Complex Number Calculator

What this calculator does

Complex numbers stretch ordinary arithmetic to include the square root of -1, which opens the door to answers that plain numbers cannot give. This does the four basic operations on two complex numbers: add, subtract, multiply and divide.

Enter the two numbers, choose the operation, and it returns the result. It runs right here in the browser.

Using the calculator

1. Enter the real part and imaginary part of the first number.
2. Choose the operation: addition, subtraction, multiplication or division.
3. Enter the second number's parts and press Calculate.

It returns the answer as a complex number, its own real and imaginary parts. Reset clears everything.

What a complex number is

A complex number has two pieces stitched together: a real part, an ordinary number, and an imaginary part, a multiple of the special quantity i. It is written a + bi, where a is the real part and b is the imaginary part. The number 3 + 2i, for example, is "3 plus 2 lots of i".

Everything hinges on one rule that defines i: i squared equals -1. That is the thing ordinary numbers cannot do, since any real number squared is positive or zero. By allowing this one new quantity whose square is negative, you can suddenly take square roots of negative numbers and give answers to equations that otherwise have none. That single rule is what powers all four operations below.

Adding and subtracting

These are the easy ones. You simply combine the real parts together and the imaginary parts together, keeping the two kinds separate, just like collecting like terms. So (2 + 3i) + (1 + 4i) adds the reals, 2 + 1 = 3, and the imaginaries, 3 + 4 = 7, giving 3 + 7i. Subtraction works the same way, taking one set of parts from the other.

Multiplying, where i squared earns its keep

Multiplying is where the rule i squared equals -1 does its work. You multiply the two numbers out term by term, exactly as you would expand two ordinary brackets, and then simplify any i squared to -1.

Take (2 + 3i) times (1 + 4i). Expanding gives 2 + 8i + 3i + 12i squared. Combine the i terms to get 2 + 11i + 12i squared, and now replace i squared with -1: the last term becomes -12, which merges into the real part. The result is -10 + 11i. That flip from 12i squared to -12 is the whole character of complex multiplication.

Dividing, and the conjugate trick

Division needs one clever move, because you cannot tidily divide by something with an i in it. The trick uses the conjugate: the conjugate of a + bi is a − bi, the same number with the sign of its imaginary part flipped. The magic of it is that a complex number times its own conjugate is always a plain real number, since the i terms cancel.

So to divide, you multiply both the top and the bottom of the fraction by the conjugate of the bottom. That turns the denominator into an ordinary real number, and then dividing is straightforward. The tool does this for you, but the conjugate is the idea to remember: it is the key that unlocks division.

Why "useless" numbers turned out to matter

Complex numbers had a rocky start. When they first cropped up in the 1500s, in the middle of solving cubic equations, mathematicians distrusted them. Cardano, who ran into them first, called them "as subtle as they are useless" and a "mental torture". The name "imaginary" was originally a put-down, meant to suggest they were not proper numbers at all.

And yet those supposedly useless numbers are now indispensable. They are the natural language of anything that oscillates or rotates, which makes them essential in electrical engineering, where they describe alternating current and impedance, in signal processing, in control systems, and in quantum mechanics, where the whole theory is built on them. A curiosity that early mathematicians wanted to throw away turned out to be one of the most useful ideas in science.

Questions people ask

What is a complex number?

A number of the form a + bi, with a real part a and an imaginary part b, where i is the square root of -1.

What is i?

The imaginary unit, defined by the rule that i squared equals -1. It is the quantity that lets you take square roots of negative numbers.

How do you multiply complex numbers?

Expand the two numbers term by term as you would ordinary brackets, then replace any i squared with -1 and combine the real and imaginary parts.

How do you divide complex numbers?

Multiply the top and bottom by the conjugate of the denominator. This makes the denominator a real number, after which the division is simple.

What is a conjugate?

The conjugate of a + bi is a − bi, the same number with the imaginary part's sign flipped. A number times its conjugate is always real.

References

A note on where this comes from. Complex numbers extend the reals with the imaginary unit i, defined by i squared equal to -1. They first arose in the 1500s in the work of Cardano on cubic equations, were given a proper arithmetic by Rafael Bombelli in 1572, had the symbol i introduced by Euler, and were placed on firm foundations by Gauss. For further reading, see Complex number.

1. Rafael Bombelli, Algebra (1572), the first systematic set of rules for calculating with complex numbers.
2. Leonhard Euler, who introduced the symbol i for the square root of -1.
3. Carl Friedrich Gauss, who put complex numbers on a firm footing and proved that every polynomial has a complex root.