Complex Numbers

A complex number is an expression of the form $a+bi$, where $a$ and $b$ are real numbers, and $i^2=-1$. We add and multiply complex numbers as follows:

$$(a+bi) + (c+di) = (a+c) + (b+d)i$$

$$(a+bi) \cdot (c+di) = (ac-bd) + (ad+bc)i$$

The complex conjugate of a complex number is

$$\overline{a+bi} = a-bi.$$

Note that

$$(a+bi) (\overline{a+bi}) = a^2 + b^2$$

is a real number (has no complex part).

If $c+di \neq 0$, then

$$\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2 + d^2} = \frac{1}{c^2+d^2}((ac+bd) + (bc-ad)i).$$

Example 4.3.1   $(1-2i)(8-3i) = 2 - 19i$ and $1/(1+i) = (1-i) / 2 = 1/2 - (1/2) i$.

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 $f(x)$ factors as a product of linear factors $(x-\alpha)$, if we allow the $\alpha$'s in the factorization to be complex numbers. For example,
  $$f(x) = x^2+1 = (x-i)(x+i).$$
  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 $(x,y) \leftrightarrow x+iy$. Via this correspondence we obtain a way to add and multiply points in the plane.
• Similarly, points in polar coordinates correspond to complex numbers:
  $$(r,\theta) \leftrightarrow r (\cos(\theta) + i \sin(\theta)).$$

• Complex numbers provide a very nice way to remember and understand trig identities.