# Construction of The Real Numbers

**DEFINITION1:** The set with at least two distinct elements and satisfying the following five axioms is called the set of real numbers and each element of is called a real number:

**I. AXIOMS OF ADDITION:**

The function defined as for each in satisfies the following properties:

I,

I,

I ( is called the additive identity element),

I ( is called the additive inverse of ).

A pair of satisfying the properties is called a commutative additive group (or called an Abelian group). According to this, is a commutative additive group.

**II. AXIOMS OF MULTIPLICATION:**

The function defined as for each in satisfies the following properties:

II,

II,

II ( is called the multiplicative identity element),

II ( is called the multiplicative inverse of ).

The multiplication of and is generally denoted by instead of .

**III. DISTRIBUTIVITY OF MULTIPLICATION OVER ADDITION:**

For each in , .

A trilogy of satisfying the axioms **I, II, III** is called a field. According to this, is a field.

**IV. ORDER AXIOMS:**

The relation "" is defined over . For each distinct in , one and only one of the propositions and is true. By the help of the relation "", the "at most" relation "" is defined as or . Furthermore, the relation "" satisfies the following properties:

IV and ,

IV,

IV and .

is a totally ordered set with the relation "".

**V. COMPLETENESS AXIOM:**

If it holds , , for any two nonempty subsets and of , then there exists a real number such as , , .

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 , the additive identity element is unique.

**2.** The additive inverse of each real number is unique. (The additive inverse of a real number is denoted by . By the help of this, the substruction over is defined by )

**3.** For each in , the unique solution of the equation is .

**4.** In , the multiplicative identity element is unique.

**5.** The multiplicative inverse of each nonzero real number is unique. (The multiplicative inverse of a nonzero real number is denoted by or . By the help of this, the fraction over is defined by . Where is nonzero)

**6.** For each in (), the unique solution of the equation is .

**7.** , .

**8.** .

**9.** , .

**10.** , .

**11.** , .

**12.** For any in ,

,

.

**13.** For any in ,

,

,

.

**14.** For any in ,

,

,

,

.

**15.** .

**16.** .

The set of the real numbers satisfying the inequality (or ) is called the positive real numbers and the set of the real numbers satisfying the inequality is called the negative real numbers. They are denoted by and , respectively.

**DEFINITION2:** Let be a nonempty subset of .

**(i)** If there exists a in such that , then is called an upper bound of .

**(ii)** If there exists an in such that , then is called an lower bound of .

**(iii)** If has an upper and a lower bound i.e., , then is called a bounded set.

**(iv)** If there exists an in such that , then is called the maximum element of and denoted by

or

**(v)** If there exists an in such that , then is called the minimum element of and denoted by

or

For example, the set has no minimum and maximum elements. However, the set has minimum and maximum elements. They are and , respectively.

When the set has an upper bound, the set

is nonempty. Similarly, when the set has a lower bound, the set

is nonempty.

**DEFINITION3:** Let be a nonempty subset of .

**(i)** if has an upper bound, then the minimum of the set is called the supremum of and denoted by .

**(ii)** if has a lower bound, then the maximum of the set is called the infimum of and denoted by .

According to this definition, it holds,

,

.

There may not be the minimum and the maximum of any nonempty subset of . 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.

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 and . Then,

**(i)**

(a) ,

(b) , .

**(ii)**

(a) ,

(b) , .

**DEFINITION4:** Let and be two real numbers and . The set is called an open interval with the endpoint and and denoted by (or ). Similarly, the set is called a closed interval with the endpoint and and denoted by . A left-open right-closed interval and a left-closed right-open interval are defined by the follows:

left-open right-closed interval

left-closed right open interval

For and in , the unbounded intervals , , , and are defined by the follows:

,

,

,

,

.

The set obtained by adding two elements and read as "negatif infinity" and "positive infinity" to is called the extended real number line or the extended real number system and denoted by . Henceforth, we assume that the extended real number system satisfies the following properties:

**1.** ,

**a)** ,

**b)** ,

**c)** ,

**d)** ,

**2.**

**a)** ,

**b)** ,

**3.** ,

**a)** ,

**b)** ,

**4.** ,

**a)** ,

**b)** ,

**5.**

**a)** ,

**b)** ,

**c)** ,

**6.** ,

**a)** ,

**b)** .

Let be a nonempty subset of . If has no a lower bound, then the infimum is defined as and if has no an upper bound, then the supremum is defined as . According to this, each nonempty subset of has both the infimum and the supremum.

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

**THEOREM5:**

**a)** , ,

**b)** ,

**c)** , and ,

**d)** , and ,

**e)** , ,

**f)** , ,

**g)** ,

**h)** ,

**i)** ,

**j)** .

For two real numbers , the number is called the distance between and and denoted by .

Let and be two real numbers and . The number is called the length, width, measure or size of the intervals , , and .