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Complex Number Modulus Calculator

Find a complex number’s modulus.

How it is calculated

A complex number z = a + bi can be drawn as a point (a, b) on a plane, with the real part a on the horizontal axis and the imaginary part b on the vertical axis. Its modulus is the distance from the origin, |z| = √(a² + b²). Its argument is the angle from the positive real axis, arg z = atan2(b, a), which the tool shows in degrees. Because atan2 looks at the signs of both parts, it places the angle in the right quadrant, with values from −180° to 180°. Results are rounded to two decimals. For z = 0 the modulus is 0 and the angle is shown as "—", since it is undefined.

Worked example

Take z = 3 + 4i. Modulus: √(3² + 4²) = √(9 + 16) = √25 = 5.00. Argument: atan2(4, 3) = about 0.9273 radians, which is 53.13 °. The point lies in the first quadrant, on a 3-4-5 right triangle, and the number can be written in polar form as 5 at 53.13 °.

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Common mistakes

  • Computing arctan(b ÷ a) by hand, which puts numbers in the second and third quadrants on the wrong side. For −3 + 4i the answer is about 126.87 °, not −53.13 °.
  • Including the i when entering the imaginary part. Enter only the coefficient, so 4 for 4i.
  • Forgetting that the result is in degrees, while many textbooks and programming languages use radians.
  • Adding the real and imaginary parts instead of using Pythagoras. |3 + 4i| is 5, not 7.

About this calculator

This calculator finds the modulus and argument of a complex number a + bi. The modulus is its distance from the origin on the complex plane, calculated as √(a² + b²), and the argument is the angle it makes with the positive real axis, calculated as atan2(b, a).

Engineering and math students use it to check homework on complex numbers, and electrical engineers use the same modulus-argument form to work with AC circuit impedance and phasors. The angle is reported in degrees alongside the modulus.

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Comments (1)

  • Editorial team

    For AC circuit work, enter the resistance as the real part and the reactance as the imaginary part. The modulus is then the impedance magnitude in ohms, and the angle is the phase shift in degrees, positive when the circuit is inductive and negative when capacitive. A quick sanity check: the modulus is never smaller than the larger of the two parts. Which range does your course use for the argument, 0 to 360 degrees or minus 180 to 180?

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