How to use the complex number calculator
- Choose an Operation such as a x b, a / b, a to the power n or all n-th roots of a.
- Type Complex number a (and b for operations that need two) in the form 3+4i, -2i, 1.5-2.5i or just 7. The letter j works as well as i.
- For a power or root, type n. Roots need a whole number from 1 to 60. A power can be any real number, including a fraction or a negative.
- Read the answer in rectangular form (a + bi) and polar form (modulus and angle). Roots are listed in a table with all of them.
Formula
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
(a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)
modulus r = square root of (a^2 + b^2) angle theta = atan2(b, a)
n-th roots: r^(1/n) at angles (theta + 360 k) / n for k = 0 to n - 1
De Moivre's theorem gives powers as r^n at angle n theta, and the same idea with fractional exponents gives roots. Whole-number powers are computed by repeated multiplication so that no angle rounding creeps in.
Arithmetic with 3+4i and 1-2i
The product (3+4i)(1-2i) is 11 - 2i, and the quotient (3+4i) / (1-2i) is -1 + 2i. The cube (3+4i)^3 is -117 + 44i. The number 3+4i has modulus 5 and angle 53.130102 degrees.
The three cube roots of 3+4i are 1.6289371 + 0.5201745i, -1.2649529 + 1.1506137i and -0.3639842 - 1.6707882i. They sit evenly around a circle of radius 5^(1/3) and each one cubed gives 3+4i back. Multiplying in polar form (sizes multiply, angles add) gives the same 11 - 2i as the rectangular method.
Rectangular and polar forms
Rectangular form a + bi is best for adding and subtracting, because you combine real and imaginary parts separately. Polar form, a size and an angle, is best for multiplying, dividing, powers and roots, because sizes multiply and angles add. The page shows both for every answer.
The angle is the principal one, between -180 and 180 degrees. Other angles that differ by 360 degrees describe the same number, which is why a number has several roots. Square roots of a negative number work too: enter -4 and ask for the square roots (n = 2) to get 2i and -2i.
Using it with other tools
If you came from solving an equation, the Polynomial Root Finder lists complex roots of a polynomial directly, and the Quadratic Equation Solver handles the second-degree case. To work with polynomials themselves, the Polynomial Arithmetic Calculator adds, multiplies and divides them exactly.
More math tools
For significant figures, polynomials, repeating decimals, differential equations and graphs, try the Polynomial Arithmetic Calculator, Repeating Decimal to Fraction Converter and Differential Equation Solver.
Assumptions and limits
- Numbers are written with real and imaginary parts as plain decimals, with i or j for the imaginary unit.
- Parts are limited to a size of 1e100 so that results stay readable and finite.
- Roots and non-integer powers use the principal angle. Other branches give other values.
- Tiny parts left over by binary arithmetic, such as 1e-16, are shown as 0.
- The example numbers are an illustration. Replace them with your own.
Frequently asked questions
What does it mean to raise a number to a fractional power?
The page uses the principal value: take the size to the power and multiply the angle by the exponent. For example (3+4i)^0.5 is 2 + i, the principal square root. The other square root is its negative.
Why are there n roots of a number?
Because the angle of a number is only known up to full turns. Dividing the angle plus each full turn by n gives n different directions, all with the same size, and each one raised to the n-th power gives the number back.
How do I enter a purely imaginary number?
Write it as 2i, -3.5i or just i. A plain number such as 7 is read as 7 + 0i.
Is the answer exact?
It uses ordinary floating-point numbers with about 15 digits. Whole-number products and powers of numbers with integer parts are exact within that precision, and anything involving angles is rounded to about 8 significant digits for display.