Order Axiom

Definition of a Negative Real Number

• The real number \(a\) is negative if and only if \(-a\) is positive.

Definition of “Less Than” and “Greater Than”

• \(a < b\) if and only if \(b-a\) is positive.
• \(a > b\) if and only if \(a-b\) is positive.

Law of Trichotomy

For any real numbers \(a\) and \(b\), exactly one of the following is true :

• \(a = b\)
• \(a > b\)
• \(a < b\)

Theorem

** Let \(a\) and \(b\) be real numbers such that \(a < b\).

• If \(c\) is any real number, then \(a+c < b+c\).
• If \(c\) is a positive number, then \(ac < bc\).
• If \(c\) is a negative number, then \(ac > bc\).

Transitive Property

Let \(a,b,c\) be real numbers such that \(a <b\) and \(b<c\). Then \(a < c\).

Theorem

If \(a\) is a real number, then \(a^2 >0\).

Exercises 1.1