# What is my complex number answer?

Adds, subtracts, multiplies and divides two complex numbers a + bi exactly, and gives the result’s modulus, argument, conjugate and polar form.

- Page: https://www.acalculator.org/math/complex-number-calculator
- JSON spec: https://www.acalculator.org/math/complex-number-calculator.json
- Version: ec7a701226a7

## Default answer

Example with the default inputs (Real part of z₁ (a) 4, Imaginary part of z₁ (b) 3, Operation z₁ × z₂, Real part of z₂ (c) 2, Imaginary part of z₂ (d) -5): The result is 23 − 14i.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Real part of z₁ (a) | The real part a of the first number z₁ = a + bi. |
| b | Imaginary part of z₁ (b) | The imaginary part b of the first number z₁ = a + bi. |
| op | Operation | What to do with z₁ and z₂. |
| c | Real part of z₂ (c) | The real part c of the second number z₂ = c + di. |
| d | Imaginary part of z₂ (d) | The imaginary part d of the second number z₂ = c + di. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Result | The answer in the form x + yi, to 10 significant figures. |
| exact | Exact result | The answer with exact decimals, or fractions in lowest terms where a decimal does not end. |
| re | Real part x | The real part of the result. |
| im | Imaginary part y | The imaginary part of the result (the number in front of i). |
| modulus | Modulus \|z\| | The distance of the result from 0: √(x² + y²). |
| argument | Argument θ (degrees) | The angle of the result from the positive real axis, from −180° to 180°. |
| radians | Argument θ (radians) | The same angle in radians, from −π to π. |
| polar | Polar form | r(cos θ + i sin θ), with r the modulus and θ the argument in degrees, to 6 significant figures. |
| conjugate | Conjugate | The result with the sign of its imaginary part flipped: x − yi. |

## 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).

## Assumptions

- Each part is read as the exact decimal you typed, so +, −, × and ÷ have no rounding error.
- The argument is the principal value, from −180° to 180°.
- Each part is from −10¹² to 10¹².

## Worked examples

1. a = 4, b = 3, op = mul, c = 2, d = -5 gives result = 23 − 14i, re = 23, im = -14, conjugate = 23 + 14i, modulus = 26.925824. Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02): (4 + 3i)(2 − 5i) = 23 − 14i.
2. a = 2, b = 5, op = div, c = 4, d = -1 gives exact = 3/17 + 22/17i, result = 0.1764705882 + 1.294117647i, re = 0.176471, im = 1.294118. Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02): (2 + 5i) ÷ (4 − i) = 3/17 + 22i/17.
3. a = 3, b = -4, op = add, c = 2, d = 5 gives result = 5 + i, re = 5, im = 1. Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02): (3 − 4i) + (2 + 5i) = 5 + i.
4. a = 0.1, b = 0.2, op = sub, c = 0.3, d = 0.2 gives result = −0.2, exact = −0.2, re = -0.2, im = 0, argument = 180. Source: OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers, retrieved 2026-10-02).
5. a = 1, b = 1, op = mul, c = 1, d = 0 gives modulus = 1.414214, argument = 45, polar = 1.41421(cos 45° + i sin 45°). Source: OpenStax, Algebra and Trigonometry 2e, §10.5 Polar Form of Complex Numbers (https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-5-polar-form-of-complex-numbers, retrieved 2026-10-02).

## FAQ

### How do I multiply complex numbers?

Multiply out the brackets and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (4 + 3i)(2 − 5i): 8 − 20i + 6i − 15i² = 8 + 15 − 14i = 23 − 14i.

### How do I divide complex numbers?

Multiply the top and the bottom by the conjugate of the bottom, which makes the bottom a real number. (2 + 5i) ÷ (4 − i) = (2 + 5i)(4 + i) ÷ (16 + 1) = (3 + 22i) ÷ 17 = 3/17 + 22/17 i.

### How do I add or subtract complex numbers?

Add or subtract the real parts and the imaginary parts separately: (3 − 4i) + (2 + 5i) = (3 + 2) + (−4 + 5)i = 5 + i.

### What is the modulus of a complex number?

Its distance from 0 in the complex plane: |x + yi| = √(x² + y²). For 23 − 14i it is √725 ≈ 26.9258.

### What is the argument?

The angle θ from the positive real axis to the point (x, y), measured counterclockwise. The calculator gives the principal value, from −180° to 180°. For 1 + i it is 45°.

### What is the conjugate?

The same number with the sign of the imaginary part flipped: the conjugate of x + yi is x − yi. A number times its conjugate is the real number x² + y².

### Why can I not divide by 0 + 0i?

Division by z₂ means multiplying by 1 ÷ z₂ = (c − di) ÷ (c² + d²), and c² + d² is 0 only when z₂ is 0. Like dividing by the real number 0, there is no answer.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §2.4 Complex Numbers (adding, subtracting, multiplying and dividing; (4 + 3i)(2 − 5i) = 23 − 14i; (2 + 5i) ÷ (4 − i) = 3/17 + 22i/17), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-4-complex-numbers
- OpenStax, Algebra and Trigonometry 2e, §10.5 Polar Form of Complex Numbers (|z| = √(x² + y²), z = r(cos θ + i sin θ)), CC BY 4.0, retrieved 2026-10-02. https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-5-polar-form-of-complex-numbers