# Arithmetic
---
1. natural numbers
- addition
- subtraction
- multiplication
2. integers
- addition
- subtraction
- multiplication
3. rational numbers
- fraction
- cake-pie analogy
- addition
- subtraction
- multiplication
- division
- remainder
- relation of division to fractions
- FOIL method
- Order of operations
4. real numbers
- exponentiation
- radicals
- logarithms

\part{Natural numbers}

Mathematics is well known for dealing with numbers, so let's discuss them!

\chapter{Natural numbers}

The natual numbers are used to count 'how many' of something there is. How many fingers do I have? 10. How many arms? 2. How many teeth? 32.

\begin{definition}
The natural numbers are the whole numbers starting from 0. The set of natural numbers is denoted as $\mathbb{N}$.
\[ \mathbb{N} = \{0,1,2,3,4,5,...\} \]
\[ \mathbb{N} = \{ n+1 :n \in \mathbb{N} \} \cup \{ 0 \} \]
\end{definition}

One important thing to note is that each natural number has a \emph{sucessor}; a natural number directly after that number. For example, 4 is the next number after 3.

\chapter{Addition}

Pretend you are having a picnic with 3 friends. There are 4 people at the picnic; you (1 person) and your friends (3 people). To calculate 4, you realize that on top of the three people, you have yourself to count, so the number of people must be the next number after 3. 

A more complicated situation is if I have 4 apples and 3 bananas in a bag; what is the \emph{total} number of fruits altogether? Is there a way to put numbers together to find the total amount of things?

This leads us to \emph{addition}, an operation which takes 2 numbers together to calculate what amount the numbers they make together. We calculate this by counting the first lot of objects, and then continuing from that number when counting the second lot of objects.

\begin{definition}
The \emph{addition operator} takes 2 numbers and calculates the total that they make. This is also called the sum.
\end{definition}

\section{Basic rules}
- Associative
\[1+(2+3)=(1+2)+3\]
- Commutative
\[7+5=5+7\]



\chapter{Subtraction}
\chapter{Multipliation}

\chapter{Natural numbers}

Though numbers can be used to describe the amount of something, 

The theory of variables allows us to talk about 'algebraic structures'; ways in which objects in mathematics relate to one another.
Indeed, numbers are perhaps the most renown objects of mathematical discourse, and therefore elementary algebra deals with how real numbers relate to one another under the 4 operations.

What are real numbers? Are numbers that aren't real numbers 'fake numbers'? Here's a refresher:

\begin{definition}
The \emph{set of real numbers} \mathbb{R} is a set of numbers containing numbers that are either rational (can be represented as a fraction of integers $5,0,-7, \frac{22}{7}$ etc.) and irrational (numbers like $\sqrt{2},\pi$ etc.).
\end{definition}

These numbers basically encompass any number that a highschool student can think of. At the end of this book, imaginary numbers will be introduced; the most basic numbers that are not real numbers.

Now that the real numbers are established, one can introduce the 4 operations. The real numbers together with these 4 operations create an algebraic structure called a \emph{field}, hence this is known as \emph{the field of real numbers}. One can think of the field of real numbers as the 'playing field' which sets the rules for how numbers can 'play around'.


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

\chapter{Addition}
\chapter{Subtraction}
- anticommutative
\[9-6=-(6-9)\]
\chapter{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
\[5 \cdot (3-4)=5\cdot3-5\cdot4\]
\chapter{Division}
- right-distributive over addition
\[\frac{(3+4)}{5}=\frac{3}{5}+\frac{4}{5}\]
- right-distributive over subtraction
\[\frac{(3-4)}{5}=\frac{3}{5}-\frac{4}{5}\]


\chapter{FOIL method}

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



\chapter{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 
\[\frac{(3-4)}{5}=\frac{3}{5}-\frac{4}{5}\]

