Construction of The Real Numbers

DEFINITION1: The set $\mathbb{R}$ with at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of $\mathbb{R}$ is called a real number:

The function $+:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined as $\left( x,y \right)\to x+y\in \mathbb{R}$ for each $\left( x,y \right)$ in $\mathbb{R}\times \mathbb{R}$ satisfies the following properties:

I${{}_{1}}.\,\forall a,b\in \mathbb{R},a+b=b+a$,

I${{}_{2}}.\,\forall a,b,c\in \mathbb{R},a+(b+c)=(a+b)+c$,

I${{}_{3}}.\,\exists 0\in \mathbb{R}:\forall a\in \mathbb{R},a+0=a$ ($0$ is called the additive identity element),

I${{}_{4}}.\,\forall a\in \mathbb{R},\exists b\in \mathbb{R}:a+b=0$ ($b$ is called the additive inverse of $a$).

A pair of $\left( X,+ \right)$ satisfying the properties ${{I}_{1}},{{I}_{2}},{{I}_{3}},{{I}_{4}}$ is called a commutative additive group (or called an Abelian group). According to this, $\mathbb{R}$ is a commutative additive group.

II. AXIOMS OF MULTIPLICATION:

The function $\cdot:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined as $\left( x,y \right)\to x\cdot{y}\in \mathbb{R}$ for each $\left( x,y \right)$ in $\mathbb{R}\times \mathbb{R}$ satisfies the following properties:

II${{}_{1}}.\,\forall a,b\in \mathbb{R},a\cdot b=b\cdot a$,

II${{}_{2}}.\,\forall a,b,c\in \mathbb{R},a\cdot (b\cdot c)=(a\cdot b)\cdot c$,

II${{}_{3}}.\,\exists 1\in \mathbb{R}:\forall a\in \mathbb{R},a\cdot 1=a$ ($1$ is called the multiplicative identity element),

II${{}_{4}}.\,\forall a\in \mathbb{R}\backslash \left\{ 0 \right\},\exists b\in \mathbb{R},a\cdot b=1$ ($b$ is called the multiplicative inverse of $a$).

The multiplication of $a$ and $b$ is generally denoted by $ab$ instead of $a\cdot{b}$.

III. DISTRIBUTIVITY OF MULTIPLICATION OVER ADDITION:

For each $a,b,c$ in $\mathbb{R}$, $(a+b)\cdot c=ac+bc$.

A trilogy of $(X,+,\cdot)$ satisfying the axioms I, II, III is called a field. According to this, $\mathbb{R}$ is a field.

IV. ORDER AXIOMS:

The relation "$<$" is defined over $\mathbb{R}$. For each distinct $a,b$ in $\mathbb{R}$, one and only one of the propositions $a and $b is true. By the help of the relation "$<$", the "at most" relation "$\le$" is defined as $a\le b$ $\Leftrightarrow$ $a or $a=b$. Furthermore, the relation "$<$" satisfies the following properties:

IV${{}_{1}}.\,a and $b,

IV${{}_{2}}.\,a,

IV${{}_{3}}.\,a and $0.

$\mathbb{R}$ is a totally ordered set with the relation "$\le$".

V. COMPLETENESS AXIOM:

If it holds $\forall a\in A$, $\forall b\in B$, $a\le b$ for any two nonempty subsets $A$ and $B$ of $\mathbb{R}$, then there exists a real number $c$ such as $\forall a\in A$, $\forall b\in B$ , $a\le c\le b$.

All the properties of the real numbers except for I, II, III, IV, V can be easily proved by using the axioms I, II, III, IV, V. Now, we give some of them as a theorem:

THEOREM1:

1. In $\mathbb{R}$, the additive identity element $0$ is unique.

2. The additive inverse of each real number is unique. (The additive inverse of a real number $a$ is denoted by $-a$. By the help of this, the substruction over $\mathbb{R}$ is defined by $a-b:= a+\left( -b \right)$)

3. For each $a,b$ in $\mathbb{R}$, the unique solution of the equation $x+a=b$ is $x=b+(-a)=b-a$.

4. In $\mathbb{R}$, the multiplicative identity element $1$ is unique.

5. The multiplicative inverse of each nonzero real number is unique. (The multiplicative inverse of a nonzero real number $a$ is denoted by $a^{-1}$ or $\displaystyle{\frac{1}{a}}$. By the help of this, the fraction over $\mathbb{R}$ is defined by $\displaystyle{\frac{b}{a}}=b\cdot\frac{1}{a}:=b{{a}^{-1}}$. Where $a$ is nonzero)

6. For each $a,b$ in $\mathbb{R}$ ($a\ne{0}$), the unique solution of the equation $ax=b$ is $\displaystyle{x=b\cdot {{a}^{-1}}=b\cdot \frac{1}{a}=\frac{b}{a}}$.

7. $\forall{a}\in \mathbb{R}$, $a\cdot 0=0$.

8. $a\cdot b=0\,\Rightarrow \,a=0\lor{b=0}$.

9. $\forall{a}\in \mathbb{R}$, $(-1)\cdot a=-a$.

10. $\forall{a}\in \mathbb{R}$, $(-1)\cdot (-a)=a$.

11. $\forall{a}\in \mathbb{R}$, $(-a)\cdot (-a)=a\cdot a={{a}^{2}}$.

12. For any $a,b$ in $\mathbb{R}$,

$a,

$a\le b\wedge b.

13. For any $a,b,c,d$ in $\mathbb{R}$,

$0,

$a\le b\wedge c\le d\Rightarrow a+c\le b+d$,

$a\le b\wedge c.

14. For any $a,b,c$ in $\mathbb{R}$,

$0,

$a<0\wedge 0,

$a<0\wedge b<0\Rightarrow 0,

$a.

15. $0.

16. $0.

PROOF:

The set of the real numbers satisfying the inequality $0 (or $a>0$) is called the positive real numbers and the set of the real numbers satisfying the inequality $a<0$ is called the negative real numbers. They are denoted by ${{\mathbb{R}}^{+}}=\left\{ x\in \mathbb{R}\:|\:x>0 \right\}$ and ${{\mathbb{R}}^{-}}=\left\{ x\in \mathbb{R}\:|\:x<0 \right\}$, respectively.

DEFINITION2: Let $X$ be a nonempty subset of $\mathbb{R}$.

(i) If there exists a $b$ in $\mathbb{R}$ such that $\forall{x}\in{X}, x\le{b}$, then $b$ is called an upper bound of $X$.

(ii) If there exists an $a$ in $\mathbb{R}$ such that $\forall{x}\in{X}, a\le{x}$, then $a$ is called an lower bound of $X$.

(iii) If $X$ has an upper and a lower bound i.e., $\exists{a,b}\in{\mathbb{R}}: a\le{x}\le{b}$, then $X$ is called a bounded set.

(iv) If there exists an $M$ in $X$ such that $\forall{x}\in{X}, x\le{M}$, then $M$ is called the maximum element of $X$ and denoted by

$M=\underset{x\in X}{\mathop{\max }}\,\left\{ x \right\}$ or $M=\max \left\{ x|x\in X \right\}$

(v) If there exists an $m$ in $X$ such that $\forall{x}\in{X}, m\le{x}$, then $m$ is called the minimum element of $X$ and denoted by

$m=\underset{x\in X}{\mathop{\min }}\,\left\{ x \right\}$ or $m=\min \left\{ x|\,x\in X \right\}$

For example, the set $X=\left\{ x\in \mathbb{R}|\,-1 has no minimum and maximum elements. However, the set $Y=\left\{ x\in \mathbb{R}|\,-1\le x\le 1 \right\}$ has minimum and maximum elements. They are $-1$ and $1$, respectively.

When the set $X\subset \mathbb{R}$ has an upper bound, the set

$B=\left\{ b\in \mathbb{R}|\,\forall{x}\in{X}, x\le{b} \right\}$

is nonempty. Similarly, when the set $X\subset \mathbb{R}$ has a lower bound, the set

$A=\left\{ a\in \mathbb{R}|\,\forall{x}\in{X}, a\le{x} \right\}$

is nonempty.

DEFINITION3: Let $X$ be a nonempty subset of $\mathbb{R}$.

(i) if $X$ has an upper bound, then the minimum of the set $B=\left\{ b\in \mathbb{R}|\,\forall{x}\in{X}, x\le{b} \right\}$ is called the supremum of $X$ and denoted by $\sup{X}$.

(ii) if $X$ has a lower bound, then the maximum of the set $A=\left\{ a\in \mathbb{R}|\,\forall{x}\in{X}, a\le{x} \right\}$ is called the infimum of $X$ and denoted by $\inf{X}$.

According to this definition, it holds,

$\sup{X}=\min \left\{ b\in \mathbb{R}|\,\forall{x}\in{X}, x\le{b} \right\}$,

$\inf{X}=\max \left\{ a\in \mathbb{R}|\,\forall{x}\in{X}, a\le{x} \right\}$.

There may not be the minimum and the maximum of any nonempty subset of $\mathbb{R}$. Is the property valid for the infimum and supremum? The following theorem is the answer of this question:

THEOREM2 (Least Upper Bound Property): Any subset of Real Numbers with an upper bound has unique supremum.

PROOF:

Similarly, the following theorem can be proved:

THEOREM3 (Greatest Lower Bound Property): Any subset of Real Numbers with an lower bound has unique infimum.

THEOREM4: Let $\varnothing\ne{X}\subset{\mathbb{R}}$ and $l,L\in{\mathbb{R}}$. Then,

(i) $\sup{X}=L$ $\iff$

(a) $\forall{x}\in{A}, x\le{L}$,

(b) $\forall{\varepsilon}>0$, $\exists{x_{\varepsilon}}\in{X}:$ $L-\varepsilon.

(ii) $\inf{X}=l$ $\iff$

(a) $\forall{x}\in{X}, l\le{x}$,

(b) $\forall{\varepsilon}>0$, $\exists{x_{\varepsilon}}\in{X}:$ $x_{\varepsilon}.

PROOF:

DEFINITION4: Let $a$ and $b$ be two real numbers and $a. The set $\left\{ x\in \mathbb{R}\,|\,a is called an open interval with the endpoint $a$ and $b$ and denoted by $\left( a,b \right)$ (or $\left] a,b \right[$). Similarly, the set $\left\{ x\in \mathbb{R}\,|\,a\le x\le b \right\}$ is called a closed interval with the endpoint $a$ and $b$ and denoted by $\left[ a,b \right]$. A left-open right-closed interval and a left-closed right-open interval are defined by the follows:

$\left( a,b \right]=\left] a,b \right]=\left\{ x\in \mathbb{R}\,|\,a left-open right-closed interval

$\left[ a,b \right)=\left[ a,b \right[=\left\{ x\in \mathbb{R}\,|\,a\le x left-closed right open interval

For $a$ and $b$ in $\mathbb{R}$, the unbounded intervals $(a,+\infty)$, $[a,+\infty)$, $(-\infty,b)$, $(-\infty,b]$ and $(-\infty,+\infty)$ are defined by the follows:

$(a,+\infty)=\left\{ x\in \mathbb{R}\,|\,x>a \right\}$,

$[a,+\infty)=\left\{ x\in \mathbb{R}\,|\,x\ge a \right\}$,

$(-\infty,b)=\left\{ x\in \mathbb{R}\,|\,x,

$(-\infty,b]=\left\{ x\in \mathbb{R}\,|\,x\le b \right\}$,

$(-\infty,+\infty)=\mathbb{R}$.

The set obtained by adding two elements $-\infty$ and $+\infty$ read as "negatif infinity" and "positive infinity" to $\mathbb{R}$ is called the extended real number line or the extended real number system and denoted by $\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}$. Henceforth, we assume that the extended real number system satisfies the following properties:

1. $\forall x\in \mathbb{R}$,

a) $-\infty ,

b) $x-(+\infty )=x-\infty =-\infty$,

c) $x+(+\infty )=x+\infty =+\infty$,

d) $x-(-\infty )=x+\infty =+\infty$,

2.

a) $+\infty +(+\infty )=+\infty$,

b) $-\infty +(-\infty )=-\infty$,

3. $\forall x\in {{\mathbb{R}}_{+}}$,

a) $x(+\infty )=+\infty$,

b) $x(-\infty )=-\infty$,

4. $\forall x\in {{\mathbb{R}}_{-}}$,

a) $x(+\infty )=-\infty$,

b) $x(-\infty )=+\infty$,

5.

a) $(+\infty )(+\infty )=+\infty$,

b) $(-\infty )(-\infty )=+\infty$,

c) $(+\infty )(-\infty )=-\infty$,

6. $\forall x\in \mathbb{R}$,

a) $\displaystyle{\frac{x}{+\infty }=0}$,

b) $\displaystyle{\frac{x}{-\infty }=0}$.

Let $X$ be a nonempty subset of $\overline{\mathbb{R}}$. If $X$ has no a lower bound, then the infimum is defined as$\inf X=-\infty$ and if $X$ has no an upper bound, then the supremum is defined as $\sup X=+\infty$. According to this, each nonempty subset of $\overline{\mathbb{R}}$ has both the infimum and the supremum.

DEFINITION5: The absolute value or modulus of a real number $x$ is defined as follows:

$|x|=\left\{ \begin{array}{rl} x, & \text{if } x\ge{0}\\ -x, & \text{if }x<0 \end{array} \right.$

THEOREM5:

a) $\forall{x}\in{\mathbb{R}}$, $|-x|=|x|\ge{0}$,

b) $|x|=0\Leftrightarrow{x=0}$,

c) $\forall{x}\in{\mathbb{R}}$, $-x\le{|x|}$ and $x\le{|x|}$,

d) $\forall{a,b}\in{\mathbb{R}}$, $|ab|=|a||b|$ and $\displaystyle{\left| \frac{a}{b} \right| =\frac{|a|}{|b|}}$ $(b\ne{0})$,

e) $\forall{a,b}\in{\mathbb{R}}$, $|a\pm{b}|\le{|a|+|b|}$,

f) $\forall{a,b}\in{\mathbb{R}}$, $\big| |a|-|b| \big| \le{|a-b|}$,

g) $|x|,

h) $|x|\le{r}\Leftrightarrow{-r\le{x}\le{r}}$,

i) $|x|>r\Leftrightarrow{x<-r\lor{x>r}}$,

j) $|x|\ge{r}\Leftrightarrow{x\le{-r}\lor{x\ge{r}}}$.

PROOF:

For two real numbers $a,b$, the number $\left| a-b \right|=\left| b-a \right|$ is called the distance between $a$ and $b$ and denoted by $d(a,b)$.

Let $a$ and $b$ be two real numbers and $a. The number $b-a>0$ is called the length, width, measure or size of the intervals $(a,b)$, $[a,b)$, $(a,b]$ and $[a,b]$.