Complex Number Modulus Calculator
Enter the real and imaginary parts of z = a + bi to get the modulus |z| and the argument in degrees — and see exactly where the number sits in the complex plane.
Modulus and argument at once
Enter the real part a and the imaginary part b and the calculator returns the modulus |z| = √(a² + b²) and the argument atan2(b, a) in degrees together.
Negatives welcome
Either part can be negative — the four-quadrant argument keeps the right sign, so a point below the real axis returns a negative angle.
What is the modulus of a complex number?
The size and direction of z = a + bi
The complex number modulus calculator turns the two parts of a complex number z = a + bi into its modulus and argument. The modulus |z| is the distance of the point a + bi from the origin in the complex plane — its size, always zero or positive. The argument is the angle that the line from the origin to the point makes with the positive real axis, the direction of the number. Together the modulus and argument describe a complex number in polar form, the natural way to multiply, divide, and take powers of complex numbers, and the bridge to phasors in electrical engineering and signal processing.
Enter a real part and an imaginary part to get the modulus |z| and the argument in degrees instantly.
The modulus is the square root of the sum of the squares of the two parts, and the argument is the four-quadrant arctangent of the imaginary part over the real part.
|z| = √(a² + b²)φ = atan2(b, a)The modulus comes straight from the Pythagorean theorem applied to the real and imaginary coordinates. The argument uses atan2, the two-argument arctangent, which looks at the signs of both parts and so returns the correct angle in all four quadrants — something plain arctangent cannot do.
Suppose z = 3 + 4i, so the real part a is 3 and the imaginary part b is 4.
Square each part
3² = 9 and 4² = 16 — the squared real and imaginary coordinates.
Add and take the square root
√(9 + 16) = √25 = 5 — the modulus |z|, the distance from the origin.
Find the argument
atan2(4, 3) ≈ 53.1301° — the angle above the positive real axis. So z = 3 + 4i has modulus 5 and argument 53.13°.
The two outputs answer two different questions. The modulus (5 for 3 + 4i) is the magnitude of the number — how far the point lies from the origin and, in applications, the amplitude of a phasor or the gain of a system. The argument (53.13°) is the direction — the angle of the point, the phase of a signal. The four points 3 + 4i, −3 + 4i, −3 − 4i and 3 − 4i all share a modulus of 5 but have arguments of about 53.13°, 126.87°, −126.87° and −53.13° respectively, one in each quadrant: the modulus measures size while the argument records where the number points. Read together they pin down a complex number completely, and converting to this polar view is what makes multiplying complex numbers (moduli multiply, arguments add) so much simpler than working with a + bi directly.
The formulas are exact, but a couple of conventions are worth keeping in mind.
Argument in degrees, principal value only
This calculator reports the argument in degrees, measured from the positive real axis, as the principal value between −180° and 180°. Other conventions use radians or the range 0° to 360°; add 360° to a negative result if you need the 0°–360° form. The argument is undefined at the origin, so for z = 0 + 0i the modulus is 0 and the argument is reported as 0° by convention.