Complex Number Calculator
Enter the real and imaginary parts of two complex numbers. You get the sum, difference, product and quotient, and the polar form, conjugate, square root and power of the first.
First number, a + bi
Second number, c + di
Worked examples
Multiplying
(3 + 4i) × (1 - 2i)
11 - 2i
3 - 6i + 4i - 8i², with i² = -1.
Dividing
(3 + 4i) ÷ (1 - 2i)
-1 + 2i
Multiply top and bottom by 1 + 2i, then divide by 5.
About Complex Number Calculator
A complex number has a real part and an imaginary part, written a + bi, where i is the square root of -1. They let you solve equations such as x² + 1 = 0 that have no real answer, and they are used in electrical engineering, signal processing and physics. Adding and subtracting work on the real and imaginary parts separately. Multiplying uses the fact that i² = -1, so (3 + 4i)(1 - 2i) expands to 3 - 6i + 4i - 8i², which comes to 11 - 2i.
Division multiplies the top and the bottom by the conjugate of the bottom, which removes i from the denominator. The modulus is the length of the number as a point on the plane, the square root of a² + b², and the argument is its angle from the real axis. Together they give the polar form, which makes powers and roots easy. Enter two complex numbers and the page shows the four operations between them, and for the first number the modulus, argument, polar form, conjugate, principal square root and any power you choose.
How to use Complex Number Calculator
- Enter the real and imaginary parts of the first number.
- Enter the real and imaginary parts of the second number.
- Set the power you want the first number raised to.
- Read the sum, difference, product and quotient, then the polar form and other results for the first number.
Common uses
- Checking complex number homework
- AC circuit and impedance calculations
- Finding roots of quadratic and cubic equations
- Rotations and scaling in two dimensions
Frequently asked questions
What is the modulus of a complex number?
It is its distance from zero on the complex plane, the square root of a² + b². For 3 + 4i it is 5.
What is the argument?
It is the angle between the number and the positive real axis, measured anticlockwise. The page shows it in degrees and in radians.
What is the conjugate?
The conjugate of a + bi is a - bi. A number times its conjugate is always real and equals its modulus squared.
How do I divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom number. The bottom becomes a real number and you can then split the result into real and imaginary parts.
Which square root is shown?
The principal root, the one with a positive real part. The other root is its negative.