Complex Number to Polar Form Calculator
Type an imaginary or complex number as a + bi and get its polar form r∠θ at once, or type the polar form and get a + bi back. The two fields update each other as you type, and the result card shows the modulus and argument, with the modulus as an exact surd like √2 when it is one. Everything happens in your browser.
How to convert a complex number to polar form
- Type the number in Rectangular form, for example
3 + 4ior-1 + i. The page opens with3 + 4ialready in the field. - Polar form fills in straight away. For
3 + 4iit reads5∠0.927295. - Under the fields, the result card gives Modulus r (here
5) and Argument θ (here0.927295 rad). - To get the angle in degrees, press Degrees under Angle unit. The Polar form field becomes
5∠53.130102°and the argument shows53.130102°. Press Radians to switch back.
To go the other way, type in Polar form and read Rectangular form. While you type in one field, only the other field changes.
Rectangular to polar form: modulus and argument
For a complex number a + bi the polar form r∠θ has
- Modulus:
r = √(a² + b²), the distance from zero. - Argument:
θ = atan2(b, a), the angle from the positive real axis.
The calculator uses atan2 rather than a plain arctan of b/a, so the angle lands in the right quadrant. For -1 + i a plain arctan gives -π/4, which points the wrong way; the correct argument is 3π/4.
| Rectangular form | Modulus r | Argument θ (Radians) | Argument θ (Degrees) |
|---|---|---|---|
| 3 + 4i | 5 | 0.927295 rad | 53.130102° |
| 1 + i | √2 ≈ 1.414214 | π/4 ≈ 0.785398 rad | 45° |
| -1 + i | √2 ≈ 1.414214 | 3π/4 ≈ 2.356194 rad | 135° |
| 1 - i | √2 ≈ 1.414214 | -π/4 ≈ -0.785398 rad | -45° |
| -3i | 3 | -π/2 ≈ -1.570796 rad | -90° |
| -2 | 2 | π ≈ 3.141593 rad | 180° |
| -3 - 4i | 5 | -2.214297 rad | -126.869898° |
The argument is the principal value: greater than -π and up to π in Radians, greater than -180° and up to 180° in Degrees. In Radians it is shown as a fraction of π when it is a multiple of 15°, and as a decimal otherwise. The number 0 has no direction, so its Argument θ reads Undefined for 0.
Polar form to rectangular form
Type the polar form in Polar form and Rectangular form shows a + bi, using a = r cos θ and b = r sin θ. These spellings are all read:
5∠0.9273,5<0.9273and5@0.92735 cis 0.9273- Exponential form:
5e^(i*0.9273)or2e^(i pi/3) πorpiin the angle:5∠pi/4gives3.535534 + 3.535534i
An angle typed with no unit follows the Angle unit toggle. Type °, deg or rad after the angle to override it, so 5∠0.9273 rad works even while Degrees is selected. Choose the unit before you type the angle: pressing Radians or Degrees afterwards converts the number already entered rather than re-reading your digits, so 2∠60 typed in Radians and then switched to Degrees becomes 2∠-162.253229°, not 60°. Results are shown to 6 decimal places, so 5∠0.9273 gives 2.999981 + 4.000014i: 0.9273 is itself a rounded angle, and the exact angle gives exactly 3 + 4i.
Type arithmetic, not just a single number
The Rectangular form field evaluates simple complex arithmetic, so you can convert an expression without simplifying it first. The field keeps what you typed; Polar form and the result card show the answer.
| Type in Rectangular form | Equals | Modulus r | Argument θ (Degrees) |
|---|---|---|---|
| (1+i)^2 | 2i | 2 | 90° |
| 2(1+i) | 2 + 2i | 2√2 ≈ 2.828427 | 45° |
| (1+i)(1-i) | 2 | 2 | 0° |
| sqrt(-4) | 2i | 2 | 90° |
| i^3 | -i | 1 | -90° |
| e^(i*pi) | -1 | 1 | 180° |
Decimals can use a comma or a point, so 1,5 + 2i and 1.5 + 2i are the same number.
Phasors and AC circuit work
For a phasor, press Degrees, then type the magnitude and phase angle in Polar form. 10∠-90 becomes -10i, and 2∠60 becomes 1 + 1.732051i. In the other direction, 3 + 4j converts to 5∠53.130102°: j is read as the imaginary unit like i.
What this calculator does not do
- One number at a time: each field holds a single value, so there is no list or batch mode.
- 6 decimal places: results are rounded for display. The modulus is shown as an exact surd only for short integer and decimal inputs.
- No steps, no graph: it gives the answer, not a worked solution, and it does not draw the number on a diagram.
- Private: the numbers you type stay in your browser and are never sent anywhere.
Related math tools
To see complex numbers as color, try the complex function plotter. For quadratics, the factor quadratics calculator shows every step.
Frequently Asked Questions
How do I convert a complex number to polar form?
Type it in the Rectangular form field, for example 3 + 4i. The Polar form field fills in as you type, and the result card shows the Modulus r and the Argument θ. By hand, r = √(a² + b²) and θ = atan2(b, a), and the polar form is r∠θ.
What is the polar form of 3 + 4i?
The modulus is √(3² + 4²) = 5 and the argument is atan2(4, 3) ≈ 0.927295 radians, which is 53.130102°. So 3 + 4i = 5∠0.927295 in radians, or 5∠53.130102° in degrees.
How do I find the modulus and argument of a complex number?
The modulus is the distance from zero, r = √(a² + b²). The argument is the angle from the positive real axis, θ = atan2(b, a). Enter the number in the Rectangular form field and the Modulus r and Argument θ rows give both values, with the modulus as an exact surd such as √2 when it is one.
What is the argument of a complex number?
It is the angle the number makes with the positive real axis. This calculator shows the principal argument: greater than -π and up to π in radians, or greater than -180° and up to 180° in degrees. For -1 + i that is 3π/4 (135°), not -π/4.
What is the argument of zero?
Zero has no argument, because it has no direction. Enter 0 and the Argument θ row shows "Undefined for 0" while the modulus is 0.
How do I convert polar form to rectangular form?
Type the polar form in the Polar form field, for example 2∠60 with Degrees selected. The Rectangular form field fills in as a + bi, here 1 + 1.732051i. By hand, a = r cos θ and b = r sin θ.
How do I convert a complex number to polar form in radians or degrees?
Press Radians or Degrees under Angle unit. The Polar form field and the Argument θ row switch at once. In Radians the argument is also shown as a fraction of π when it is a multiple of 15°, such as π/4 for 1 + i.
How do I write a complex number in exponential form?
Exponential form is r·e^(iθ), using the same r and θ as the polar form. For 1 + i that is √2·e^(iπ/4). This calculator reads exponential form as input, for example 2e^(i pi/3) in the Polar form field, but it displays the result as r∠θ.
Does it work for phasors and AC circuits?
Yes. Select Degrees and type a phasor such as 10∠-90 in the Polar form field to get -10i, or enter 3 + 4j to get 5∠53.130102°. You can write the imaginary unit as i or j.
Does it show the steps?
No. It shows the answer, the modulus and argument, not a worked solution. The formulas are on this page, so you can check your own working against the result.
Recommended
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Casio fx-991ES Plus 2 Scientific CalculatorCheck your a + bi to r∠θ answers in complex mode, then carry it into the exam.
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FNIRSI DSO152 Pocket OscilloscopeSee the phase shift between two AC signals as an angle, the same θ you just converted.
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AstroAI Auto-Ranging MultimeterMeasure the AC voltages and currents behind your phasor numbers on a real circuit.
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