Complex Number Calculator guide
Complex Number Calculator: Rectangular, Polar and Exponential Form
Use our complex number calculator to add, multiply, divide and raise complex numbers, and see the modulus, argument and polar form of the answer.
By the AllCalculator editorial teamUpdated 15 October 20262 min read
Complex numbers combine a real part and an imaginary part, written a + bi, where i is the square root of −1. They're used in further maths, electrical engineering and physics. Our Complex Number Calculator works out any expression and shows the answer in every common form.
How to use the calculator (Complex Number Calculator)
Type an expression using i (or j, as engineers often write it). For example:
- (3 + 4i) × (1 − 2i)
- (2 + 3i) / (1 − i)
- (1 + i)^8
- sqrt(−4)
- e^(i × pi)
The forms of a complex number
- Rectangular: a + bi, the real and imaginary parts.
- Polar: r ∠ θ, where r is the modulus (distance from 0) and θ is the argument (angle).
- Exponential: re^(iθ), which makes multiplying and dividing easy.
The modulus is √(a² + b²) and the argument is the angle from the positive real axis.
A worked example
With the calculator's default setting, (3 + 4i)(1 − 2i):
- Multiply out: 3 − 6i + 4i − 8i²
- Since i² = −1, that's 3 − 2i + 8 = 11 − 2i
- Modulus: √(121 + 4) ≈ 11.18
- Argument: about −10.3°
Useful facts
- Conjugate: a − bi. A number times its conjugate is real: (a + bi)(a − bi) = a² + b².
- Euler's identity: e^(iπ) + 1 = 0.
- Multiplying complex numbers multiplies the moduli and adds the arguments.
For matrices, see our Matrix Calculator, and for everyday sums our Scientific Calculator.
Frequently asked questions
Can I use j instead of i?
Yes. Both are treated as the imaginary unit.
Is the argument in degrees or radians?
The calculator shows both.
What is the square root of a negative number?
It's imaginary. For example, sqrt(−4) = 2i. The calculator gives the principal root.
Complex numbers become much easier once you can see each form side by side. Try the Complex Number Calculator.
Sources and further reading
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.