Next: Miscellaneous Properties Up: C. Prerequisites Previous: Algebraic Laws

## Order Laws

There is a relation (less than) defined on the real numbers such that for each pair of real numbers, the statement '' is either true or false, and such that the following conditions are satisfied:

Trichotomy law: For each pair of real numbers exactly one of the following statements is true:

 (C.49)

We say that a real number is positive if and only if and we say that a real number is negative if and only if Thus as a special case of the trichotomy law we have:

If is a real number, then exactly one of the following statements is true:

 (C.50)

Sign laws: Let be real numbers. Then

 (C.51) (C.52) (C.53) (C.54) (C.55) (C.56)

Also,
 (C.57)

and
 (C.58)

 (C.59)

 (C.60)

 (C.61)

 (C.62)

It follows immediately from the sign laws that for all real
numbers
 (C.63)

Here, as usual means

Transitivity of : Let be real numbers. Then

 (C.64)

We write as an abbreviation for either or '', and we write to mean . We also nest inequalities in the following way:

means

Addition of Inequalities: Let be real numbers. Then

 (C.65) (C.66) (C.67) (C.68) (C.69) (C.70)

Multiplication of Inequalities: Let be real numbers.

 (C.71) (C.72) (C.73) (C.74) (C.75) (C.76)

Discreteness of Integers: If is an integer, then there are no integers between and , i.e. there are no integers satisfying A consequence of this is that

 (C.77)

If and are real numbers such that then there is an integer such that
 (C.78)

Archimedean Property: Let be an arbitrary real number. Then

 (C.79)

Next: Miscellaneous Properties Up: C. prerequisites Previous: Algebraic Laws
Ray Mayer 2007-08-31