Every complex number is a point on the plane, and two numbers pin it down: the modulus (how far from the origin) and the argument (in which direction). Together they're the polar description that a complex number calculator reports as r∠θ.
On this page
Modulus |z|
The modulus is the straight-line distance from the origin to the point (a, b), straight from Pythagoras:
|a + bi| = √(a² + b²)|3 + 4i| = √(9 + 16) = √25 = 5. (The classic 3–4–5 triangle.)
The modulus is never negative, and |z| = 0 only for z = 0.
Argument arg(z)
The argument is the angle, measured anticlockwise from the positive real axis, to the line joining the origin and z.
arg(3 + 4i) = arctan(4/3) ≈ 53.13°.
Principal value & quadrants
An angle is ambiguous by multiples of 360°, so we report the principal argument in the range −180° < θ ≤ 180°. Because arctan(b/a) can't tell quadrant II from IV, apply the same correction used when converting a+bi to polar form:
| z | Quadrant | arg(z) |
|---|---|---|
| 1 + i | I | 45° |
| −1 + i | II | 135° |
| −1 − i | III | −135° |
| 1 − i | IV | −45° |
Conjugates & symmetry
A conjugate z̄ = a − bi mirrors z across the real axis. So it has the same modulus and the negated argument: |z̄| = |z| and arg(z̄) = −arg(z). Also handy: z · z̄ = |z|², a real number.
On the calculator
In CMPLX mode use Abs(z) for the modulus and arg(z) for the argument (set Deg or Rad as needed):
Abs(3+4i) → 5
arg(3+4i) → 53.13° (Deg mode)For the complete walkthrough of complex features, see the complex number calculator pillar guide.
Frequently asked questions
What is the modulus?
The distance from the origin: |a+bi| = √(a²+b²). For 3+4i it's 5.
What is the argument?
The anticlockwise angle from the positive real axis. For 3+4i it's about 53.13°.
What is the principal argument?
The argument restricted to −180° < θ ≤ 180° (−π to π in radians).
Find |z| and arg(z)
Enter your complex number in CMPLX mode and read both at once.
Open the complex number calculator →