\magnification=\magstep1
\baselineskip=12pt
\parskip=5pt
\def\ds{\displaystyle}
\nopagenumbers
\leftline{Gottlieb}

\centerline{\bf Rules of Algebra}

\rightline{Addition}

\noindent
Let $a,\ b,\ c$ be any real numbers

\item{1)}$a+b=b+a$
\item{2)}$a+(b+c)=(a+b)+c$
\item{3)}$a+(-1)a=0$ (also written $a-a=0$)
\item{4)}$a+0=a$

\line{\hrulefill}

\rightline{Multiplication}

\item{5)}$ab=ba$
\item{6)}$a(bc)=(ab)c$
\item{7)}$aa^{-1}=1$ (also written $\ds{a\over a}=1$) [if $a\not= 0$]
\item{8)}$1\cdot a=a,\ 0\cdot a=0,\ (-a)(-b)=ab$
\item{9)}$\ds{{a\over b}+{c\over d}={ad+bc\over bd}=(ad+bc)(bd)^{-1}}$
\item{10)}$a(b+c)=ab+ac$

\line{\hrulefill}

\rightline{Exponents}

\item{11)}$a^n=a\cdot a\ldots a$ ($n$--times) where $n$ is a positive integer
\item{12)}$a^0=1$
\item{13)}$a^{-n}\colon ={1\over a^n}$ [$a\not= 0$]
\item{14)}$a^{n/m}\colon =\root{m/n}\of a$ (i.e.~$(a^{n/m})^m=a^n)$ [$a\geq 0$ if $m$ is even]
\item{15)}$a^{r+s}=a^r a^s$ [for $r,s$ any numbers and $a > 0$]
\item{16)}$(a^r)^s=a^{rs}$ [for $r,s$ any numbers and $a > 0$]
\item{17)}$(ab)^r=a^r b^r$ [for $r$ any number and $a > 0,\ b > 0$]
\item{18)}$1^r=1$
\item{19)}$0^r=0$ [if $r > 0$]

\noindent
Note that

\item{20)}$a-b\colon =a+(-1) b$

\item{21)}$\ds{a\over b}\colon =ab^{-1}$

\item{22)}$a^{m/n}\colon = \root{m/n}\of a$ where $(a^{m/n})^n=a^m$

\noindent
So $\sqrt{\ \ }$, subtraction, and $\ds{a\over a}$ are not necessary.

\bye
