Real Number System

The natural numbers, or counting numbers, are the positive integers:

1, 2, 3, 4, 5, . . . . The negative integers are -1, -2, -3, . . .. .

A number in the form a/b, where a and b are integers, b ­ 0, is a rational number. A real number that cannot be written as the quotient of two integers is called an irrational number, e.g., , , , Π, e, .

There is a one-to-one correspondence between the set of real numbers and the set of points on an infinite line (coordinate line). Order among Real Numbers; Inequalities

a > b means that a - b is a positive real number.
If a <>
If a <>
If a <> 0, then ac <>
If a <> bc.
If a <>
If 0 <>
If a <> 0, then 1/a > 1/b.
If a <>

Absolute Value For any real number x, |x| = if x ³ 0, -x if x <>

Properties

If |x| = a, where a > 0, then x = a or x = -a.
|x| = |-x|; -|x| ≤ x ≤ |x|; |xy| = |x| |y|.
If |x| <> 0.
||x| - |y|| ≤ |x + y| ≤ |x| + |y|.
= |x|.

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