My Digital Garden

Real numbers

All the numbers from - \infin to \infin

Real number line

Graph of real numbers R\R

real number line drawio

Absolute value

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

For any xRx \in \R

x={x,if x isnonnegativex,if x is negative|x| = \begin{cases} \enspace\: x, &\text{if }\enspace x \enspace\ is \enspace non-negative \\ -x, &\text{if }\enspace x\enspace\ is\enspace\ negative \end{cases}

Basic algebra rules

With equalities

  • If a=ba=b then a+c=b+ca+c = b+c
  • If aa, bb, and cc are numbers, and c0c\neq0, and a=ba = b, then ca=cbc · a = c · b.

With inequalities

  • If a<ba < b, then a+c<b+ca + c < b + c.

  • but with multiplication:

    Suppose a<ba < b.
    If c>0c > 0, then ac<bca · c < b · c.
    If c<0c < 0, then ac>bca · c > b · c.

Intervals

Closed intervals

Include the endpoints

[2,3.1]={xR:2x3.1}[2, 3.1] = \lbrace x \in \R : 2 \le x \le 3.1 \rbrace

1[2,3.1]\:\enspace 1 \notin [2, 3.1]
2[2,3.1]\:\enspace 2 \in [2, 3.1]
2.7[2,3.1]2.7 \in [2, 3.1]
3.1[2,3.1]3.1 \in [2, 3.1]
3.2[2,3.1]3.2 \notin [2, 3.1]

Open intervals

Don't include the endpoints

(5,8)={xR:5<x<8}(5, 8) = \lbrace x \in \R : 5 \lt x \lt 8 \rbrace

5.5(5,8)\quad 5.5 \in (5,8)
5.001(5,8)5.001 \in (5,8)
  5(5,8)\quad\enspace\; 5 \notin (5,8)

Half-open intervals

Include one endpoint

(7.1,15]={xR:7.1<x15}(-7.1, 15] = \lbrace x \in \R : -7.1 \lt x \le 15 \rbrace

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

Referred in


Copyright

Unless otherwise noted all content on this site is (c) copyright Julian Elve 2020 onwards, released under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence.

Real numbers