\part{Fundamentals} \part{Natural numbers} Mathematics is well known for dealing with numbers, so let's discuss them! \chapter{Natural numbers} The natual numbers are the numbers used to describe 'how many' of something there is. How many heads do I have? 1. How many arms? 2. \begin{definition} The natural numbers are the whole numbers starting from 0. \end{definition} \section{Counting up} 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, in other words, 4 is the sucessor of 3. This gives us a way to count towards larger numbers. If I have a room of 7 people and another person comes in, the amount of people jumps to 8. This is called counting up. \section{Counting down} We can also count towards smaller numbers. Imagine if in our room of 7 people someone left; the amount of people falls to 6. This is called counting down. Counting up is always possible, but sometimes we can't count down. Imagine a room with no people (0 people); what could we count down to? In this case, counting down makes no sense because having a 'room with less than 0 people' makes no sense. \section{Place value} Humans have had many different ways of writing numbers, however today we use the \emph{Hindu-Arabic numeral system}; a system with ten symbols called \emph{digits}. \begin{itemize} \item 0 (zero) counts nothing \item 1 (one) counts a single object \item 2 (two) is the number after 1 \item 3 (three) is the number after 2 \item 4 (four) is the number after 3 \item 5 (five) is the number after 4 \item 6 (six) is the number after 5 \item 7 (seven) is the number after 6 \item 8 (eight) is the number after 7 \item 9 (nine) is the number after 8 \end{itemize} After 9, we use several digits to represent new numbers. After 9 is 10 (ten); the '1' is said to be in the 'tens place' and the '0' in the 'ones place'. The tens place is a like a 'counter' for how many times we have had to cycle through all the digits in the ones place. We can then count 10,11,12,13, all the way to 99. The next number after this is 100 (one hundred); which has '0' in the ones and tens place, and '1' in the 'hundreds place'. When the hundreds place is used up, we create a fourth digit place, and so on. \section{Addition of numbers} Imagine that 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? Let's try using counting! We know that I have 4 fruits from the apples, and 3 more fruits from the bananas. If I start at 4 and count 3 numbers up, I can find the total amount of fruits; 5,6,\emph{7}. I have 7 fruits! This idea is called \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 operation} takes the first number and counts up as much as the second number. We represent this with the symbol $+$. \end{definition} \subsection{Counting method} The simplest way to add two numbers is to count up the second number, starting from the first. \begin{example} To solve 9+6, we count 6 up, starting from 9. So counting 6 numbers after 9, we have 10,11,12,13,14,\emph{15}. So 9+6=15! \end{example} Another way of thinking of this same idea is with a \emph{number line}. \subsection{Basic rules} - Commutative \[7+5=5+7\] - Associative \[1+(2+3)=(1+2)+3\] \subsection{Friends of 10} These basic rules make addition a little easier, but to really get good at addition, it is useful to remember a few 'addition facts'. The most basic of these is the idea of 'friends of 10'. This looks at all the single digit numbers that add together to get 10; 1+9=10 2+8=10 3+7=10 4+6=10 5+5=10 10 is a nice number to work with; adding 10 just adds 1 to the tens place. \begin{example} To solve 7+5, we remember that 7+3=10; so we know that after counting 3 more than 7 we have 10. We need to count 2 more numbers up to get our answer, which is the same as 10+2, which is 12. So we have calculated that 7+5=12. \end{example} \subsection{Addition algorithm} Addition problems can get very large; how can we quicly calculate that 9237+2385=11622. This section will use the tricks we've learned to make a method of addition that we can use for any problem. \section{Subtraction of numbers} \begin{definition} The \emph{subtraction operation} takes the first number and counts down as much as the second number. \end{definition} \section{Multipliation of 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{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{Less than and greater than} \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 \[\frac{(3-4)}{5}=\frac{3}{5}-\frac{4}{5}\]