What this calculator does
Work out sums, products, powers and roots of complex numbers, with every step shown. A complex number has two parts, written a + bi. a is the real part, and b is the imaginary part. i is a special number whose square is −1.
You can picture each complex number as a point on a flat grid. The real part goes across, and the imaginary part goes up. You can describe that point by its across-and-up position (rectangular form). Or you can give its distance from the center, 0, and its direction (polar form).
Choose an operation. Enter the first number as a + bi, or as a size (modulus) and an angle. The calculator gives the answer as a + bi. It also gives the size, the angle (called the argument), the polar form and the conjugate, the mirror-image number. The steps show how the real and imaginary parts combine, so you can follow or check each line.
Where it's used
Why this math matters outside the classroom:
- AC circuits. Electrical engineers combine a 30 Ω resistance and a 40 Ω reactance as 30 + 40i, an impedance of size 50 Ω.
- Signal phase. Audio engineers write a wave's amplitude and phase as one complex number, such as 2(cos 45° + i sin 45°) ≈ 1.414 + 1.414i.
- Rotating points. Multiplying 3 + 2i by i gives −2 + 3i, the same point turned 90° counterclockwise, a shortcut used in computer graphics.
- Dividing by conjugates. Students divide 4 + 2i by 1 − i by multiplying top and bottom by 1 + i, which gives (2 + 6i) ÷ 2 = 1 + 3i.
- Fractal images. Fractal programs repeat z² + c thousands of times from z = 0 for each pixel's complex number c to decide its color.
Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSN.CN.A.2, HSN.CN.A.3, HSN.CN.B.4. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Pick the operation: convert, add, subtract, multiply, divide, raise to a whole-number power, or find all k-th roots.
- Choose how to enter the first number, z₁. In rectangular form, type the real part a and the imaginary part b. In polar form, type the size r and the angle θ (theta), in degrees or radians.
- For adding, subtracting, multiplying or dividing, enter the second number z₂ as c + di. For a power, enter n. For roots, enter k.
- Read the result at the top, written a + bi. Below it are its size, angle, polar form and conjugate, or the full list of roots.
What the inputs mean
- z₁ imaginary part (b)
- The number multiplied by i. For 3 − 4i, enter 3 for a and −4 for b. For a number with no real part, such as 5i, a is 0.
- z₁ angle (θ)
- Measured counterclockwise, starting from the right-hand side of the across (real) line. Negative angles go clockwise; 30° and 390° describe the same number.
- Exponent (n)
- Any whole number from −20 to 20. A negative power means one divided by the number: z⁻¹ = 1 ÷ z.
- Root index (k)
- How many roots to find. Every non-zero number has exactly k different k-th roots.
Formula and method
(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) r = √(a² + b²), θ = atan2(b, a), a = r cos θ, b = r sin θ zⁿ = rⁿ(cos nθ + i sin nθ) k-th roots: r^(1/k) · [cos((θ + 360°j) ÷ k) + i sin((θ + 360°j) ÷ k)], j = 0 … k − 1
In the formulas, the first number is a + bi and the second is c + di. r is a number's size (its distance from 0), and θ is its angle. atan2 is a form of the inverse tangent that finds the angle from b and a. n is a power, k is how many roots you want, and j counts the roots from 0.
Adding and subtracting work part by part. Multiplying opens the brackets, as with any two-part product. Then it swaps i² for −1. Dividing multiplies the top and bottom by the conjugate of the bottom number, c − di. That's because a number times its conjugate is the plain real number c² + d².
In polar form, multiplying multiplies the sizes and adds the angles. That's why powers and roots are easiest there. De Moivre's theorem raises the size to the power and multiplies the angle. Roots divide the angle by k. Adding whole turns of 360° before dividing gives the other k − 1 roots, spaced evenly around a circle. Whole-number powers use exact repeated multiplication, so results such as i³ = −i have no rounding errors.
Worked example
Multiply z₁ = 3 + 4i by z₂ = 1 − 2i.
- Real part: ac − bd = 3 × 1 − 4 × (−2) = 3 + 8 = 11.
- Imaginary part: ad + bc = 3 × (−2) + 4 × 1 = −6 + 4 = −2.
- So the product is 11 − 2i.
- Its modulus is √(11² + (−2)²) = √125 ≈ 11.18034, and its argument is atan2(−2, 11) ≈ −10.304846°.
Result: (3 + 4i)(1 − 2i) = 11 − 2i, about 11.18034∠−10.304846° in polar form.
Understanding the result
The modulus is the size of the number: its distance from 0 on the grid. The argument is its direction, given here between −180° and 180°. Adding or subtracting any multiple of 360° describes the same number.
The conjugate is the mirror image across the across (real) line. A number and its conjugate have the same size. Their sum and product are both real. That's why conjugates show up in division, and why the complex roots of a polynomial come in pairs.
Assumptions and limitations
- Angles are in degrees unless you choose radians for the polar input. Results list the argument in both.
- Results are worked out with standard computer math (double precision) and rounded for display. A part smaller than a trillionth of the other part is shown as 0. So tiny rounding leftovers don't appear as, for example, 2 + 0.0000000000000001i.
- Only whole-number powers are supported. For fractional powers, use the roots option.
- Logarithms, exponentials and trig functions (such as sine) of complex numbers aren't included.
- The second number in arithmetic must be entered in rectangular form.
Frequently asked questions
What is the difference between rectangular and polar form?
Rectangular form, a + bi, gives how far across and how far up the point is. Polar form, r∠θ, gives its distance from 0 and its angle. They describe the same number. Rectangular is handier for adding; polar is handier for multiplying, powers and roots.
Why does dividing use the conjugate?
Multiplying the bottom number c + di by c − di gives c² + d², a plain real number. Doing the same to the top keeps the fraction's value the same. Then each part can be divided by c² + d².
How many square roots does a complex number have?
Every complex number except 0 has exactly two square roots, and they're negatives of each other. It has three cube roots. In general, it has k different k-th roots, spread evenly around a circle. The main (principal) root is the one whose angle is the original angle divided by k. The calculator lists it first.
What are the powers of i?
They repeat every four steps: i¹ = i, i² = −1, i³ = −i and i⁴ = 1. Then the cycle starts again. Enter a = 0, b = 1 and choose a power to see any of them.
How this calculator is checked
Method: Standard complex arithmetic; polar conversion with atan2; De Moivre's theorem for powers and roots. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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