How to use this calculator
- Enter or choose real part a.
- Enter or choose imaginary part b.
- The modulus |z| updates as you type. Use Copy result or Share to keep it.
Method
|a + bi| = √(a² + b²); z̄ = a − bi; z·z̄ = a² + b²
- a
- the real part
- b
- the imaginary part
Worked example
- a² + b² = 9 + 36 = 45.
- √45 = √(9 × 5) = 3√5 ≈ 6.7082039325.
- The conjugate flips the sign of the imaginary part: 3 − 6i.
|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.