A complex number calculator handles numbers with a real and an imaginary part, the kind that appear everywhere in engineering and physics. Ours works in both rectangular (a+bi) and polar (r∠θ) form, and converts freely between them. Here's how to use every feature.
On this page
Entering complex numbers
Switch to CMPLX mode and type the number using i for the imaginary unit:
3+4iElectrical engineers write j to avoid clashing with current; it's the same imaginary unit. On the calculator, enter complex numbers with i.
Arithmetic
Add, subtract, multiply and divide just like real numbers, the calculator keeps real and imaginary parts straight for you.
(3+4i)+(1+2i) → 4+6i
(3+4i)(1+2i) → −5+10i. By hand: 3 + 6i + 4i + 8i², and since i² = −1 that's 3 + 10i − 8 = −5 + 10i. ✔
(3+4i)/(1+2i) → 2.2−0.4i. (Multiply top and bottom by the conjugate 1−2i.)
Modulus, argument & conjugate
Three operations describe a complex number geometrically:
- Modulus
Abs(z), its distance from the origin, |z| = √(a²+b²). - Argument
arg(z), its angle from the positive real axis. - Conjugate
Conjg(z), the reflection a−bi.
For z = 3+4i: Abs(3+4i) → 5, arg(3+4i) → 53.13° (Deg mode), Conjg(3+4i) → 3−4i.
Full detail in modulus and argument explained.
Rectangular ↔ polar
After any calculation, switch the display form:
►r∠θshows the result in polar form r∠θ.►a+bishows it back in rectangular form.
You can also type polar input with the ∠ separator, e.g. 2∠45 (the angle uses the current angle mode, so set Deg for degrees).
3+4i then ►r∠θ → 5∠53.13°. The full method is in converting a+bi to polar form.
Why two forms?
Each form makes one operation easy:
| Operation | Easiest form | Rule |
|---|---|---|
| Add / subtract | Rectangular a+bi | Add real & imaginary parts |
| Multiply / divide | Polar r∠θ | Multiply/divide r, add/subtract θ |
(2∠30°)(3∠40°) = 6∠70°, multiply the magnitudes, add the angles. This is why engineers love polar form, as covered in complex arithmetic for engineering students.
Frequently asked questions
How do I enter a complex number?
In CMPLX mode, type it with i, for example 3+4i.
How do I convert between rectangular and polar?
Use ►r∠θ for polar and ►a+bi for rectangular; type polar input with ∠, e.g. 2∠45.
Are j and i the same?
Yes, engineers write j, but it's the same imaginary unit. Enter i.
Why is polar better for multiplication?
You multiply magnitudes and add angles, much faster than expanding brackets.
Try the complex number calculator
Switch to CMPLX mode and reproduce any example, then hit ►r∠θ to see polar form.
Open the complex number calculator →