**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

If and are

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.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) |

(C.96) |

(C.97) |

(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.104) |

(C.105) |

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

then we can conclude that

and that

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.