Complex Number to Polar Form Calculator
This complex number to polar form calculator turns a + bi into a distance and a direction — and keeps the distance exact, as a whole number where it can be and as a surd like 2√2 where it cannot.
It also does the thing most calculators skip: it works out which quadrant the number is in before touching the arctangent, because the arctangent on its own gets a quarter of all cases wrong by half a turn.
Two Numbers Instead of Two Numbers
Rectangular form says how far right and how far up: 3 + 4i means three across and four up. Polar form says the same thing as how far out and which way: five units from the origin, at an angle of about 53°.
Neither is more correct. They suit different jobs, and the reason polar form exists is that multiplying is far easier in it — the distances multiply and the angles simply add, which is much less work than expanding brackets. (If the numbers are still in rectangular form and you simply want the product, the complex number calculator multiplies them out directly.)
The distance is called the modulus and written |z| or r. The angle is called the argument and written arg z or θ. Those two numbers are the whole of polar form; everything below is about finding them and writing them down.
The Modulus Is Pythagoras
r = √(a² + b²)
That is the distance formula, and nothing more: a right triangle with legs a and b. So |3 + 4i| = √25 = 5 exactly, and |5 − 12i| = √169 = 13.
Those two are lucky. Usually the sum of squares is not a perfect square, and then the exact answer is a surd: |1 + i| = √2, |2 + 4i| = 2√5, |2 + 2i| = 2√2. Writing 1.4142 instead of √2 is an approximation of the answer rather than the answer, and this page keeps the surd.
One detail worth noticing: a² + b² is always an ordinary number, however awkward a and b are. It is only the square root that can turn irrational. That is why the modulus can always be given exactly here, in one form or the other.
The Argument, and the Mistake Everybody Makes
The angle is where the trouble is, and it is worth being blunt about why.
The usual instruction is θ = arctan(b/a). But the arctangent only ever sees the ratio of the two parts — and −1 ÷ −1 is the same ratio as 1 ÷ 1. So it cannot tell a point from the one directly opposite it.
Take −1 − i. The arctangent says 45°. The point is in the third quadrant, so the true angle is −135° — out by half a turn. The rule is short: if the real part is negative, add or subtract 180° to swing the answer round to the right side.
That is why this page names the quadrant first and only then applies the arctangent, showing you what the uncorrected value would have been. A calculator that hands back the raw arctangent is wrong for every number with a negative real part — which is half of them.
When the Angle Is Exact, and Why It Usually Is Not
Here is something rarely said plainly. Type an ordinary a + bi and the argument comes out exactly in eight directions only:
0, ±π/4, ±π/2, ±3π/4, π — that is, 0°, ±45°, ±90°, ±135° and 180°.
The reason is short. The tangent of the angle is b ÷ a, which is an ordinary number whenever a and b are — and the only ordinary tangents belonging to neat fractions of π are 0 and ±1, with the vertical directions where the tangent does not exist at all. Every other angle is genuinely irrational.
So 1 + i is exactly π/4, and −3 + 3i is exactly 3π/4, but 3 + 4i is not any neat fraction of π at all — 53.1301° is a rounded figure and always will be. If you have seen a calculator print π/3 for a typed a + bi, it was either given polar input to begin with or rounding without telling you.
This page marks which you are looking at every single time: the modulus stays exact, and the angle is labelled exact or rounded rather than left for you to guess.
The Four Ways of Writing It
Once you have r and θ you have the number, and it gets written in four ways depending on who is writing:
| r∠θ | Angle or phasor notation. Common in electrical engineering |
| r(cos θ + i sin θ) | Trigonometric form. The one that shows why it works |
| r cis θ | The same again, abbreviated. cis is just cos + i sin |
| reiθ | Exponential form, from Euler's formula |
These are four notations for one pair of numbers, not four different things. Euler's formula, eiθ = cos θ + i sin θ, is what joins the last two to the middle one — and it is the exponential form that makes the multiplication rule obvious, since multiplying powers of e adds the exponents, which is exactly the statement that the angles add.
How to Use This Calculator
Two boxes. Type the real part and the multiple of i; an empty box counts as 0. Either box will take a fraction such as 3/4 or a decimal. The angle is given in degrees and radians together, so whichever your course wants is already there.
| real | How far right. Empty means 0 |
| imaginary (× i) | How far up. Empty means 0 |
| ← → ↑ ↓ | Move between the two boxes |
| ± | Flips the sign of the box you are in |
| ⁄ and . | Fraction bar and decimal point |
| ⌫ / AC | Delete one character, or clear both boxes |
| = | Works it out |
Open the working and you get the whole route: r² before r, the quadrant named before the arctangent is used, what the uncorrected arctangent would have said, whether the angle is exact or rounded, all four notations, and the answer turned back into a + bi as a check.
One number has no polar form at all: zero. It sits at the origin, so its modulus is 0 and every angle describes it equally well — which is why the argument of 0 is left undefined rather than given as 0.
Polar Form FAQ
How do you convert a complex number to polar form?
What is the modulus of a complex number?
Why is the arctangent not enough to find the argument?
When is the argument an exact angle like pi over 4?
What is the exponential form of a complex number?
What does cis mean?
Distance, then direction — and the quadrant before the arctangent, every time. That one habit removes most of the wrong answers in this topic.