A complex number is an expression of the form , where and are real numbers, and . We add and multiply complex numbers as follows:

The complex conjugate of a complex number is

If , then

Complex numbers are incredibly useful in providing better ways to understand ideas in calculus, and more generally in many applications (e.g., electrical engineering, quantum mechanics, fractals, etc.). For example,

- Every polynomial
**factors**as a product of linear factors , if we allow the 's in the factorization to be complex numbers. For example, - Complex numbers are in
**correspondence**with points in the plane via . Via this correspondence we obtain a way to add and*multiply*points in the plane. - Similarly, points in
**polar coordinates**correspond to complex numbers: - Complex numbers provide a very nice way
to remember and
**understand trig identities**.