๐งฎ Complex Number Calculator: Add, Multiply and Divide a + bi
By Shihab Mia ยท Updated 2026-07-02
Enter two complex numbers and choose an operation to see the result, its modulus and its argument.
A complex number calculator adds, subtracts, multiplies and divides two numbers written as a + bi and returns the answer in a + bi form in a single step. For example, (3 + 2i) times (1 - 4i) = 11 - 10i, with modulus about 14.866 and argument about -42.27 degrees. Enter the real and imaginary parts of each number, pick an operation, and this complex number calculator instantly shows the result plus its modulus (size), argument (angle) and polar form. Division by 0 + 0i is handled gracefully, so you always get a clear message 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 i squared = -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, so a complex number calculator handles real numbers, pure imaginary numbers and everything in between with one set of rules.
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 i squared 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. This is called the complex plane, or Argand diagram. 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 (real squared + imaginary squared); and the argument, the angle that line makes with the positive real axis, found with the two-argument arctangent. This complex number calculator reports both, in degrees and radians, for every result.
The modulus and argument together give the polar form of a complex number, written as r(cos theta + i sin theta) or the shorter r cis theta, where r is the modulus and theta is the argument. Polar form is what makes multiplication and division feel simple on the complex plane: to multiply two numbers you multiply their moduli and add their arguments, and to divide you divide the moduli and subtract the arguments. That is why engineers reach for polar form when combining phasors, and why the calculator shows it alongside the rectangular a + bi answer.
Complex numbers are not just an abstract puzzle. Electrical engineers use them to describe alternating current, where the real part is resistance and the imaginary part is reactance, combining into impedance. Signal processing, control theory, quantum mechanics and computer graphics all lean on complex arithmetic because rotation and oscillation are naturally described by i. Whenever you need to rotate a point, model a wave or solve a quadratic with no real roots, a complex number calculator turns the bookkeeping into a one-click task so you can focus on the problem.
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.
- Finding the modulus and argument of a complex number quickly without a scientific calculator.
- Confirming the roots of a quadratic equation that has a negative discriminant.
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.
- Use the polar form output if you need modulus-argument notation instead of rectangular a + bi.
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 = sqrt(11^2 + (-10)^2) = sqrt(121 + 100) = sqrt(221) = about 14.866
- Argument = atan2(-10, 11) = about -0.7378 rad = about -42.27 degrees
Result: 11 - 10i, modulus about 14.866, argument about -42.27 degrees
Divide (5 + 3i) by (2 + i) using the conjugate.
- Denominator c^2 + d^2 = 2^2 + 1^2 = 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 = sqrt(2.6^2 + 0.2^2) = sqrt(6.8) = about 2.6077
- Argument = atan2(0.2, 2.6) = about 0.0768 rad = about 4.40 degrees
Result: 2.6 + 0.2i, modulus about 2.6077, argument about 4.40 degrees
Add (7 - 2i) and (3 + 5i).
- Real part = a + c = 7 + 3 = 10
- Imaginary part = b + d = -2 + 5 = 3
- Result = 10 + 3i
- Modulus = sqrt(10^2 + 3^2) = sqrt(109) = about 10.440
- Argument = atan2(3, 10) = about 0.2915 rad = about 16.70 degrees
Result: 10 + 3i, modulus about 10.440, argument about 16.70 degrees
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) / (c^2 + d^2) | (bc - ad) / (c^2 + d^2) |
Powers of the imaginary unit i (the pattern repeats every four)
| Power | Value |
|---|---|
| i^1 | i |
| i^2 | -1 |
| i^3 | -i |
| i^4 | 1 |
| i^5 | i (cycle repeats) |
Rectangular and polar form for common complex numbers
| Rectangular (a + bi) | Modulus |z| | Argument (degrees) |
|---|---|---|
| 1 + 0i | 1 | 0 |
| 0 + 1i | 1 | 90 |
| 1 + 1i | about 1.414 | 45 |
| -1 + 0i | 1 | 180 |
| 3 + 4i | 5 | about 53.13 |
Common mistakes to avoid
- Forgetting that i squared equals -1. When you multiply complex numbers, the product of the two imaginary parts gives bd times i squared, and i squared 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 c squared + d squared.
- 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.
- Confusing degrees and radians. The argument can be shown in degrees or radians, and they are not interchangeable. An angle of 90 degrees is about 1.5708 radians, so always confirm which unit your homework or software expects.
- Assuming multiplication distributes over the imaginary unit incorrectly. Treat i like a variable while multiplying out the brackets, then substitute i squared = -1 only at the end. Substituting too early or applying the rule to i by itself leads to sign errors.
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 i squared = -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 sqrt(real squared + imaginary squared).
- Argument
- The angle a complex number makes with the positive real axis, measured from the origin.
- Polar form
- A complex number written as r(cos theta + i sin theta), where r is the modulus and theta is the argument.
- Complex plane
- A two-dimensional plane, also called an Argand diagram, where a complex number is plotted using its real and imaginary parts.
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 i squared with -1. The result of (a+bi)(c+di) is (ac - bd) + (ad + bc)i. This complex number calculator applies that rule automatically.
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the divisor, which is the divisor with the sign of its imaginary part flipped. That makes the denominator the real number c squared + d squared, 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 part squared + imaginary part squared). For 11 - 10i it is sqrt(121 + 100) = sqrt(221), about 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.
How do you convert a complex number to polar form?
Find the modulus r = sqrt(real squared + imaginary squared) and the argument theta = atan2(imaginary, real), then write the number as r(cos theta + i sin theta). For 3 + 4i the polar form is 5(cos 53.13 degrees + i sin 53.13 degrees).
What is the conjugate of a complex number?
The conjugate of a + bi is a - bi, formed by flipping the sign of the imaginary part. Multiplying a number by its conjugate gives a real result, a squared + b squared, which is why conjugates are used for division.
Can a complex number be a real number?
Yes. Any real number is a complex number with a zero imaginary part, such as 5 = 5 + 0i. Likewise, a pure imaginary number like 4i is 0 + 4i, so real and imaginary numbers are both special cases of complex numbers.
How do you square a complex number?
Multiply the number by itself: (a + bi) squared = (a squared - b squared) + 2abi. For example, (3 + 2i) squared = (9 - 4) + 12i = 5 + 12i. Just enter the same number in both fields and choose multiply.