².
Examples
Modulus of 3 + 4i
Find |3 + 4i|.
|3 + 4i| = √(9 + 16) = √25 = 5. This is the classic 3-4-5 Pythagorean triple.
Key takeaway: The modulus has a geometric meaning: it is the straight-line distance from the origin to the point (3, 4) on the Argand diagram.
Using the Modulus
Practical applications of the modulus:
- Use |z|² = z·z̅ to avoid square roots when possible
- In signal processing, |z| represents the signal amplitude
- The triangle inequality |z₁ + z₂| ≤ |z₁| + |z₂| bounds the modulus of a sum
Frequently Asked Questions about Complex Number Modulus Calculator
Is the modulus the same as the absolute value?
Yes. For complex numbers, "modulus", "absolute value", and "magnitude" all refer to |z| = √(a² + b²). For real numbers, this reduces to the ordinary absolute value.
What is |z|²?
|z|² = a² + b² = z·z̅ (the product of z and its conjugate). This avoids the square root and is often more convenient in calculations.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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