Complex Number Argument Calculator
The argument of a complex number a + bi is its phase angle: the direction it points from the origin of the complex plane, measured counterclockwise from the positive real axis. It is computed with the two-argument arctangent atan2(b, a), which returns the correct quadrant by keeping the signs of both parts. Enter the real and imaginary parts and this calculator returns the principal argument in radians and degrees, plus the modulus, so you have the full polar description of the number.
Argument formula
z = a + b*i
arg(z) = atan2(b, a) in radians, principal value in (-pi, pi]
degrees = radians * 180 / pi
|z| = sqrt(a^2 + b^2)
The atan2 function uses the signs of both a and b to place the angle in the correct quadrant. Multiplying by 180 / pi converts radians to degrees.
About the argument
- The argument and modulus together give the polar form of a complex number.
- For 1 + i the argument is pi/4 radians, which is 45 degrees.
- The principal argument lies in (-pi, pi], so angles below the real axis are negative.
- Multiplying complex numbers adds their arguments and multiplies their moduli.
- The argument of a positive real number is 0; of a negative real number it is pi.
Complex argument: frequently asked questions
What is the argument of a complex number?
The argument is the angle that the complex number a + bi makes with the positive real axis in the complex plane, measured counterclockwise. It is the phase angle theta in the polar form z = r(cos theta + i sin theta).
How is the argument calculated?
The principal argument uses the two-argument arctangent: arg(z) = atan2(b, a). This returns an angle between -pi and pi (excluding -pi), correctly handling every quadrant, unlike a plain arctangent of b divided by a.
What is the principal value range?
The principal argument lies in the interval (-pi, pi] radians, equivalently (-180, 180] degrees. Adding any multiple of 2 pi gives an equally valid argument, which is why the argument is multivalued.
Why use atan2 instead of arctan(b/a)?
A plain arctan of b divided by a cannot tell the second quadrant from the fourth, or the third from the first, because it loses the individual signs of a and b. atan2 keeps both signs and returns the correct quadrant.
What is the argument when the number is zero?
The argument of 0 is undefined, since the zero complex number has no direction. This calculator shows n/a 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: Inverse Trigonometric Functions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.