The word {\it algebra} has a wide meaning in mathematics. In
general, algebra deals with the topic of defining and using rules of
arithmetic on some set to investigate what the effects of the rules
are. In this treatise we will investigate the effects of the rules of
arithmetic for the real numbers which were under discussion earlier.

{\it An expression} or {\it formula}) is a collection of symbols,
each of which can be numbers or alphabetical characters, connected
together with the operators $+$, $-$, $\cdot$, $:$, fraction signs or
parentheses. The alphabetical characters are viewed as symbols to
denote real numbers These can be fixed numbers which have specific
values or be variables which can take on any value within a subset of
the real numbers.

As an example, consider
$$
\dfrac{(3x(y+2)^2-z)^3-2\frac 34(2x^2y+z)^4}
{(ax^2-2xyz)^2+1.53}
$$
This is a mixture of a variety of parentheses, power symbols, general
fractions and decimal fractions. There are no limits to the
complexity of this notation but rules must be observed on how the
operations are used and the parentheses, $($ \ \ \ $)$, must pair
up. When computing the value of an expression, four rules of {\it
  operator precedence} must be followed:
\begin{description}
\item{{\bf 1.}} {\it Anything inside a pair of parentheses must be
    evaluated without using anything outside the parentheses.}
\item{{\bf 2.}} {\it The addition (plus) $+$, and subtraction (minus)
    $-$, operators split the expressions into terms and each term must
    be evaluated completely before addition or subtraction is conducted.}
\item{{\bf 3.}} {\it The symbols for multiplicatioon, the dot $\cdot$
and cross $\times$, and for division, colon $:$, slash $/$ and
fraction symbol, split the above terms into components. They only
apply to the component which immediately follows}
\item{{\bf 4.}} {\it Operations are conducted from left to right when
    terms are evaluated.}
\end{description}


\textbf{Example 3.1.1 }\\
Simplify
\[
    \frac{\frac{14}{4} + 27.5 + (2^2 + 1)^2 - 12x(1 - \frac{1}{3} ) }{(1-x)\frac{1}{2} + 3 }
\]\\
\textbf{Solution:} We start by calculating in the parenthesis and taking together convenient components and get
\[
    \frac{\frac{14}{4} + 27.5 + (2^2 + 1)^2 - 12x(1 - \frac{1}{3} ) }{(1-x)\frac{1}{2} + 3 }
    =\frac{\frac{7}{2} + 27.5 + (4+1)^2 - 12x \frac{2}{3} }{(1-x)\frac{1}{2} + 3 }
    = \frac{ \frac{7 + 55}{2} + 5^2 - 8x ) }{\frac{1}{2} - \frac{x}{2} + 3 }.
\]
We then extend by 2 and draw together similar components and cancel out:
\[
    = \frac{62+ 50 -16x}{7-x} = \frac{112 - 16x}{7-x} = 16 \frac{7-x}{7-x} = 16.
\]

\textbf{Example 3.1.2 }\\
Simplify
\[
    \left(2 - \frac{5}{6} \right)^2 +1 \cdot \frac{5}{2}.
\]
\textbf{Solution:} We start by calculating in the parenthesis and the right component. The rest is self explanatory.
\[
    \left(2 - \frac{5}{6} \right)^2 +1 \cdot \frac{5}{2}
    = \left( \frac{7}{6} \right)^2 + \frac{5}{2}
    = \frac{49}{36} + \frac{5}{2}
    = \frac{49 + 90}{36} = \frac{139}{36}.
\]