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
7. ABSOLUTE VALUE
The absolute value of a real number x, denoted by ÷ x÷ is defined by the formula
|x|. = x if x > 0
-x if x < 0
that is, if x is positive or zero then |x| equals x, and if x is negative then
|x| equals – x. Consequently, the absolute value of any number is always nonnegative, i.e. |x| > 0 for every x € R.
Geometrically speaking, the absolute value of x is the distance between the point x on the real line and the origin, i.e. the point 0. Moreover, the distance between any two points, i.e. real numbers, a and b is |a - b| = |b - a|.
Example 2.1: |-2| = 2, |7| = 7. |-p| = p
Example 2.2: The statement |x| < 5 can be interpreted to mean that the distance between x and the origin is less than 5, i.e. x must lies between -5 and 5 on the real line. In other words,
|x| < 5 and -5 < x < 5
have identical meaning. Similarly,
|x| < 5 and -5 < x < 5
have identical meaning.