Complex Number Calculator: Polar, Roots
This complex number calculator handles any expression built from numbers of the form a + bi. Type something like (3 + 4i)(1 − 2i), (2 + 3i)/(1 − i), (1 + i)^8 or sqrt(−16) and it returns the result in rectangular form, with exact fractions and radicals when possible. It also gives the modulus, the argument in radians and degrees, the polar form r(cos ฮธ + i sin ฮธ), the exponential form re^(iฮธ), the conjugate and the reciprocal. Choose a root option to list every square, cube or nth root of the number, spaced evenly around the circle, as De Moivre’s theorem predicts.
Complex Number Calculator
Add, multiply, divide and raise complex numbers, and convert to polar and exponential form.
How to use the Complex Number Calculator
- Type a complex number or expression, using i for √−1.
- Pick a root option if you want nth roots.
- Read the result in rectangular form.
- Check the polar and exponential forms below it.
Complex number formulas
(a + bi)(c + di) = (ac − bd) + (ad + bc)i · |a + bi| = √(a² + b²) · a + bi = r(cos ฮธ + i sin ฮธ) = re^(iฮธ) · nth roots: r^(1/n)·e^(i(ฮธ + 2ฯk)/n), k = 0 … n − 1
Features
- Add, subtract, multiply, divide and powers
- Exact fractions and radicals
- Polar and exponential form
- Nth roots by De Moivre
- Conjugate, modulus and argument
Frequently asked questions
How do you multiply complex numbers?
Use FOIL and replace i² with −1. (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i.
How do you divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom. (2 + 3i)/(1 − i) = (2 + 3i)(1 + i)/2 = (−1 + 5i)/2 = −1/2 + (5/2)i.
What is the modulus of a complex number?
The modulus |a + bi| is √(a² + b²), the distance from the origin. For 3 + 4i it is √(9 + 16) = 5.
How do I convert a complex number to polar form?
Find r = √(a² + b²) and ฮธ = atan2(b, a). Then a + bi = r(cos ฮธ + i sin ฮธ). For 1 + i, r = √2 and ฮธ = ฯ/4, which is 45°.
How many nth roots does a complex number have?
Every nonzero complex number has exactly n nth roots. They have the same modulus and are spaced 360°/n apart, so the 4th roots are 90° apart.