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
The two formulas
For a complex number z = a + bi, two formulas locate it on the plane:
|z| = √(a² + b²)
arg(z) = tan⁻¹(b/a), adjusted for the quadrant a and b place z inThe first is Pythagoras' theorem: it gives the straight-line distance from the origin. The second gives the direction, measured anticlockwise from the positive real axis. That "adjusted for the quadrant" clause matters: tan⁻¹ (also written arctan) only ever returns a value between −90° and 90°, so used on its own it gives the wrong answer for any point left of the imaginary axis. The fix is a short table, in the next section.
Worked example: z = 3 + 4i
|3 + 4i| = √(3² + 4²) = √25 = 5. (The classic 3-4-5 triangle.)
arg(3 + 4i) = tan⁻¹(4/3) ≈ 53.13° (0.927 rad). Both a = 3 and b = 4 are positive, so z sits in the first quadrant and tan⁻¹(4/3) is already correct: no adjustment needed.
Try abs(3+4i) on the calculator · Try arg(3+4i) on the calculator
What the modulus means
|z| is the straight-line distance from the origin to the point (a, b), the same Pythagoras' theorem you would use for any right triangle with legs a and b. It is never negative, and |z| = 0 only when z = 0 itself. Scale a and b together, say by doubling both, and |z| scales by the same factor: the modulus depends on the point as a whole, not on either coordinate read alone.
What the argument means
arg(z) is the angle between the positive real axis and the line from the origin to z, always measured anticlockwise. It describes direction only, never distance. 2+2i and 5+5i point the same way and share an argument of 45°, even though their moduli are completely different.
The argument in each quadrant
tan⁻¹(b/a) is only correct as it stands for two of the four quadrants. The table below covers all four, using the principal value range explained underneath it.
| Quadrant | Signs of a and b | Rule | Worked example |
|---|---|---|---|
| First (I) | a > 0, b ≥ 0 | arg(z) = tan⁻¹(b/a) | 1 + i: tan⁻¹(1/1) = 45° (π/4) |
| Second (II) | a < 0, b ≥ 0 | arg(z) = tan⁻¹(b/a) + π, or +180° | −1 + i: tan⁻¹(1/−1) + 180° = 135° (3π/4) |
| Third (III) | a < 0, b < 0 | arg(z) = tan⁻¹(b/a) − π, or −180° | −1 − i: tan⁻¹(−1/−1) − 180° = −135° (−3π/4) |
| Fourth (IV) | a > 0, b < 0 | arg(z) = tan⁻¹(b/a) | 1 − i: tan⁻¹(−1/1) = −45° (−π/4) |
Try arg(1+i) · Try arg(-1+i) · Try arg(-1-i) · Try arg(1-i)
Only the second and third quadrants need the +180° or −180° correction. tan⁻¹ cannot tell a point from the one directly opposite it through the origin: 1/1 and −1/−1 are the same ratio, so tan⁻¹ returns 45° for both 1 + i and −1 − i unless you check the sign of a first and add the correction back in by hand.
The range used in the table, from −180° up to and including 180° (written (−π, π] in radians), is the principal value: the one answer a calculator reports, since adding or subtracting any whole multiple of 360° (2π rad) points in exactly the same direction and would otherwise leave the argument ambiguous. Some courses and textbooks use the range 0° to 360° instead (written [0, 2π) in radians), where the same direction as −45° is written 315°. Both describe the identical direction; only the label changes.
Special cases: real, imaginary and zero
The quadrant rules assume a and b are both nonzero. Five cases sit outside them.
| z | Type | arg(z) |
|---|---|---|
| 3 | Positive real | 0 |
| −3 | Negative real | π, or 180° |
| 2i | Positive imaginary | π/2, or 90° |
| −2i | Negative imaginary | −π/2, or −90° |
A positive real number points along the positive real axis, so its argument is 0. A negative real number points the opposite way, at 180°. A positive imaginary number points straight up the imaginary axis, at 90°; a negative one points straight down, at −90°. None of these four needs tan⁻¹ at all, since b/a is either 0 or undefined by division.
Try arg(-3) · Try arg(2i) · Try arg(-2i)
z = 0 is the fifth case, and a genuinely special one. The point (0, 0) has no direction, so its argument is undefined, mathematically. This calculator still has to display something rather than nothing: enter arg(0) and it returns 0 by convention, not an error.
Common mistakes
- Using tan⁻¹(b/a) without the quadrant fix. A calculator's tan⁻¹ button only ever returns a value between −90° and 90°. For any z in the second or third quadrant, that raw answer is off by 180°: not a rounding slip, a genuinely wrong direction. Check the sign of a before you trust the answer.
- Mixing radians and degrees. tan⁻¹(4/3) is 53.13 in one unit and 0.927 in the other, and both are correct, only for their own unit. Set the angle mode (Deg or Rad) before you compute, not after you read the result, and write down which unit an answer is in.
- Sign errors on a or b. arg(z) depends on the sign of a and b separately, not just their ratio. −1 + i and 1 − i both reduce to a ratio of −1, yet they are 180° apart, 135° against −45°, because one has a negative real part and the other a negative imaginary part. Write down the signs before you divide.
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, press hyp to open the CMPLX Operations menu, where arg( gives the argument. Press SHIFT then hyp (labelled Abs above the key) to type abs( directly for the modulus. Set Deg or Rad first, since arg( reports its answer in whichever one is active:
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 of a complex number?
The modulus |z| is the distance from the origin to the point (a, b) in the complex plane: |z| = √(a² + b²). It is never negative, and |z| = 0 only when z = 0.
What is the argument of a complex number?
The argument arg(z) is the angle measured anticlockwise from the positive real axis to the line joining the origin and z. It is found from tan⁻¹(b/a), then adjusted for the quadrant that a and b place z in.
What is the modulus of 3+4i?
|3 + 4i| = √(3² + 4²) = √25 = 5, the classic 3-4-5 right triangle with the origin and the point (3, 4) as two of its corners.
Can the argument be negative?
Yes. Using the principal value range (−π, π], any complex number in the third or fourth quadrant, or on the negative imaginary axis, has a negative argument. For example, arg(1 − i) is −45°, or −π/4 rad.
What is the principal value of the argument?
It is the single value of the argument in the range (−π, π], or −180° to 180°, that a calculator reports by default, since adding or subtracting a whole multiple of 360° points in the same direction. Some courses use the range [0, 2π) instead, where the same direction as −45° is written 315°.
What is the argument of a real or purely imaginary number?
A positive real number has argument 0. A negative real number has argument π, or 180°. A positive imaginary number like 2i has argument π/2, or 90°. A negative imaginary number like −2i has argument −π/2, or −90°.
What is the argument of 0?
It is undefined, since the point (0, 0) has no direction to measure an angle to. On this calculator, entering arg(0) returns 0 by convention rather than an error.
Do you need CMPLX mode to find the modulus and argument?
That is where the tools live. In CMPLX mode, pressing hyp opens the CMPLX Operations menu, where arg( gives the argument; SHIFT then hyp types abs( for the modulus. Set the angle mode to Deg or Rad first, since arg( reports its answer in whichever one is active.
Find |z| and arg(z)
Enter your complex number in CMPLX mode and read both at once.
Open the complex number calculator →