MATH 4220 - Lec. 2

# Lecture 2

Algebra & Point Representation

## Complex numbers

We describe a set of numbers with form $$a+bi$$, where both $$a$$ and $$b$$ are real numbers, and $$i$$ is the square root of $$-1$$. These numbers are called the complex numbers, and the set of them is denoted $$\mathbb{C}$$. Traditionally, an arbitrary complex number is denoted with the letter $$z$$. In symbols: $\mathbb{C} = \{z=a+bi~|~i=\sqrt{-1};~a,b\in\mathbb{R}\}$

We can utilize this representation of a complex number in order to define multiplication: $(a+ib)(c+id) := ac-bd+i(ad+bc)$

Definition. The complex conjugate of $$z=c+id$$ is $$\bar{z}=c-id$$.

Definition. For a complex number $$z=a+bi$$ we call $$a$$ the real part, and $$b$$ the imaginary part. In symbols, the real part is denoted $$\Re[z]$$ and the imaginary part $$\Im[z]$$.

## Point representation of complex numbers

You should be familiar with typical graphing of points on a Cartesian plane. As an example, we take two real numbers $$(x,y)$$ and plot them:

Now for a complex number: $$z=2+3i$$. Notice $$z$$ is completely described by two real numbers, in this case $$(2,3)$$. Generally, think of a complex number as being represented by two reals $$(x,y)$$. We plot our complex number much the same: