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
is a real number (has no complex part).
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.
This will provide an easier to use variant of
the ``partial fractions'' integration technique,
which we will see later.
- 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.