Complex Number Calculator
Add, subtract, multiply, divide or raise complex numbers a + bi to a power, with the modulus, argument, polar form and conjugate of the result.
Result
Result: −11 + 23i
Exact arithmetic on the parts entered; decimals shown to ten significant figures.
- Real part
- -11
- Imaginary part
- 23
- Modulus, |z|
- 25.495098
- Argument, θ
- 115.559965 ° (2.016902 rad)
- Polar form
- 25.49509757(cos 115.5599652° + i sin 115.5599652°)
- Complex conjugate
- −11 − 23i
https://calcgallery.com/algebra-numbers/complex-number-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Complex Number Calculator
- Enter the first number as its real part a and imaginary part b, so 3 + 2i is a = 3 and b = 2.
- Choose the operation, then enter the second number, or the power for raising to a power.
- Read the result as a + bi, with its modulus, argument, polar form and conjugate.
The rules
(a + bi) + (c + di) = (a + c) + (b + d)i(a + bi)(c + di) = (ac − bd) + (ad + bc)i(a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)|z| = √(a² + b²), θ = the angle of (a, b)Polar form
A complex number is also a length and a direction: z = r(cos θ + i sin θ), with r the modulus and θ the argument. In that form multiplying is simple, since the lengths multiply and the angles add, and the diagram shows the product turning by the sum of the two angles. The polar coordinates calculator converts any point between the two forms.
Powers
The page raises a number to a whole power by repeated multiplication, which keeps whole-number answers exact: (1 + i)⁸ is exactly 16. De Moivre’s theorem, zⁿ = rⁿ(cos nθ + i sin nθ), gives the same value through trigonometry but leaves a trace of rounding in the imaginary part. A negative power is 1 divided by the positive power. Complex roots of a quadratic or a cubic are found by the quadratic formula and cubic equation calculators.
With your numbers
- (3 + 2i)(1 + 7i) = (ac − bd) + (ad + bc)i, since i² = −1
Quick reference
Exact arithmetic on the parts entered; decimals shown to ten significant figures.
| Example | Result |
|---|---|
| (3 + 2i)(1 + 7i) | −11 + 23i |
| (3 + 2i) ÷ (1 + 7i) | 0.34 − 0.38i |
| (3 + 2i) + (1 + 7i) | 4 + 9i |
| (3 + 2i) − (1 + 7i) | 2 − 5i |
| i² | −1 + 0i |
| (1 + i)⁸ | 16 + 0i |
| 3 + 4i, modulus and argument | 3 + 4i |
Common questions about the Complex Number Calculator
- How do I multiply complex numbers?
- Multiply out the brackets as with any two binomials, then replace i² with −1. (3 + 2i)(1 + 7i) = 3 + 21i + 2i + 14i² = 3 + 23i − 14 = −11 + 23i.
- How do I divide complex numbers?
- Multiply the top and bottom by the conjugate of the bottom, which turns the bottom into a real number. For (3 + 2i) ÷ (1 + 7i), multiply by (1 − 7i): the top becomes 17 − 19i and the bottom 1 + 49 = 50, so the answer is 0.34 − 0.38i.
- What are the modulus and the argument?
- The number’s position on the complex plane in polar terms. The modulus |z| = √(a² + b²) is its distance from zero, and the argument is the angle from the positive real axis. For 3 + 4i they are 5 and about 53.13°. The page gives the principal argument, between −180° and 180°; zero has no argument.
- What is the complex conjugate?
- The same number with the sign of its imaginary part flipped: the conjugate of 3 + 4i is 3 − 4i. A number times its conjugate is always real, a² + b², which is why conjugates are used to divide.
- Why is i² equal to −1?
- Because that is how i is defined: the number whose square is −1, which no real number can be. Everything else follows from it. i³ = −i, i⁴ = 1, and the pattern repeats every four powers.