REAL NUMBERS, R

6. INEQUALITIES

The concept of “order” is introduced in the real number system by the

Definition: The real number a is less than the real number b,

                  written a < b

If b – a is a positive number.

The following properties of the relation a < b can be proven. Let a, b and c be real numbers; then:

    P1: Either a < b, a = b or b < a.

     P2: If a < b and b < c, then a < c.

      P3: If a < b, then a + c < b + c

       P4: If a < b and c is positive, then ac < bc

P5: If a < b and c is negative, then bc < ac.

Geometrically, if a < b then the point a on the real line lies to the left of the point b.

We also denote a < b by b > a

Which reads “b is greater then a”. Furthermore, we write

a < b or b > a

if a < b or a = b, that is, if a is not greater than b.

Example 1.1: 2 < 5; -6 < -3 and 4 < 4; 5 > -8

Example 1.2: The notation x < 5 means that x is a real number which is less than 5; hence x lies to the left of 5 on the real line.

The notation 2 < x < 7; means 2 < x and also x < 7; hence x will lie between 2 and 7 on the real line.

Remark 3.1: Notice that the concept of order, i.e. the relation a < b, is defined in terms of the concept of positive numbers. The fundamental property of the positive numbers which is used to prove properties of the relation a < b is that the positive numbers are closed under the operations of addition and multiplication. Moreover, this fact is intimately connected with the fact that the natural numbers are also closed under the operations of addition and

multiplication.

Remark 3.2: The following statements are true when a, b, c are any real numbers:

           1.   a < a

            2. if a < b and b < a then a = b.

             3. if a < b and b < c then a < c.