Converting a+bi to polar form means rewriting a complex number as r∠θ — a distance and an angle. It's a two-step calculation, and a complex number calculator will confirm every result with the ►r∠θ key.
On this page
The two steps
- Modulus r = √(a² + b²) — the distance from the origin.
- Argument θ — the angle from the positive real axis to the point (a, b).
Then the polar form is simply r∠θ.
1+i: r = √(1+1) = √2 ≈ 1.414, θ = 45°. So 1+i = √2∠45°.
Getting the quadrant right
The trap: arctan(b/a) alone only gives the correct angle in quadrants I and IV. Adjust for the others:
| Quadrant | Signs (a, b) | Adjustment to arctan(b/a) |
|---|---|---|
| I | (+, +) | none |
| II | (−, +) | add 180° |
| III | (−, −) | subtract 180° |
| IV | (+, −) | none |
The calculator's arg and ►r∠θ already apply the correct quadrant — they use the two-argument arctangent internally — so you never have to remember this table when checking your work.
Examples in all four quadrants
3+4i (I) → 5∠53.13°
−1+i (II) → r = √2, arctan(1/−1) = −45°, +180° = √2∠135°
−1−i (III) → r = √2, arctan gives 45°, −180° = √2∠−135°
1−i (IV) → √2∠−45°
Doing it on the calculator
In CMPLX mode (angle mode Deg for degrees), type the rectangular number and press ►r∠θ:
-1+i → ►r∠θ → √2∠135°To go the other way, type polar input directly, e.g. 2∠45, then press ►a+bi. The concepts behind r and θ are unpacked in modulus and argument explained, and the full feature tour is the complex number calculator pillar guide.
Frequently asked questions
How do I convert a+bi to polar form?
Find r = √(a²+b²) and the angle θ, then write r∠θ. On the calculator, press ►r∠θ.
How do I get the quadrant right?
arctan(b/a) works in I and IV; add 180° in II and subtract 180° in III. The calculator does this for you.
What is the modulus?
The distance from the origin, r = √(a²+b²). For 3+4i it's 5.
Convert it in one keystroke
Enter a+bi in CMPLX mode and press ►r∠θ to read the polar form.
Open the complex number calculator →