# Lecture 1: All Real Numbers & the Real Line

Real Number - Can be represented as a decimal

• Integers - ﻿$-3$﻿,﻿$-2$﻿, ﻿$-1$﻿, ﻿$0$﻿, ﻿$1$﻿...
• Non Negative Numbers - ﻿$0$﻿, ﻿$1$﻿, ﻿$2$﻿, ﻿$3$﻿...
• Natural Numbers - ﻿$1$﻿, ﻿$2$﻿, ﻿$3$﻿
• Rational Numbers - ﻿$\frac{p}{q}$﻿ where ﻿$p$﻿ & ﻿$q$﻿ are integers and ﻿$q \neq 0$﻿
• Irrational Numbers - Cannot be represented as a rational (ex. ﻿$\sqrt 2$﻿, ﻿$\pi$﻿, ﻿$\sqrt 5$﻿)

Interval - a set of all real numbers can be represented as a line on the number line

• [include]
• ﻿$\leqslant$﻿ ﻿$\geqslant$﻿
• (exclude)
• ﻿$<$﻿ ﻿$>$﻿

### Number Lines

Absolute Value - denoted by ﻿$|x|$﻿, defined by formula : ﻿$\left | x \right |=\left\{\begin{matrix} x, &x\geq 0 \\ -x, & x\leq 0 \end{matrix}\right.$﻿

• ﻿$|-a| = |a|$﻿
• ﻿$|ab| = |a||b|$﻿
• ﻿$|\frac{a}{b}| = \frac{|a|}{|b|}$﻿
• ﻿$|a + b| \leq |a| + |b|$﻿ triangle equality

### Solving Inequality

﻿$x^2 - 2x \geq 15$﻿

﻿$x^2 - 2x - 15 \geq 0$﻿ thus we want positive value of values of x ﻿$[+ \cdot + = + , - \cdot - = +]$﻿

﻿$(x + 3)(x - 5) \geq 0$﻿

Case 1 : Both positive

﻿$x + 3 \geq 0$﻿

﻿$x \geq -3$﻿

and

﻿$x - 5 \geq 0$﻿

﻿$x \geq 5$﻿

Case 2 : Both negative -

﻿$x + 3 \leq 0$﻿

﻿$x \leq -3$﻿

and

﻿$x - 5 \leq 0$﻿

﻿$x \leq 5$﻿

All together ﻿$\rightarrow$﻿ ﻿$(-\infty$﻿, ﻿$3]$﻿ ﻿$\cup$﻿ ﻿$[5$﻿, ﻿$\infty)$﻿