The Real numbers $ℝ$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 19 November 2011

The real numbers $ℝ$

The real numbers $ℝ$ is the set $$ℝ={ℝ}_{\le 0}\cup {ℝ}_{\ge 0}$$ where $${ℝ}_{\ge 0}=\{a_l{a}_{l-1}\dots a_1{a}_0.a_{-1}{a}_{-2}{a}_{-3}\dots |l\in {\mathbb{Z}}_{\ge 0},a_i\in \{0,1,\dots ,9\}\}$$ $${ℝ}_{\le 0}=\{-a_l{a}_{l-1}\dots a_1{a}_0.a_{-1}{a}_{-2}{a}_{-3}\dots |l\in {\mathbb{Z}}_{\ge 0},a_i\in \{0,1,\dots ,9\}\}$$ and $$a=b\text{if}a-b=0.00000\dots ,$$ where addition and multiplication are given by

1. the $m^{\mathrm{th}}$ decimal place of $a+b$ is the same as the $m^{\mathrm{th}}$ decimal place of $$a_{\le 2m}+b_{\le 2m}\text{in}\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q},\text{and}$$
2. the $m^{\mathrm{th}}$ decimal place of $ab$ is the same as the $m^{\mathrm{th}}$ decimal place of $$a_{\le 2m}{b}_{\le 2m}\text{in}\mathbb{Q}\times \mathbb{Q}\to \mathbb{Q}.$$

THESE OPERATIONS ARE REALLY MUCKED UP AND NEED TO BE FIXED.

Define a relation $\le$ on $ℝ$ by $$x\le y\text{if}y-x\in {ℝ}_{\ge 0}.$$

Define the absolute value on $ℝ$ $$\begin{array}{rrcl}\left|\phantom{T}\right|:& ℝ& ⟶& {ℝ}_{\ge 0}\\ & x& ⟼& \left|x\right|\end{array}\text{by}\left|x\right|=\{\begin{array}{cc}x,& \text{if}x>0,\\ 0,& \text{if}x=0,\\ -x,& \text{if}x<0.\end{array}$$

Define the distance on $ℝ$ $$d:ℝ\times ℝ⟶{ℝ}_{\ge 0}\text{by}d(x,y)=|y-x|.$$

Let $\epsilon \in {ℝ}_{>0}$. The $\epsilon$-ball at $x$ is $${ℬ}_{\epsilon }\left(x\right)=\{y\in ℝ\left|d\right(x,y)<\epsilon \}.$$

Let $E$ be a subset of $ℝ$. The set $E$ is open if

$E\text{is a union of}\epsilon \text{-balls.}$

(1.1)

1. The set $ℝ$ with the operations of addition, multiplication, the order $\le$ and open sets as in (1.1) is an ordered field and a topological field.
2. The set $\mathbb{Q}$ is a ordered topological subfield of $ℝ$.

Notes and References

Decimal expansions (real numbers) are introduced to a child to read and write numerical values. Long division (the Eucidean algorithm) is the algorithm that converts rational numbers to real numbers. The integers $\mathbb{Z}$ is the free group on one generator, the rationals $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$, and the real numbers $ℝ$ is a completion of $\mathbb{Q}$ (a decimal expansion is no different than a Cauchy sequence of rational numbers). Thus, all three number systems are universal objects (in the sense of category theory).

References

[BouTop] N. Bourbaki, General Topology, Chapter IV, Springer-Verlag, Berlin 1989. MR?????

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