Complex Number Calculator NZ

This calculator works with two complex numbers, returning their sum, their product, and the modulus and argument of the first, the essential operations for anyone studying or applying complex numbers. A complex number has a real part and an imaginary part, written a plus bi, where i is the square root of minus one. Far from being merely abstract, complex numbers are indispensable in electrical engineering, signal processing, control systems, quantum physics and advanced mathematics, where they neatly describe oscillations, waves and rotations. The basic operations have simple rules but are easy to slip on by hand. Adding combines the real parts and the imaginary parts separately. Multiplying uses the distributive rule, remembering that i squared equals minus one, which mixes the parts in a way that catches people out. The modulus is the distance of the number from the origin in the complex plane, found like a vector magnitude, and the argument is the angle it makes, which together give the polar form used so widely in engineering. You enter the real and imaginary parts of each number, and the calculator returns the sum, the product, the modulus of the first number, and its argument in degrees. The results update as you type, so you can explore how the parts interact, particularly the sign changes in multiplication. Use it to check homework, to verify an engineering calculation, or to convert between rectangular and polar thinking. The arithmetic is exact for your inputs, with the modulus and argument rounded for display.

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-5 + 10i
product (a+bi)(c+di)
Sum4 + 6i
Modulus of first5
Argument of first53.13°

For complex numbers a+bi and c+di. Multiplication uses i squared = -1. Modulus and argument are rounded for display.

How it works

The sum adds the real parts and the imaginary parts separately. The product uses (a+bi)(c+di) = (ac - bd) + (ad + bc)i, because i squared is minus one. The modulus of a+bi is the square root of a squared plus b squared, and the argument is the inverse tangent of b over a, in degrees.

Worked example

For 3 + 4i and 1 + 2i, the sum is 4 + 6i. The product is (3 times 1 minus 4 times 2) plus (3 times 2 plus 4 times 1)i, which is minus 5 plus 10i. The modulus of 3 + 4i is the square root of 9 plus 16, equals 5, and its argument is about 53.13 degrees.

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Frequently asked questions

How do you multiply complex numbers?

Use the distributive rule and remember that i squared is minus one. So (a+bi)(c+di) equals (ac minus bd) plus (ad plus bc)i. The real and imaginary parts mix, which is why doing it by hand is error-prone. This calculator does it instantly.

What are the modulus and argument?

The modulus is the distance of the complex number from the origin in the complex plane, the square root of the real part squared plus the imaginary part squared. The argument is the angle it makes with the positive real axis. Together they give the polar form.

Where are complex numbers used?

Throughout electrical engineering and signal processing, where they describe alternating current, impedance and waves, as well as in control systems, quantum physics and advanced mathematics. They make rotations and oscillations far easier to handle than real numbers alone.