# Complex Number Modulus Calculator

> Find the modulus (absolute value) of a complex number as an exact simplified surd and a decimal, plus its conjugate, unit vector and argument.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/complex-number-modulus-calculator

The modulus |a + bi| = √(a² + b²) is the distance of the number from 0 on the complex plane. It is given here exactly, such as 3√5, and as a decimal.

Also called: Complex absolute value calculator, Complex magnitude calculator.

## Inputs
- Real part a (default 3)
- Imaginary part b (default 6)

## Results
- Modulus |z|
- Exact value
- |z|² = z·z̄
- Conjugate z̄
- Unit complex number z ÷ |z|
- Argument

## Formula

```
|a + bi| = √(a² + b²);  z̄ = a − bi;  z·z̄ = a² + b²
```

Where:
- a: the real part
- b: the imaginary part

## Worked example
1. a² + b² = 9 + 36 = 45.
2. √45 = √(9 × 5) = 3√5 ≈ 6.7082039325.
3. The conjugate flips the sign of the imaginary part: 3 − 6i.

**Result:** |3 + 6i| = 3√5 ≈ 6.7082.

## What the result means

The modulus is always zero or positive and equals 0 only for z = 0. Dividing z by its modulus gives the unit complex number with the same direction.

## Assumptions and limits
- The exact form is given when a and b are integers or terminating decimals; otherwise the decimal is shown.
- The exact surd is simplified for a² + b² (after clearing decimals) up to 10¹³.

## Questions

### How is the modulus related to the Pythagorean theorem?

a and b are the legs of a right triangle on the complex plane and |z| is its hypotenuse.

### Why is z times its conjugate real?

(a + bi)(a − bi) = a² − (bi)² = a² + b², which is |z|².

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Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/complex-number-modulus-calculator