\documentclass{article}
\usepackage{amsmath}
\author{William Labbett}

\title{title}



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\maketitle

\tableofcontents

% REMS
% Must leave interest and things to do for the reader (Is that Delusional of me to think I might not?}
% How to put greek letter pi in text
% explain Real







\newpage



\section{Prerequisite Terms for Understanding This Document} 

\begin{itemize}
   \item WHOLE NUMBER
   \item PROPER FRACTION 
   \item REAL NUMBER
   \item $\lbrack x \rbrack$
\end{itemize}

   \subsection{WHOLE NUMBER}
      A WHOLE NUMBER is a number like  0, 1, 2, 3, 10, 308, \ldots
      i.e the numbers used for counting.
   \end{WHOLE NUMBER}
   \subsection{REAL NUMBER}
      A PROPER FRACTION is a fraction whose numerator which musn't be zero 
      is less than the denominator. e.g $\frac{4}{11}$

   \end{REAL NUMBER}

\newpage
\section{Finding the Decimal Representation of Powers of Real Numbers}



When going about finding the decimal representation of the square
of a real number like $\pi$ it is helpful to express the number as
a sum of a WHOLE NUMBER and a real number from 0 to 1 i.e on the set (0,1).

Suppose we have a number X. If X = n + x where n is the whole
part of X and x is the number on the set (0,1) then 
\begin{equation*}
   X^2 = (n + x)^2 = n^2 + 2nx + x^2
\end{equation*}

To calculate $2nx$ to some specific number of decimal places is straightforward.

Calculating $x^2$ to a specified number of decimal places is more involved.
 
For an example, let X = 6.873416873640587613458072364......

In this case we have n = 6 and x = 0.873468736405876......

The meaning of 0.873468736405876 is \newline

\begin{equation*}

 
\frac{8}{10^1}+\frac{7}{10^2}+\frac{3}{10^3}+\frac{4}{10^4}+\frac{6}{10^5}

\end{equation*}











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