Real numbers

All the numbers from $-\text{infin}$ to $\text{infin}$

Real number line

Graph of real numbers $\text{R}$

Absolute value

Absolute value of number $x$, written $|x|$ is distance from $x$ to 0.

For any $x\in \text{R}$

$$|x|=\{\begin{array}{ll}x,& \text{if }xisnon-negative\\ -x,& \text{if }xisnegative\end{array}$$

Basic algebra rules

With equalities

• If $a=b$ then $a+c=b+c$
• If $a$, $b$, and $c$ are numbers, and $c\ne 0$, and $a=b$, then $c·a=c·b$.

With inequalities

• If $a<b$, then $a+c<b+c$.

• but with multiplication:

  Suppose $a<b$.
  If $c>0$, then $a·c<b·c$.
  If $c<0$, then $a·c>b·c$.

Intervals

Closed intervals

Include the endpoints

$[2,3.1]=\{x\in \text{R}:2\le x\le 3.1\}$

$1\notin [2,3.1]$
$2\in [2,3.1]$
$2.7\in [2,3.1]$
$3.1\in [2,3.1]$
$3.2\notin [2,3.1]$

Open intervals

Don't include the endpoints

$(5,8)=\{x\in \text{R}:5<x<8\}$

$5.5\in (5,8)$
$5.001\in (5,8)$
$5\notin (5,8)$

Half-open intervals

Include one endpoint

$(-7.1,15]=\{x\in \text{R}:-7.1<x\le 15\}$

Based on course material by Paul Bendich and Daniel Egger from Data Science Maths Skills