Complex Number Calculator
Perform complex number arithmetic, including addition, subtraction, multiplication, division, and powers, with clear step-by-step solutions that show exactly how each answer is calculated.
Getting started
How to use the complex number calculator
Enter your expression directly in the input box and use i for the imaginary unit. For example, you can type (3 + 2i) * (1 - 4i). As you type, the calculator shows your expression in proper mathematical notation below the box, making it easy to check that everything looks right before you calculate the answer.
If you prefer clicking instead of typing, the built-in math keyboard makes it easy to enter operators, powers, roots, variables, and grouping symbols. When your expression is ready, click Evaluate to see the answer along with a clear step-by-step breakdown, from the parsed expression to the intermediate simplification and final rectangular form. You can also turn on Live Solve to see the result update as you make changes.
Once you’ve got your result, you can copy it as LaTeX or Markdown, export or print it, or share it with a link. This is useful for homework, notes, proofs, or documentation when you need to show the working as well as the final answer.
The basics
Understanding complex number operations
A complex number is written as a + bi, where a is the real part, b is the imaginary part, and i is defined by i² = -1. This simple rule is the key to working with complex numbers and is used throughout every operation below.
Addition and subtraction
These are the simplest operations. Add or subtract the real parts together, then do the same with the imaginary parts. It works much like combining like terms in regular algebra.
Multiplication
Multiply complex numbers the same way you would multiply two binomials using distribution or FOIL. The main difference is that whenever you get i², replace it with -1. This often turns a complicated expression into a much simpler result.
Division
Division can be a little trickier because you don’t want an i left in the denominator. The usual solution is to multiply both the numerator and denominator by the conjugate of the denominator. This changes the sign of the imaginary part and removes the imaginary term from the denominator, leaving you with a real number to divide by.
Powers
Working with powers of complex numbers, such as (1 + i)⁸, means multiplying the number by itself several times. While the process is straightforward, it’s easy to make a small sign or arithmetic mistake along the way. That’s why step-by-step working is helpful — you can see exactly how each part is simplified and spot mistakes before they affect the final answer.
See it in action
A few worked examples
To see the calculator in action, try dropping in expressions like these:
(3 + 2i)(1 - 4i)— a straightforward multiplication that resolves to11 - 10i(2 + 3i)/(1 - i)— a division that walks through multiplying by the conjugate(1 + i)(1 + i)⁸— a power expression that simplifies all the way down to a real number,16
Each of these is available as a one-tap example inside the tool, and your last few calculations are saved under Recent so you can revisit or tweak them without retyping the whole expression.
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Frequently asked questions
What is the imaginary unit i?
i is defined as the square root of -1, so i² = -1. It lets us work with the square roots of negative numbers, which have no solution among real numbers alone.
Why do I multiply by the conjugate when dividing complex numbers?
Multiplying by the conjugate turns the denominator into a real number (since (a+bi)(a-bi) = a² + b²), which removes i from the bottom of the fraction so the division can be completed normally.
Can this calculator handle powers of complex numbers?
Yes — enter an expression like (1 + i)^8 and the calculator expands it step by step down to its simplified rectangular form.
Does the calculator show its work?
Yes. Every result comes with a derivation panel showing the parsed expression and each simplification step, and you can copy the working as LaTeX or Markdown.
Is this complex number calculator free to use?
Yes, it’s completely free with no sign-up required.
Have an expression that’s giving you trouble?
Scroll back up and drop it into the calculator — the step-by-step breakdown will show you exactly where it simplifies.