Complex Number Modulus Calculator
The modulus of a complex number a + bi is its distance from the origin of the complex plane, found from the Pythagorean expression sqrt(a squared + b squared). It is the absolute value of the number and is always zero or positive. Enter the real part a and the imaginary part b and this calculator returns the modulus, the squared modulus (which equals a squared plus b squared and avoids a square root), and the separate squares of each component so you can follow the calculation step by step.
Complex modulus formula
z = a + b*i
|z| = sqrt(a^2 + b^2)
|z|^2 = a^2 + b^2 = z * conjugate(z)
The modulus is the hypotenuse of a right triangle with legs a and b. The squared modulus equals the product of z with its complex conjugate and needs no square root.
About the complex modulus
- The modulus is the radius r in the polar form z = r(cos theta + i sin theta).
- For 3 + 4i the modulus is exactly 5, a classic Pythagorean triple.
- The modulus of a product equals the product of the moduli.
- The squared modulus is widely used in signal power and quantum probability amplitudes.
- A complex number with modulus 1 lies on the unit circle in the complex plane.
Complex modulus: frequently asked questions
What is the modulus of a complex number?
The modulus of a complex number a + bi is its distance from the origin in the complex plane, computed as the square root of (a squared + b squared). It is written |z| and is always a non-negative real number.
How do I calculate the modulus?
Square the real part a, square the imaginary part b, add them and take the positive square root: |z| = sqrt(a^2 + b^2). For 3 + 4i the modulus is sqrt(9 + 16) = sqrt(25) = 5.
Is the modulus the same as the absolute value?
Yes. For complex numbers the modulus is the absolute value. For a real number (b = 0) the modulus reduces to the ordinary absolute value |a|.
What is the squared modulus used for?
The squared modulus |z|^2 = a^2 + b^2 equals z times its complex conjugate. It avoids a square root, so it is often preferred in signal processing where it represents power.
Can the modulus be negative?
No. The modulus is a length, so it is always zero or positive. It is zero only when both the real and imaginary parts are zero.
Official sources
- NIST Digital Library of Mathematical Functions: Complex Variables.
- NIST Digital Library of Mathematical Functions: DLMF home.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.