Miscellaneous Properties

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

$$r = \frac{m}{n} \mbox{ where } m,n \mbox{ are integers and } n \not= 0,$$

and without loss of generality we may take $n>0.$ In general, a rational number has many different names, e.g. $\frac{2}{3} , \frac{-10}{-15} ,$ and $\frac{34}{51}$ are different names for the same rational number. If I say ``let $x = \frac{2}{3}$'', I mean let $x$ denote the rational number which has ``$\frac{2}{3}$'' as one of its names. You should think of each rational number as a specific point on the line of real numbers. Let $m,n,p,q$ be integers with $n \not= 0 \mbox{ and }\ q \not= 0.$ Then

$$\frac{m}{n} = \frac{p}{q}\index{equality in \mbox{{\bf Q}}} \mbox{ if and only if } mq = np.$$ (C.80)

If $n$ and $q$ are positive, then

$$\frac{m}{n} < \frac{p}{q} \mbox{ if and only if } mq < np.$$ (C.81)

Equations C.80 and C.81 hold for arbitrary real numbers $m,n,p,q.$ 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 $x$ is a real number, then the absolute value of $x$, denoted by $\vert x\vert$, is defined by

$$\vert x\vert =\cases{x&if $x>0$,\cr 0&if $x=0$,\cr -x&if $x<0$.\cr} \glossary{$\vert x\vert$}$$ (C.82)

For all real numbers $x$ and all positive numbers $a$ we have

$$(\vert x\vert<a) \mbox{if and only if} (-a<x<a),$$ (C.83)

$$(\vert x\vert \leq a) \mbox{if and only if} (-a \leq x\leq a).$$ (C.84)

For all real numbers $x,y,z$ with $z\neq 0$,

$$\vert x\vert = \vert-x\vert$$ (C.85)

$$-\vert x\vert \leq x\leq \vert x\vert$$ (C.86)

$$\vert xy\vert = \vert x\vert\cdot \vert y\vert$$ (C.87)

$$\left\vert {x \over z}\right\vert = {{\vert x\vert}\over {\vert z\vert}}.$$ (C.88)

Powers: If $a$ is a real number, and $n$ is a non-negative integer, then the power $a^n$ is defined by the rules

$$a^0 = 1\index{power laws}$$ (C.89)

$$a^{n+1} = a^n \cdot a \mbox{ for } n \geq 0.$$ (C.90)

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

$$a^n = (a^{-n})^{-1} = \frac{1}{a^{-n}}.$$ (C.91)

If $a$ is a non-negative number and $n$ is a positive integer, then $a^{\frac{1}{n}}$ is defined by

$$a^{\frac{1}{n}} \mbox{ is the unique non-negative number $b$\ such that $b^n=a$.}$$ (C.92)

If $a$ is a non-negative number and $m$ is an arbitrary integer and $n$ is a positive integer, then $a^\frac{m}{n}$ is defined by

$$a^{\frac{m}{n}} = \cases{ (a^{\frac{1}{n}})^m & if $a>0$. \... ...\ and $m > 0$\ \cr \mbox{undefined} & if $a=0$\ and $m < 0$.}$$ (C.93)

If $m,n,p,q$ are integers such that $n \not = 0$ and $q \not = 0$ and $\frac{m}{n} = \frac{p}{q}$, then

$$(a^{\frac{1}{n}})^m = (a^m)^{\frac{1}{n}} = (a^p)^{\frac{1}{q}} = (a^{\frac{1}{q}})^p.$$ (C.94)

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

$$x < y \mbox{ if and only if } x^r < y^r.$$ (C.95)

If $r$ is a negative rational number, and $x$ and $y$ are positive real numbers, then

$$x < y \mbox{ if and only if } x^r > y^r.$$ (C.96)

If $a$ is a positive real number greater than 1, and $p$ and $q$ are rational numbers, then

$$p < q \mbox{ if and only if } a^p < a^q.$$ (C.97)

If $a$ is a positive real number less than 1, and $p$ and $q$ are rational numbers, then

$$p < q \mbox{ if and only if } a^p > a^q.$$ (C.98)

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

$$a^ra^s = a^{r+s},$$ (C.99)

$$(a^r)^s = a^{(rs)},$$ (C.100)

$$(ab)^r = a^rb^r.$$ (C.101)

$$a^{-r} = \frac{1}{a^r}$$ (C.102)

Remarks on equality: If $x,y \mbox{ and }\ z$ are names for mathematical objects, then we write $x=y$ to mean that $x$ and $y$ are different names for the same object. Thus

$$\mbox{ if }\ x = y \mbox{ then }\ y = x,$$ (C.103)

and it is always the case that

$$x=x.$$ (C.104)

It also follows that

$$\mbox{ if }\ x=y \mbox{ and }\ y=z \mbox{ then }\ x = z,$$ (C.105)

and more generally,

$$\mbox{ if }\ x = y = z = t = w \mbox{ then }\ x = w.$$ (C.106)

If $x=y,$ then the name $x$ can be substituted for the name $y$ in any statement containing the name $x$. For example, if $x,y$ are numbers and we know that

$$x=y,$$ (C.107)

then we can conclude that

$$x+1 = y+1,$$ (C.108)

and that

$$x+x = x+y.$$ (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.