\chapter{Algebraic laws}

Maybe variables seem simple, especially after a bit of practice with them. Though this may be true, variables are still possibly the most powerful tool for the mathematician. 

Here we revisit many definitions from arithmetic, and take a look at how these mix with variables, providing the 'laws of algebra'.



This book deals with \emph{the field of real numbers}; it describes how the 4 operations interact with real numbers. You can think of the word 'field' as a 'playing field of numbers'!

\begin{definition}
A \emph{real number} is a number that is either rational or irrational. The set of real numbers is denoted as $\mathbb{R}$.
\end{definition}



Let's quickly review the behaviour of these 4 operations. Also recall the order in which operations are conducted:
1. Anything in brackets
2. Division
3. Multiplication
4. Subtraction
5. Addition

\section{Addition}
- Associative
\[1+(2+3)=(1+2)+3\]
- Commutative
\[7+5=5+7\]
\section{Subtraction}
- anticommutative
\[9-6=-(6-9)\]
\section{Multiplication}
- Associative
\[1 \cdot (2 \cdot 3)=(1 \cdot 2) \cdot 3\]
- Commutative
\[7\cdot 5=5\cdot 7\]
- Distributive over addition
\[5 \cdot (3+4)=5\cdot3+5\cdot4\]
- Distributive over subtraction
\[5 \cdot (3-4)=5\cdot3-5\cdot4\]
\section{Division}
- right-distributive over addition
\[\frac{(3+4)}{5}=\frac{3}{5}+\frac{4}{5}\]
- right-distributive over subtraction
\[\frac{(3-4)}{5}=\frac{3}{5}-\frac{4}{5}\]


\section{FOIL method}

It will be useful for our further study to take a closer look at the consequences of distributivity of multiplication



\section{Exponentiation}

In addition to the field of real numbers, one may wish for a set of rules for squaring, cubing, and other indices. These operations are examples of \emph{exponentiation}; repeated multiplication. These rules can be added as a 'side dish' for our field of real numbers.

- right-distributive over multiplication 
\[\frac{(3+4)}{5}=\frac{3}{5}+\frac{4}{5}\]
- right-distributive over division 
\[\frac{(3-4)}{5}=\frac{3}{5}-\frac{4}{5}\]