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## Miscellaneous Properties

Names for Rational Numbers: Every rational number can be written as a quotient of integers:

and without loss of generality we may take In general, a rational number has many different names, e.g. and are different names for the same rational number. If I say let '', I mean let denote the rational number which has '' as one of its names. You should think of each rational number as a specific point on the line of real numbers. Let be integers with Then
 (C.80)

If and are positive, then
 (C.81)

Equations C.80 and C.81 hold for arbitrary real numbers It will be assumed that if you are given two rational numbers, you can decide whether or not the first is less that the second. You also know that the sum, difference, and product of two integers is an integer, and the additive inverse of an integer is an integer.

Absolute value:    If is a real number, then the absolute value of , denoted by , is defined by

 (C.82)

For all real numbers and all positive numbers we have

 (C.83) (C.84)

For all real numbers with ,

 (C.85) (C.86) (C.87) (C.88)

Powers: If is a real number, and is a non-negative integer, then the power is defined by the rules
 (C.89) (C.90)

If is a non-zero number and is a negative integer, then is defined by

 (C.91)

If is a non-negative number and is a positive integer, then is defined by

 (C.92)

If is a non-negative number and is an arbitrary integer and is a positive integer, then is defined by

 (C.93)

If are integers such that and and , then

 (C.94)

Monotonicity of Powers: If is a positive rational number, and and are non-negative real numbers, then

 (C.95)

If is a negative rational number, and and are positive real numbers, then
 (C.96)

If is a positive real number greater than 1, and and are rational numbers, then
 (C.97)

If is a positive real number less than 1, and and are rational numbers, then
 (C.98)

Laws of exponents: Let and be real numbers, and let and be rational numbers. Then the following relations hold whenever all of the powers involved are defined:

 (C.99) (C.100) (C.101) (C.102)

Remarks on equality: If are names for mathematical objects, then we write to mean that and are different names for the same object. Thus

 (C.103)

and it is always the case that
 (C.104)

It also follows that
 (C.105)

and more generally,
 (C.106)

If then the name can be substituted for the name in any statement containing the name . For example, if are numbers and we know that
 (C.107)

then we can conclude that
 (C.108)

and that
 (C.109)

When giving a proof, one ordinarily goes from an equation such as C.107 to equations such as C.108 or C.109 without mentioning the reason, and the properties C.103-C.106 are usually used without mentioning them explicitly.

Next: Area formulas Up: C. Prerequisites Previous: Order Laws
Ray Mayer 2007-08-31