$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 0 Introduction

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Mathematical objects come into existence by definitions. These definitions must give an absolutely clear picture of the object or concept. We don't need to prove them, simply to clearly define them.

We are going to state some basic facts that needed in this course:

Basic facts:

The collection of  counting numbers otherwise known as the collection of  natural numbers is usually denoted by $$\mathbb{N}.$$  We write $$\mathbb{N} = \{ 1,2,3,4, \dots\}.$$

The collection of the integers is usually denoted by $$\mathbb{Z}$$ and
we write $${\mathbb{Z}} = \{ \dots,-3,-2,-1,0,1,2,3,4, \dots\}.$$

The collection of all rational numbers (fractions) is usually denoted by $$\mathbb{Q}$$ and
We write $${\mathbb{Q}} = \left\{ \frac{a}{b}: a \mbox{ and }b \mbox{ are integers}, \, b \ne 0 \right\}.$$

The collection of all irrational numbers is denoted by $${\mathbb{Q^c}}$$.

The collection of all real numbers is denoted by $$\mathbb{R}$$. This set contains all of the rational numbers and all of the irrational numbers.

We shall assume the use of the usual addition, subtraction, multiplication, and division as operations and, inequalities ($$<,>,\leq,\geq)$$and equlity ($$=$$),  are relations on $$\mathbb{R}$$.

Recall that, if $$a$$ and $$b$$ are real numbers, or $$a, \, b \in \mathbb{R}$$ as written in mathematical language, then

1.  $$a<b$$ means that $$a$$ is less than $$b.$$
2. $$a>b$$ means that $$a$$ is greater than $$b.$$

Definitions

1. A real number is called positive if it is greater than $$0$$.
2. A real number is called non-negative if it is greater than or equal to $$0$$.
3. An integer $$n$$ is an even number if there is an integer $$m$$ such that $$n=2m$$.
4. An integer $$n$$ is an odd number if there is an integer $$m$$ such that $$n=2m+1$$.
5. An integer $$a$$ is said to be divisible by an integer $$b$$ if there is an integer $$m$$ such that $$a=bm$$. In this case, we can say that $$b$$ divides $$a$$ and denoted $$b|a$$. Further, $$b$$ is called a divisor (factor) of $$a$$.
6. A positive  integer $$p$$ is called prime if $$p>1$$ and the only positive divisors of $$p$$  are $$1$$ and $$p$$.
7. A positive integer $$n$$ is called composite if there is a positive integer $$m$$ such that $$1<m< n$$ and  $$m|n$$.

Note that $$1$$ is neither prime nor composite.