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.
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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 →