🧮 Complex Number Calculator
By ToolNimba Math Team · Updated 2026-06-19
Enter two complex numbers and choose an operation to see the result, its modulus and its argument.
This complex number calculator adds, subtracts, multiplies and divides two complex numbers written in the form a + bi. Enter the real and imaginary parts of each number, pick an operation, and you instantly get the result in a + bi form along with its modulus (size) and argument (angle). Division by zero is handled gracefully, so you always get a clear answer rather than a broken result.
What is the Complex Number Calculator?
A complex number has two parts: a real part and an imaginary part, written together as a + bi. The symbol i is the imaginary unit, defined by the single rule i² = -1. That one rule is what makes complex arithmetic different from ordinary arithmetic, because it lets you take the square root of a negative number, something the real numbers alone cannot do. Numbers like 3 + 2i, -1 - 4i and even plain 5 (which is 5 + 0i) are all complex numbers.
Addition and subtraction are the easy operations: you simply combine the real parts with the real parts and the imaginary parts with the imaginary parts, exactly like collecting like terms in algebra. Multiplication expands the brackets like any binomial product, but every time two imaginary parts meet you replace i² with -1, which is why (a+bi)(c+di) becomes (ac - bd) + (ad + bc)i. Division is the trickiest step: you multiply the top and bottom by the conjugate of the divisor (the same number with the sign of its imaginary part flipped), which turns the denominator into a real number you can divide by.
Every complex number can also be pictured as a point on a plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. From that picture come two more useful quantities: the modulus, the straight-line distance from the origin to the point, found with the Pythagorean theorem as the square root of (real² + imaginary²); and the argument, the angle that line makes with the positive real axis, found with the two-argument arctangent. The calculator reports both, in degrees and radians, for every result.
When to use it
- Checking homework on complex arithmetic in an algebra, precalculus or engineering course.
- Working out impedance and phasor calculations in AC electrical and electronics problems.
- Converting a result into modulus-argument (polar) form for signal processing or control systems.
- Verifying the conjugate-multiplication method when dividing one complex number by another.
How to use the Complex Number Calculator
- Enter the real part a and imaginary part b of the first number (use 0 for a part you do not need).
- Enter the real part c and imaginary part d of the second number.
- Choose the operation: add, subtract, multiply or divide.
- Read off the result in a + bi form, plus its modulus and argument in degrees and radians.
Formula & method
Worked examples
Multiply (3 + 2i) by (1 − 4i).
- Real part = (a·c) − (b·d) = (3·1) − (2·−4) = 3 + 8 = 11
- Imaginary part = (a·d) + (b·c) = (3·−4) + (2·1) = −12 + 2 = −10
- Result = 11 − 10i
- Modulus = √(11² + (−10)²) = √(121 + 100) = √221 ≈ 14.866
- Argument = atan2(−10, 11) ≈ −0.7378 rad ≈ −42.27°
Result: 11 − 10i, modulus ≈ 14.866, argument ≈ −42.27°
Divide (5 + 3i) by (2 + i) using the conjugate.
- Denominator c² + d² = 2² + 1² = 4 + 1 = 5
- Real part = (a·c + b·d) ÷ 5 = (5·2 + 3·1) ÷ 5 = 13 ÷ 5 = 2.6
- Imaginary part = (b·c − a·d) ÷ 5 = (3·2 − 5·1) ÷ 5 = 1 ÷ 5 = 0.2
- Result = 2.6 + 0.2i
- Modulus = √(2.6² + 0.2²) = √6.8 ≈ 2.6077
- Argument = atan2(0.2, 2.6) ≈ 0.0768 rad ≈ 4.40°
Result: 2.6 + 0.2i, modulus ≈ 2.6077, argument ≈ 4.40°
The four operations on complex numbers a + bi and c + di
| Operation | Real part of result | Imaginary part of result |
|---|---|---|
| Add | a + c | b + d |
| Subtract | a − c | b − d |
| Multiply | ac − bd | ad + bc |
| Divide | (ac + bd) ÷ (c² + d²) | (bc − ad) ÷ (c² + d²) |
Powers of the imaginary unit i (the pattern repeats every four)
| Power | Value |
|---|---|
| i¹ | i |
| i² | −1 |
| i³ | −i |
| i⁴ | 1 |
| i⁵ | i (cycle repeats) |
Common mistakes to avoid
- Forgetting that i² equals −1. When you multiply complex numbers, the product of the two imaginary parts gives bd·i², and i² is −1, not +1. Leaving it as +1 flips the sign of the real part and gives the wrong answer.
- Dividing without using the conjugate. You cannot divide by an imaginary part directly. Multiply the numerator and denominator by the conjugate of the divisor (flip the sign of its imaginary part) so the denominator becomes the real number c² + d².
- Mixing up modulus and argument. The modulus is the distance from the origin, found with the Pythagorean theorem. The argument is the angle, found with arctangent. They answer different questions, so do not swap them.
- Ignoring the quadrant when finding the argument. A plain arctangent of imaginary over real cannot tell which quadrant the number is in. The two-argument arctangent (atan2) uses the signs of both parts to return the correct angle, which is what this tool uses.
Glossary
- Complex number
- A number of the form a + bi, with a real part a and an imaginary part b.
- Imaginary unit (i)
- The number defined by i² = −1, used to build the imaginary part of a complex number.
- Real part
- The component a in a + bi, plotted on the horizontal axis of the complex plane.
- Imaginary part
- The coefficient b in a + bi, plotted on the vertical axis of the complex plane.
- Conjugate
- The complex number a − bi formed by flipping the sign of the imaginary part of a + bi, used to divide complex numbers.
- Modulus
- The size or absolute value of a complex number, equal to √(real² + imaginary²).
- Argument
- The angle a complex number makes with the positive real axis, measured from the origin.
Frequently asked questions
How do you add two complex numbers?
Add the real parts together and the imaginary parts together. For example, (7 − 2i) + (3 + 5i) = (7 + 3) + (−2 + 5)i = 10 + 3i. Subtraction works the same way but with subtraction of each part.
How do you multiply complex numbers?
Expand the brackets like any binomial product and replace i² with −1. The result of (a+bi)(c+di) is (ac − bd) + (ad + bc)i. This calculator applies that rule automatically.
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the divisor (the divisor with the sign of its imaginary part flipped). That makes the denominator the real number c² + d², so you can divide both parts by it.
What is the modulus of a complex number?
The modulus is the distance from the origin to the point on the complex plane, equal to the square root of (real part² + imaginary part²). For 11 − 10i it is √(121 + 100) = √221 ≈ 14.866.
What is the argument of a complex number?
The argument is the angle the number makes with the positive real axis. It is found with the two-argument arctangent of the imaginary and real parts, and this tool shows it in both degrees and radians.
What happens if I divide by zero?
Dividing by 0 + 0i is undefined because zero has no reciprocal. The calculator detects this and shows a clear message instead of an error, so you know the operation cannot be performed.