Math
Complex Number Calculator
Type two complex numbers such as 3 + 4i and 1 - 2i and the complex number calculator adds, subtracts, multiplies, or divides them exactly, or raises one to a power with De Moivre's theorem.
Every answer comes with its modulus, argument (in degrees and radians, with exact π fractions when they exist), conjugate, polar and exponential forms, and a plot on the complex plane.
Write it as a + bi. Fractions and decimals are fine: 1/2 + 3i, 2.5 - 1.5i, or just 5.
The second number in the operation.
Try an example
Result
Result (a + bi)
—
- Result (decimal)
- —
- Modulus |z|
- —
- Argument (radians)
- —
- Argument (degrees)
- —
- Conjugate
- —
- Polar form
- —
More details (2 more)
- Exponential form
- —
- Location on the plane
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the complex number calculator solves
A complex number z = a + bi has a real part a and an imaginary part b, where i^2 = -1. Adding and subtracting work part by part; multiplying uses FOIL and i^2 = -1; dividing multiplies the top and bottom by the conjugate of the The bottom number of a fraction; it says how many equal parts make one whole. so the denominator becomes the real number c^2 + d^2.
Rectangular, polar, and exponential form
| Form | Looks like | Best for |
|---|---|---|
| Rectangular | a + bi | Adding and subtracting |
| Polar | r(cos θ + i sin θ) | Multiplying, dividing, powers, roots |
| Exponential | r·e^(iθ) | Compact notation; same information as polar |
The modulus r = |z| is the distance from the origin to the point (a, b). The argument θ is the angle measured counterclockwise from the positive real axis; the calculator reports the principal value in (-180°, 180°]. When the angle is a multiple of 30° or 45° it is shown as an exact fraction of π, such as 3π/4.
Reading the complex plane
The Argand plane plots a + bi as the point (a, b): the real part along the horizontal axis and the imaginary part along the vertical axis. Adding complex numbers adds the arrows tip to tail; multiplying multiplies the lengths and adds the angles.
How to use it
- Choose the operation: add, subtract, multiply, divide, power, or analyze one number.
- Type z₁ (and z₂) in a + bi form. Fractions such as 1/2 + (3/4)i are kept exact.
- Read the result at the top, then the modulus, argument, conjugate, and polar form below it. Open the steps to see the FOIL or conjugate work.
How to read the answer
The result is written in standard a + bi form with exact fractions and, when they differ, a decimal version. The argument is positive above the real axis and negative below it; a result on an axis has an argument of 0, 90°, 180°, or −90°. The modulus of a product is the product of the moduli, and the argument of a product is the sum of the arguments.
Common mistakes and edge cases
- Forgetting i^2 = -1 when multiplying: (2i)(3i) = 6i^2 = -6, not 6i.
- Dividing by 0 + 0i is undefined; the calculator refuses it.
- Writing the argument in the wrong One of the four regions the x- and y-axes cut the coordinate plane into.. atan(b/a) alone cannot tell -1 - i from 1 + i; use the signs of a and b (atan2).
- Mixing up the conjugate (flip the sign of b) with the negative (flip both signs).
- The argument of 0 is undefined because the point is at the origin.
Worked examples
Add two complex numbers
(3 + 4i) + (1 - 2i) = 4 + 2i
Result (a + bi)
4 + 2i
Multiply with FOIL
(2 + 3i)(1 - i) = 5 + i
Result (a + bi)
5 + i
Divide using the conjugate
(3 + 4i) ÷ (1 - 2i) = -1 + 2i
Result (a + bi)
-1 + 2i
Power with De Moivre
(1 + i)^8 = 16 because |1 + i| = √2 and θ = π/4
Result (a + bi)
16
Negative exponent
(1 + i)^-2 = -i/2
Result (a + bi)
-(1/2)i
Fractions stay exact
(1/2 + 3/4i) - (1/4 - i) = 1/4 + (7/4)i
Result (a + bi)
1/4 + (7/4)i
Modulus and argument of a single number
-1 + i has modulus √2 and argument 3π/4
Result (a + bi)
-1 + i
Division by zero
Dividing by 0 is undefined
Result (a + bi)
—
Frequently asked questions
How do you divide complex numbers?+
Multiply the numerator and denominator by the conjugate of the denominator. The denominator becomes c^2 + d^2, a real number, so you can split the fraction into a real part and an imaginary part.
What is the modulus of a complex number?+
|a + bi| = sqrt(a^2 + b^2), the distance from the origin to the point (a, b). For 3 + 4i it is 5. The calculator keeps it exact, for example √13.
What is the argument, and why is it sometimes negative?+
The argument is the angle from the positive real axis to the number, measured counterclockwise. The principal value is in (-180°, 180°], so numbers below the real axis have a negative argument. -1 - i has argument -135° (or -3π/4).
How does De Moivre's theorem work?+
Write z in polar form r(cos θ + i sin θ). Then z^n = r^n (cos nθ + i sin nθ): raise the modulus to the power and multiply the angle by n. The calculator also multiplies out the rectangular form exactly so you can compare.
What is the difference between polar and exponential form?+
They carry the same r and θ. Polar form writes r(cos θ + i sin θ); exponential form uses Euler's formula to shorten that to r·e^(iθ).
Can I enter fractions or decimals?+
Yes. 1/2 + (3/4)i, 2.5 - 1.5i, and 4i - 3 all work. Decimals are converted to exact fractions so the answer has no rounding error.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, College Algebra 2e, Section 2.4 “Complex Numbers” — Adding, multiplying, and dividing complex numbers with the conjugate.
- OpenStax, Precalculus 2e, Section 8.5 “Polar Form of Complex Numbers” — Modulus, argument, polar form, and De Moivre's theorem.
Related calculators
Complex Root Calculator
Find all n square, cube, or nth roots of a complex number with De Moivre's formula. Exact forms such as √3 + i when the angle is special, decimals otherwise, every root's angle, and a plot of the roots on a circle in the complex plane.
Complex Conjugate Calculator
Find the conjugate of a complex number a + bi and see why z times its conjugate equals |z|^2. Also gives the modulus, the reciprocal 1/z, z + z̄, z - z̄, and the mirrored argument.
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with exact simplified radicals, reduced fractions, or complex roots, plus discriminant, vertex, factored and vertex forms, a graph, and every step.
Powers of i Calculator
Simplify i^n for any integer exponent (huge or negative) by reducing n mod 4 and reading the cycle i, -1, -i, 1, with every step shown; a second mode simplifies (c·i)^n for a rational coefficient c, giving exact answers like (2i)^5 = 32i.
Unit Circle Calculator
Pick a special angle, type any angle, or go backward from a trig value to the angles that produce it. Get exact unit-circle coordinates (cos θ, sin θ), radians, all six trig functions, both solutions in one turn, and the complete special-angle table from 0° to 360°.
Vector Magnitude and Direction Calculator
Find the magnitude (length) and direction angle of a 2D or 3D vector from its components or from two points. Exact radical magnitude, unit vector, direction cosines, and an arrow diagram.
Last updated: September 4, 2026