This document may contain unfilled definitions or examples. We will go over these during the class and the filled version will be posted after the lecture.

Complex Numbers

Definition: Imaginary unit
The square root of ____ $\underline{\blue{-1}}$ is called $i$ .
$\displaystyle i =$ ____ $\displaystyle \text{ so that }$ ____ $= -1.$
The number $i$ is called the ________ imaginary unit.

With the introduction of the number $i$ , the equation \underline{\blue{ $x^{2} = - 1$ }} has two solutions: $x =$ ___________ $=$ ___________ $\underline{\blue{\pm i}}$ .

Definition: Complex Number
A complex number _________ $\underline{\blue{z}}$ is a number of the form:
$\displaystyle z =$ ________
where $a$ and $b$ are _____________ real numbers and $i =$ $\sqrt{-1}$ .

The number $a$ is called the real part of $z$ and the number $b$ is called the imaginary part of $z$ .

Definition: Square Root of a Negative Number
For any positive number $b$ ,
$\displaystyle \sqrt{-b} =$ $\sqrt{b} \sqrt{-1}$ $=$ $\sqrt{b}i$ .