REAL NUMBERS, R
One of the most important properties of the real numbers is that points on a straight line that can represent them. As in Fig 3.1, we choose a point, called the origin, to represent 0 and another point, usually to the right, to represent 1.
Then there is a natural way to pair off the points on the line and the real numbers, that is, each point will represent a unique real number and each real number will be represented by a unique point. We refer to this line as the real line. Accordingly, we can use the words point and number interchangeably.
Those numbers to the right of 0, i.e. on the same side as 1, are called the positive numbers and those numbers to the left of 0 are called the negative numbers. The number 0 itself is neither positive nor negative
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.