How the operations work
A complex number combines a real part and an imaginary part, written x + yi, where i is the square root of −1. Addition and subtraction simply combine the real parts together and the imaginary parts together.
Multiplication uses the distributive rule and the fact that i squared equals −1, which mixes the parts. Division multiplies the top and bottom by the conjugate of the denominator to clear the imaginary part from below.
Rectangular and polar forms
The result is shown in rectangular form (real + imaginary) and also summarized in polar terms so you can see its size and direction.
- Magnitude is the distance from the origin, found with the Pythagorean formula on the two parts.
- Argument is the angle from the positive real axis, reported here in degrees.
- Together, magnitude and argument locate the number uniquely on the complex plane.
Tips and caveats
Keep track of which number is A and which is B, since subtraction and division are not symmetric. Dividing by 0 + 0i is undefined and the calculator will stop you.
- A purely real number has an imaginary part of 0 and an argument of 0 degrees.
- A purely imaginary number sits on the vertical axis at 90 or −90 degrees.
- The argument wraps around at ±180 degrees, so a result near that boundary may flip sign.
Formula
(a+bi)(c+di) = (ac−bd) + (ad+bc)i; |z| = √(re²+im²); arg = atan2(im, re)Frequently asked questions
- What does the argument mean?
- It is the angle the number makes with the positive real axis, measured counter-clockwise. Here it is shown in degrees.