What this calculator does
A complex number can be written two ways: rectangular form, a + bi, which plots it as a point using real and imaginary coordinates, or polar form, r∠θ, which describes the same point by its distance from the origin and the angle it makes with the positive real axis. This polar form calculator converts between the two in either direction.
Rectangular form is the natural way to add and subtract complex numbers, since you just combine the real and imaginary parts separately. Polar form is the natural way to multiply, divide and raise complex numbers to a power, since those operations turn into simple arithmetic on the modulus and angle instead. Converting between the two forms is what makes it possible to pick whichever one suits the calculation at hand.
The formula
Converting rectangular to polar: the modulus r is the distance from the origin, √(a² + b²), and the angle θ is found with atan2(b, a), which correctly places the angle in the right quadrant for any combination of signs on a and b. Converting polar to rectangular: the real part is r·cos θ and the imaginary part is r·sin θ, using the angle in degrees converted to radians for the trigonometric functions.
| Term | Meaning |
|---|---|
| a + bi | Rectangular form: a is the real part, b is the imaginary part. |
| r∠θ | Polar form: r is the modulus (distance from the origin), θ is the angle from the positive real axis. |
| Modulus (r) | The magnitude of the complex number, √(a² + b²), always a non-negative value. |
| Argument (θ) | The angle the number makes with the positive real axis, found with atan2 so it accounts for the correct quadrant. |
The inputs explained
| Field | What to enter |
|---|---|
| Convert | Choose which direction to convert: rectangular to polar, or polar to rectangular. |
| Real part (a) | Real part, used when converting from rectangular form. |
| Imaginary part (b) | Imaginary part, used when converting from rectangular form. |
| Modulus (r) | Modulus (distance from the origin), used when converting from polar form. |
| Angle (θ, degrees) | Angle in degrees, used when converting from polar form. |
When to use it
Multiplying or dividing complex numbers
Converting to polar form first turns multiplication into multiplying moduli and adding angles, and division into dividing moduli and subtracting angles, both simpler than the rectangular expansion.
Electrical engineering phasor calculations
Alternating current quantities are often expressed as phasors in polar form (magnitude and phase angle), while circuit equations are frequently easier to set up in rectangular form, so converting between the two is a routine step.
Checking a conversion done by hand
Working out atan2 by hand means tracking which quadrant the angle falls in, which is an easy step to get wrong. Running the same numbers through the calculator is a quick check on that.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does the polar form look like for some common complex numbers?
The modulus and angle for a spread of rectangular-form complex numbers.
| Real part (a) | Modulus (r) | Angle (θ) |
|---|---|---|
| 0 | 4.000 | 90.000° |
| 1 | 4.123 | 75.964° |
| 3 | 5.000 | 53.130° |
| 5 | 6.403 | 38.660° |
| -3 | 5.000 | 126.870° |
| -5 | 6.403 | 141.340° |
Questions
Why use atan2 instead of a plain arctangent?
A plain arctan(b/a) cannot distinguish between a point in the first quadrant and the opposite point in the third quadrant, since both give the same ratio. atan2(b, a) uses the signs of both a and b separately, so it always returns the angle for the correct quadrant.
What is the trigonometric form?
It is polar form written out as r(cos θ + i sin θ) instead of the shorthand r∠θ. Both describe exactly the same complex number; the trigonometric form just makes the underlying cosine and sine explicit.
Can the angle be negative or greater than 360 degrees?
The angle returned from a rectangular-to-polar conversion falls between −180 and 180 degrees, which is the standard range for atan2. A negative angle simply means the point lies below the real axis rather than above it.
What happens if both the real and imaginary parts are zero?
The complex number is zero itself, with a modulus of zero and no meaningfully defined angle, since there is no direction from the origin to a point that is the origin.
For plotting or converting ordinary 2D coordinates rather than complex numbers, a general polar-to-Cartesian coordinate conversion follows the same r·cos θ and r·sin θ relationships used here.