## math

A math textbook covering a wide variety of grades.

PDF tex (see bellow)

Accompanying material:

Question Generator

The following is the LaTeX code for my math textbook. I’d upload it as a file, but wordpress won’t let me….

To compile it, run the commands (for unix).

latex math.tex
dvi2ps math.dvi
ps2pdf  math.ps


Do not use pdflatex.

The LaTeX:

\documentclass[10pt]{book}
\usepackage[greek, english]{babel}
\usepackage{makeidx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{polynom}
\usepackage{pstricks}
\usepackage{multirow}
\usepackage{pst-plot}
\usepackage{multicol}
\usepackage{xkeyval}
\usepackage{pst-xkey}
\usepackage{pst-3dplot}
\include{longdiv}

\makeindex
\title{Math}
\author{Christopher Olah}

\begin{document}
\frontmatter

\maketitle

\newpage

\begin{quotation}
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
Texts. \\
\end{quotation} %\cite{GNU}
You should’ve received a copy of the GFDL (GNU Free Documentation License) with this book. If not, you can get one from \href{http://www.gnu.org}{gnu.org}.\\
%Save the trees: no long liscense printed. Can we do this? Not all wikipedia articles have the GFDL text, but wikipedia is one work. Maybe I should put it under a CC liscense though?
%Does the FDL allow this?
This is book is a work in progress. Email comments to \href{mailto:christopherolah.co@gmail.com}{\mbox{christopherolah.co@gmail.com}}. \\

\tableofcontents

\chapter{Preface: The Logic to this Book}

You maybe starring at this book with a little bit of confusion. This is understandable. This book breaks many conventions. This preface explains why.\\

Firstly, you may have noticed that this book is released under the \emph{GNU Free Documentation License}. This is because it is made with the ideal that knowledge should be free. The GFDL, which I advise you to read, in short, gives you permission to: use this book, distribute this book to anyone, change this book, and distribute the changed versions under the GFDL. Because you can change and distribute this book, I encourage you to use it by giving it to your students, either digitally or in a printed version (you should probably get it printed at a printing shop so that it can be properly bound…) and to improve it and pass it on!\\

Secondly, this book tries to avoid a pitfall that (in the author’s opinion) many textbooks fall into: obfuscation by excessive explanation. By trying to explain things too much, these books over-complicate the topic. This book instead gives short and concise explanations. This has its own inherent dangers (easy to not explain enough, deceptively short explanations of things) but the author feels that it is still superior to the former. (That means that this book can not substitute having a teacher. (If you don’t have a teacher, you will need to rely on other resources. I’ll try to point you in the right direction.))\\

In defence of the fact that this book would be difficult for younger grades to read, I would like to point out that, for the earlier of the intended grades, any book would be impossible to read, seeing as they haven’t learnt how yet. That makes this the place of the teacher and parent to help the student and this book’s place to help them.\\

Finally, and the biggest point, this book goes into topics well beyond the grades it is intended to teach (lower grades) and is written in a way that may be difficult for younger children. Why? There are several reasons. \\

The first reason is that teachers, at this grade, often only know the bare minimum of the topic they’re teaching. (This is natural, how can a teacher be expected to be an expert in every topic? The only subjects where this seems to be dealt with at all are second languages because it is difficult for an ignorant teacher to not come off as such.) This is problematic, however, because knowing the bare minimum is not sufficient to teach a topic, in reality, because the bare minimum doesn’t include an understanding of why things happen, what to prepare students for, or topics to point advanced students towards. (This, of course, presents a challenge to the author, how do I explain things at the highest level known (to the author)? The answer seems to be with lots of care… or stopping before that point.) So this book can hopefully teach \emph{them} as well as the students.\\

The second reason was one that I hinted at before: so that students can explore beyond the curriculum.\\

I hope this book is a useful resource.\\

\emph{Christopher Olah}\\

\mainmatter

\chapter{Numbers \& Arithmetic}

\section{Math as a Standard}

Math consists of two parts. There is (as we will dub them for our purposes) the logical part and the conventional part. The logical part is static: the only changes are us fixing errors that we have made. It can and has been developed in parallel around the world. On the other hand, the conventional part is just how we represent it.\\

So why use the conventional parts? It is well explained by the celebrated physicist Richard Feynman in the book \emph{Surely You’re joking, Mr. Feynman}:

\begin{quotation}
I thought my symbols were just as good, if not better, than the regular symbols — it doesn’t make any difference what symbols you use — but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, “What the hell are those?” I realized then that if I’m going to talk to anybody else, I’ll have to use the standard symbols, so I eventually gave up my own symbols.\\
\end{quotation}

\section{Positive Whole Numbers}

A whole number is a number which represents how many of something there is.\\

\subsection{Base System}
We normally use the \emph{decimal} or base-10 system. In it there are a number of columns and a digit in each one. Each digit progresses from zero to nine (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and then increases the next digit by one and goes back to zero.\\

But base-10 isn’t the only system we can use. In fact, the base could theoretically be any whole number greater than zero. And example of this would be the unary or base-1 system which is used when we count with our fingers. We could instead use the base-2 system, binary, to count with our fingers. Coincidentally, binary is the system used by computers. But when people work with computers and need to access binary, it can be rather difficult to read so they often show it in base-8 (octal) or base-16 (hexadecimal)(the digits go 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). As you can see, many other base systems can be used. In order to ensure that a person knows what system your using, you may want to write it as $Number_{base}$. However, the standard one is decimal and it should be used and presumed to be used unless otherwise specified.\\

\subsection{What is Arithmetic?}

Arithmetic is a term used to describe several simple operations we can perform on numbers.\\

\subsection{Modular Arithmetic}

Doing arithmetic with large numbers would is very difficult, so we break doing arithmatic on large number into several smaller, easy parts. This is called \emph{modular arithmetic}.\\

In addition, two number combine. The result, or sum, is how many are in \emph{both} numbers. To add, just count up the number that you’re adding. It should be noted that $a+b$ is the same as $b+a$.\\
%Is symetrical the right term? No, come back
For more speed, one can memorize all the additions of one to nine. Go from right to left applying the chart. If there is overflow (the sum is in more than one digit) add that amount (the second digit) to the next digit. This technique is often called \emph{modular arithmetic}\\

\begin{tabular}{|l||l|l|l|l|l|l|l|l|l|l|}
\hline
+ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
\hline
\hline
0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
\hline
1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\
\hline
2 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11\\
\hline
3 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
4 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\
\hline
5 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14\\
\hline
6 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15\\
\hline
7 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16\\
\hline
8 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17\\
\hline
9 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18\\
\hline
\end{tabular}

With practice, it is possible to add multiple numbers simultaneously.\\

\subsection{Subtraction}

Subtraction is the opposite of addition. Subtraction is written as a-b where a number b is being removed or subtracted from number a. Instead of combining the numbers it removes one number. Unlike in addition, $a-b$ is not the same as $b-a$.\\

While subtraction can be performed by just counting down, a more efficient way is to use modular arithmetic: like addition, move from right to left. Instead of adding subtract each number. If you go over, decrease the next digit by one, add ten to the one your working on and continue.\\

\subsection{Multiplication}

Multiplication is a way of condensing addition: instead of writing a+a we write 2*a or 2a or 2xa or 2(a) (you may notice that there are many ways of writing 2*a).\\

Multiplication can also be done by modular arithmetic. Just memorize a chart (or do the individual ones in your head) and carry (similar to addition).\\

\newpage

\marginpar{Because the numbers are larger, memorizing the multiplication table can be a daunting task. It really is worth while though, as your ability to do mental math (which will allow you to solve problems more quickly) depends on it. Tricks (eg. rhymes) can make it easier, and practice will do the rest.}

\begin{tabular}{|l||l|l|l|l|l|l|l|l|l|l|}
\hline
* & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
\hline
\hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\hline
1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
\hline
2 & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18\\
\hline
3 & 0 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 27\\
\hline
4 & 0 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36\\
\hline
5 & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45\\
\hline
6 & 0 & 6 & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54\\
\hline
7 & 0 & 7 & 14 & 21 & 28 & 35 & 42 & 49 & 56 & 63\\
\hline
8 & 0 & 8 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72\\
\hline
9 & 0 & 9 & 18 & 27 & 36 & 45 & 54 & 63 & 72 & 81\\
\hline
\end{tabular}

\subsection{Division}

Division can be considered the opposite of multiplication. a divided by b can be written as $a/b$ or $\frac{a}{b}$. If $a*b=c$ than $a=c/b$ and $b=c/a$. $a/b$ is not the same as $b/a$.\\

%Comment on relation ship between subtraction and division.

\begin{tabular}{ll}
\longdiv{98}{2} &
\begin{tabular}{l}
\\
Large number can be divided in a modular fashion called \emph{long division}.\\
\\
\\
\\
\\
Subtract 40*2, the largest possible number using 2*10*x.\\
\\
Subract 9*2.\\
Remainder.\\
\end{tabular}\\
\end{tabular}

If a number can’t divide properly there is a remainder. The remainder is the part leftover if you subtract the divider the maximum possible number of times. For example, $21/5=4$ with a remainder of $1$.\\

\subsection{Modulo}

Modulo (represented by \%) is connected to division. Instead of returning the result of division, it returns the remainder.\\

\subsection{Exponents}

Exponents are to multiplication as multiplication is to addition. $a$ to the exponent or power $b$ is written as $a^b$. $a^b = a*a*a\ldots (b \text{times})$.\\

\section{Negative Numbers}

At some point the question must arise, what is smaller than 0? The answer is negative numbers.\\

Negative number work the same way as positive numbers. They are written as $-n$ (where $n$ is the positive version of the number). The \emph{defenition} of a negative number is $-a = 0-a$.\\

With a little bit of logic and the knowledge that -a is 0-a, it is easy to add and subtract with negative numbers.\\

Some simple rules should be followed:
a+b=a+b\\
a-b=a-b\\
-a+b=b-a\\
-a-b=-(a+b)\\
a+-b=a-b\\
a–b=a+b\\

\subsection{Multiplication/Division}
Rules for multiplying (and dividing) with negative numbers:
\begin{tabular}{|l|l||l|}
\hline
a & b & a*b or a/b\\
\hline
+ & + & +\\
+ & – & -\\
– & + & -\\
– & – & +\\
\hline
\end{tabular}

\section{Partial Numbers}
Sometimes numbers are not just representing a number of items: sometimes they are representing parts of things. In this case we use \emph{partial numbers}. Partial numbers can be both positive and negative.\\

%Ratios?

\subsection{Fractions}

The easiest way to demonstrate partial numbers is fractions. In a fraction, a number is represented as one number divided by another. For example, if I had half a pie, I would represent it as $\frac{1}{2}$ pies.\\

The top of the fraction is called the numerator and the bottom of the fraction (the dividing number) is called the denominator.\\

\subsection{Equivalent Fractions}

At any given point, one can divide or multiply both the top and the bottom of the fraction without changing the values. If to fractions have the same value they are said to be equivalent. For simplicities sake, one should always keep the denominator as small as possible.\\

It should be noted that normal’ numbers can be represented as a fraction with themselves as the numerator and one as the denominator ($x=\frac{x}{1}$).\\

To add or subtract two fractions, set them to the same denominator and add/subtract the numerators. ($\frac{a}{b}+\frac{c}{d}=\frac{d*a}{d*b}+\frac{b*c}{b*d}=\frac{d*a+b*c}{b*d}$)\\

\subsubsection{Multiplication and Division}

To multiply two fractions, just multiply the two denominators and multiply the two nominators ($\frac{a}{b}*\frac{c}{d}=\frac{a*c}{b*d}$).\\

To divide two fractions, consider this: $1/1/a=1*a$. So, it is logical that $\frac{a}{b}/\frac{c}{d}=\frac{a}{b}*\frac{d}{c}=\frac{a*d}{b*c}$.\\

\subsection{Exponents}

An observant reader may have noticed the absence of exponents in the negative number section. This is because negative exponents are connected to fractions. Look at the following table.

\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Exponent & $10^3$ & $10^2$ & $10^1$ & $10^0$ & $10^{-1}$ & $10^{-2}$ & $10^{-3}$\\
Value & 1000 & 100 & 10 & 1 & ? & ? & ?\\
\hline
\end{tabular}

Following the trend, what would the logical answer be?\\

\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
Exponent & $10^3$ & $10^2$ & $10^1$ & $10^0$ & $10^{-1}$ & $10^{-2}$ & $10^{-3}$\\
Value & 1000 & 100 & 10 & 1 & $\frac{1}{10}$ & $\frac{1}{100}$ & $\frac{1}{1000}$\\
\hline
\end{tabular}

If we generalize that, we get: $x^{-a}=\frac{1}{x^a}$\\

What about partial exponents? A partial exponent does the opposite of a whole one. $x^{\frac{1}{n}}$ is whatever number, put to the power of $n$ equals $x$. $x^{\frac{1}{n}}$ is more often represented as $\sqrt[n]{x}$ and called $n$th-root, the most common, $\sqrt[2]{x}$ or just $\sqrt{x}$ is called square root.\\

\subsection{. – The Decimal Point}

Knowing what what we do about exponents, we can reanalyze the decimal system.\\
\begin{tabular}{|l|l|l|l|}
\hline
Column & $3$ & $2$ & $1$\\
\hline
Value & $10^2$ & $10^1$ & $10^0$\\
\hline
\end{tabular}\\

Let us rename the columns such that they are the number that ten is to the power of.\\

\ldots210\\

Let us put a .’ to the right of column zero as a place holder.\\

\ldots210.\\

Let’s add columns of \emph{negative numbers} to the right of the .’.

\ldots210.-1-2\ldots\\

The dot is called the decimal point and is a common way of representing partial numbers.\\

Simply put: $\text{Value} = \text{BeforeDecimalPoint} + \frac{\text{AfterDecimalPoint}}{10^{\text{ColumnsAfterDecimalPoint}}}$\\

\subsection{Floating Points}

Another way to save numbers is as \emph{floating points}. Floating point represent the number in two parts

%\section{Infinitly Close \& Infinity}

\section{Some Handy Notations}

Sometimes we want to write down mathematical statements that would be rather long to write out. So, there are certain alternatives to writing the whole thing out.\\

If you have a logical sequence of numbers, you can just put the first few, $\ldots$ and the put the last one. For example, $1+2+3\ldots+10$\\

If you want to add a whole bunch of numbers you can use \emph{sigma notation}. In sigma notation, one writes a generic term, with a symbol representing a number, and then uses the symbol $\sum$ (the Greek letter \emph{sigma} which is its equivalent to our letter s’) with the symbol at the bottom set equal to the lowest value you want to replace it with and on top the highest. For example, $$\sum^{10}_{i=1}=1+2+3\ldots+10$$

But what about if instead you want to multiply a bunch of numbers instead of add? You can use \emph{pi notation}. It is identical in use to sigam notation except we use the Greek letter pi ($\prod$, equivalent to our p’) instead of sigma. So, $$\prod^{10}_{i=0}=1*2*3\ldots10$$

There is a special way to denote $10*9*8\ldots *1$. We can instead use a factorial. $10!=9*8*7\ldots *1$ If we only want to multiply all the way to one, we can divide by another factorial. For example, $$\frac{50!}{40!}=50*49*48\ldots *41$$

\chapter{Algebra}

\section{Variables}
Variables have been used previously in this book, though we didn’t call them variables at the time. Simply put, a variable is something that represents a number. A variable can be anything: a letter, word, symbol… It is considered bad form and annoying to use words because a common way of representing $a*b*b$ is $abc$. Generally speaking one should use symbols and letters with subscripts when necessary.

\section{Polynomials}

\section{A order of operations: BEDMAS}

Consider $2*3+4$. Should one multiply $2$ and $3$ and the add $4$? Or add $3$ and $4$ and then multiply by $2$? The results would be completely different.\\

There is no intrinsic reason why one should do one or another. Which one is correct would depend on context. In order for people to be able to understand equations written by other people there must be a standard. This standard must also provide away to override the normal way it provides. The standard we use is often represented by the acronym BEDMAS which stands for:\\

\newcounter{BEDMAScount}
\begin{list}{\arabic{BEDMAScount}. }{\usecounter{BEDMAScount}}
\item Brackets
\item Exponents
\item Division \& Multiplication
\end{list}

The top should be done first and so on. If there are multiple with the same position on the list, it doesn’t matter which is done first though going from left to right is common.

\section{Bypassing BEDMAS}
Sometimes it is undesirable to follow BEDMAS. There are certain ways to circumvent it.

\subsection{Expanding: $a(b+c) \to ab + ac$}

The shortest summary of expansion would probably be: the removal of brackets. It is very easy, if tedious. Just multiply every term in the brackets by whatever is outside the brackets ($a(b+c) \to ab + ac$).\\

What is outside the brackets can be a bracket itself: $(a+b)(c+d) \to (a+b)c + (a+b)d \to ac + bc + ad + bd$ (this is sometimes simplified to FOIL: First, Outer, Inner, Last). The important thing is that all the terms in one bracket be multiplied by all the terms in the other bracket.\\

\subsubsection{Binomial/Trinomial Theorem}

(a+b)^n = \sum_{i=0}^n \binom{n}{i} a^{n-i} b^{i}

(a+b+c)^n = \sum_{i=0}^n \binom{n}{i} a^{n-i} \sum^i_{j=0} \binom{i}{j} b^{i-j} c^j

These equations are a way of expanding polynomials. $\binom{n}{i}$ is equivalent to $\frac{n!}{i!(n-i)!}$. We will look at why this works when we get to probability.\\

\subsection{Factoring: $ab + ac \to a(b+c)$}
Factoring is the opposite of expanding: it is the removal of a common factor from a polynomial ($ab + ac \to a(b+c)$).\\

While it is easy to factor out a single constant or variable, it is much more challenging to factor out a polynomial.\\

If a polynomial is divisible by $ax-b$ without any remainder, $\frac{b}{a}$ is said to be a root of that equation. The roots of an equation are the numbers that set it to zero.\\

Polynomials can be divided using long division. For example:

\polylongdiv{x^2+3x+2}{x+2}\\

There are several ways to predict what a polynomial is divisible by.\\

If the highest degree is two you can use the \emph{quadratic formula} to find the roots. For a quadratic equation $ax^2+bx+c$, the roots are $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.\\

For other ones, it is useful to note that if a polynomial $ax^3+bx^2+cx+d$ factors to $(a_1x+b_1)(a_2x+b_2)(a_3x+b_3)$ then $a=a_1*a_2*a_3$ and $d=b_1*b_2*b_3$. So, if all of the factors contain only whole numbers, then, for a factor $ax-b$, $a$ times another whole number must be the leading coefficient and $b$ times and other whole number must be $d$.\\

Finally, it is helpful to know that if one divides a polynomial by $ax-b$, the remainder is the same as the value of the polynomial if one sets $x$ to $\frac{a}{b}$. This is called remainder theorem.\\

\subsection{Exponent Law}
The \emph{exponent laws} are:\\

\begin{tabular}{l}
$a^b*a^c=a^{b+c}$\\
$a^b/a^c=a^{b-c}$\\
$(a^b)^c=a^{b*c}$\\
$a^{-b}=\frac{1}{a^b}$\\
$a^{\frac{1}{b}}=\sqrt[b]{a}$\\
\end{tabular}

\section{Equalities, Inequalities \& Proportionalities}

There are many ways to describe the relation between to mathematical statements.\\

The most common is equivalency ($=$) which means that both sides have the same value.\\

Inequalities are when the two sides aren’t (necessarily) equal. There are several:\\

\begin{tabular}{ll}
$a>b$ & $a$ greater than $b$\\
$a\gg b$ & $a$ much greater than $b$\\
$a=b$ & $a$ greater than/equal $b$\\
$a\leq b$ or $a<=b$ & $a$ less than/equal $b$\\ $a\neq b$ or $a\thinspace!\!=\thinspace b$ & $a$ not equal to $b$\\ \end{tabular}\\ Some times nothing can be said about the actual value of the two numbers, but we know that one number being bigger/smaller implies the other being bigger. In this case we say that they are proportional ($\propto$, for example $a \propto b$). If a number being smaller implies the number being bigger, you say that it is proportional to one divided by that number ($a \propto \frac{1}{b}$).\\ \subsection{Equivelance on both sides} One may apply, on either side, any mathematical function. You may add, multiply, subtract, square, et cetera. The one exception is that if the comparison is greater or less than and you multiply or divide by a negative number you must switch the sign to its reciprocal.\\ The result of this is that you can get rid of something on one side by doing the opposite on the other. For example you can change $y=x+2$ into $y-2=x$.\\ \section{Simultaneous Truths: Substitution \& Elimination} So what about when you know that two different equations are true? You can use substitution in elimination. In substitution, you replace a variable with its value according to the other equation and in elimination you subtract one equation from another.\\ Example:\\ Both $y=2x+2$ and $y=x+4$ are true.\\ \begin{tabular}{l|l} \large{Substitution Solution}&\large{Elimination Solution}\\ \begin{tabular}{l} $2x+2=x+4$ \\ $x=2$ \\ $y=(2)+4$ \\ $y=6$\\ \\ \\ \\ \end{tabular}& $~$ \begin{tabular}{l} $~~~y=2x+2$\\ $-(y=x+4)$\\ \hline $~~~0=x-2$\\ \\ $x=2$\\ $y=(2)+4$\\ $y=6$\\ \end{tabular}\\ \end{tabular} \section{Functions} One of the keys to mathematics is functions. The definition of a function is something takes input and turns it into a single output. Since a function can only have one output, a function like a square root has to define which one it is using or it is not a function. Despite this a function may have multiple inputs. A function can take a function as it's input (eg. g(f(x)). A function that when it takes the output of a function returns that that function's input is called the inverse function (the inverse of a function f would be represented by f$^{-1}$). An inverse function is created by switching the function and input in the original function and algebraically moving the components.\\ While a function, can only return one output, it can return multiple numbers if they are different variables. For example a function my return coordinates, two numbers but only one object.\\ A operator is a symbol representing a function. For example we could write $\text{add}(a,b)$ but instead we use the operator $+$ because it's more convenient to write $a+b$ than $\text{add}(a,b)$.\\ %\subsection{Properties} %\section{Limits} \section{Sequences} A sequence or progression is a logical progression of numbers. A series is the sum of a sequence. There are two common types of sequences: arithmatic sequences and geometric sequences. The $n$th term in a sequences is denoted by $t_n$.\\ \subsection{Arithmatic Sequences} An aritmatic sequence can be described by a recursive function: \marginpar{A recursive function is one that refers to its self.} $$t_n=t_{n-1} + d$$ $$t_1=a$$\\ Or as a non-recursive function:\\ $$t_n=a + d(n-1)$$\\ Where $a$ is the original term and $d$ is the difference between terms.\\ The sum of the sequence to the $n$th term can be found by the following equation:\\ $$S_n = \sum _{i=1}^n a+d(i-1)= \frac{2an+dn(n-1)}{2} = \frac{n}{2}(t_1+t_n)$$ Proof:\\ \begin{tabular}{ccccccccc} $S_n$ & = & $a$ & $+$ & $a+d$ & $+\ldots+$ & $a+d(n-2)$ & $+$ & $a+d(n-1)$\\ $S_n$ & = & $a+d(n-1)$ & $+$ & $a+d(n-2)$ & $+\ldots+$ & $a+d$ & $+$ & $a$\\ \hline $2S_n$ & = & $2a+d(n-1)$ & $+$ & $2a+d(n-1)$ & $+\ldots+$ & $2a+d(n-1)$ & $+$ & $2a+d(n-1)$\\ \end{tabular}\\ $$2S_n = 2a + d(n-1) + 2a+d(n-1) + \ldots + 2a+d(n-1) + 2a + d(n-1)$$ $$2S_n = 2an + dn(n-1)$$ $$S_n = \frac{2an + dn(n-1)}{2}$$ $$S_n = \frac{n}{2}(a+ (a+d(n-1)))$$ $$S_n = \frac{n}{2}(t_1+t_n)$$ \subsection{Geometric Sequences} A geometric sequence can be represented recursivly by the following:\\ $$t_n = t_{n-1}r$$ $$t_1 = a$$\\ Or as a non-recursive function:\\ $$t_n = ar^{n-1}$$ Where $a$ is the original term and $r$ is the ratio ($\frac{t_n}{t_{n-1}}$) between terms.\\ The sum of the sequence to the $n$th term is:\\ $$s_n= a\frac{1-r^n}{1-r}$$ Proof:\\ $$S = a + ar + ar^2 + ar^3 + \ldots + ar^{n-1}$$ $$\frac{S}{a} = 1 + r + r^2 + r^3 + \ldots + r^{n-1}$$ This can easily be simplified by multiplying by eiher $r-1$ or $1-r$ which both create a cascade effect.\\ $$(1-r)\frac{S}{a} = 1-r^n$$ $$S = a\frac{1-r^n}{1-r}$$ $$(r-1)\frac{S}{a} = r^n -1$$ $$S = a\frac{r^n-1}{r-1}$$ \subsection{Converging Sequences} A converging sequence is one where the sum of its terms to infinity is a finite number. This requires that each succesive term becomes close to zero.\\ To find the number a sequence converges on, take an equation for its sum and limit $n$ to tend to infinity.\\ A arithmatic sequence can never converge.\\ A geometric sequence converges if $1\geq r \geq -1$.\\ \chapter{Geometry} At its surface, geometry deals with measuring the lines and areas of shapes.\\ Some may wonder why we would bother learning geometry. The fact of the matter is, geometry is incredibly useful in mathematics because we can apply the techniques we use in geometry to math as a whole.\\ \section{Lines} %How in the wide world does one define a line? %You use a picture The following is a line: \begin{pspicture}(4,1) \psline(0,0.5)(4,0.5) \end{pspicture} Two lines are said to be parallel if they never cross. %Cant define perpendicular, no angles yet \section{Circles} A circle can be formed by rotating a line segment from a fixed point.\\ \begin{tabular}{l l} \begin{tabular}{l} \begin{pspicture}(-2,-2)(2,2) \pscircle[linecolor=green]{2} \rput(1.7,1.7){\green{$p$}} \rput(0,1){$a$} \psline[linecolor=blue](-2,0)(2,0) \rput(0,0.3){\blue{$d$}} \psline[linecolor=red](0,0)(0,-2) \rput(0.2,-1){\red{$r$}} \end{pspicture} \end{tabular}& \begin{tabular}{p{6cm}} \indent The distance from a circle's center to its edge is the radius ($r$). The distance from one edge to the one across from it is the diameter ($d$) and is twice the radius ($d=2r$). The perimeter ($p$) is the edge of the circle and the area ($a$) is the amount of space it occupies.\\ \end{tabular}\\ \end{tabular}\\ There is a fixed relationship between diameter and perimeter: $a=\Pi d = 2\Pi r$. $\Pi$ or Pi is a mathematical constant. $\Pi$ is irrational (its digits never have a repeating pattern).Pi can be calculated by $$\Pi= \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} \ldots = \sum_{i=0}^{\infty}(-1)^i\frac{4}{1+2i}$$\\* $\Pi =$ 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679$\ldots$ \\* %From math.com For most purposes, one can treat $\Pi$ as $3.14$. \\ $\Pi$ is also related to the area of a circle: $a=\Pi r^2$\\ \section{Angles} When two lines intersect, they form an angle.\\ \begin{pspicture}(3,1) \psline(0,0)(3,1) \pscurve[linecolor=blue](0.6,0)(0.7,0.1)(0.6,0.2) \rput(1.5,0.2){\blue{angle}} \psline(0,0)(3,0) \end{pspicture} There are several types of angles. A square'' angle is called a \emph{right angle}. An angle smaller than a right angle is an \emph{acute angle} and one larger is an \emph{obtuse angle}.\\ \begin{pspicture}(0,-0.6)(5,1) %Acute angle: \psline(0,0)(1,0) \psline(0,0)(1,0.5) \pscurve(0.2,0)(0.23,0.05)(0.2,0.1) \rput(0.5,-0.5){acute} %Right angle: \psline(2,0)(3,0) \psline(2,0)(2,1) \psline(2.2,0)(2.2,0.2)(2,0.2) \rput(2.5,-0.5){right} %Obtuse angle: \psline(4,0)(5,0) \psline(4,0)(3.5,0.5) \pscurve(4.2,0)(4,0.15)(3.8,0.2) \rput(4.5,-0.5){obtuse} \end{pspicture} We can measure angles in radians, which are the perimeter the angle has swept out of a circle with a radius of one. \begin{pspicture}(-1,-1)(1,1) \pscircle{1} \psline(0,0)(0,1) \psline(0,0)(1,0) \pscurve(0,0.2)(0.15,0.15)(0.2,0) \rput(0.5,-0.2){1} \rput(-0.2,0.5){1} \end{pspicture} We can also measure in degrees, where $\Pi$ radians $= 180$ degrees.\\ \begin{tabular}{|l|l|l|} \hline Angle Name & Radians & Degrees\\ \hline Acute Angle & $0 < \Theta < \frac{1}{2}\Pi$ & $0 < \Theta < 90$\\ Right Angle & $\Theta = \frac{1}{2}\Pi$ & $\Theta = 90$\\ Obtuse Angle & $\frac{1}{2}\Pi <\Theta <\Pi$ & $90 ](0,0)(4,2) \end{pspicture}\\ A vector can be in many dimensions. The vector can then be split into \emph{components} for each dimension. The components are represented by using the same name with a subscript for the dimension. Using the pythagorean theorum it is logical that:\\ $$\vec{v}^2=v_x^2+v_y^2+v_z^2\ldots$$ \subsection{Addition} Two vectors can be added. If$\vec{a}=\vec{b}+\vec{c}$, then: $$a_x = b_x + c_x$$ $$a_y = b_y + c_y$$ $$a_z = b_z + c_z$$ $$\ldots$$\\ The addition of two vectors can be pictured as one continuing another.\\ \begin{pspicture}{0,-1}{4,2} \psline[arrows=->](0,0)(4,2) \psline[arrows=->](0,0)(4,2) \psline[arrows=->](0,0)(2,-1) \psline[arrows=->](2,-1)(4,2) \rput(1,1){$\vec{a}$} \rput(0.5,-0.5){$\vec{b}$} \rput(3,0){$\vec{c}$} \end{pspicture}\\ \subsection{Subtraction} Two vectors can be subtracted. If$\vec{a}=\vec{b}-\vec{c}$, then: $$a_x = b_x – c_x$$ $$a_y = b_y – c_y$$ $$a_z = b_z – c_z$$ $$\ldots$$\\ The subtraction of two vectors can be pictured as two diverging from one point.\\ \begin{pspicture}{0,-1}{4,2} \psline[arrows=<-](0,0)(4,2)%Is this right? \psline[arrows=<-](0,0)(4,2)%Is this right? \psline[arrows=](2,-1)(4,2) \rput(1,1){$\vec{a}$} \rput(0.5,-0.5){$\vec{b}$} \rput(3,0){$\vec{c}$} \end{pspicture}\\ %\subsection{Dot Product} \chapter{Booleans \& Sets} A set may contain many numbers. A boolean can only be true or false. Despite this, they are, in some ways, similar.\\ \section{Introduction to Sets} As mentioned earlier, a set may contain many numbers. One can consider them a group of things. Sets are sometimes known as classes.\\ A set can be declared in many ways. `A set foo contains all odd numbers'' is perfect valid. One can also declare classes that contain individual numbers:$a = { 1, 2,3}$would mean that a set$a$contains one two and three. Sets can contain non-numerical values.\\ There are two sets that one doesn't need to describe. The set$0$is empty and the set$1$(the \emph{universal set}) contains all numbers.\\ One can perform functions on sets.\\ \begin{tabular}{|l|l|l|} \hline Compliment &$a'$& All members of the universal set not in$a$\\ \hline Intersection &$a\cap b$& All numbers both in$a$and$b$.\\ \hline Union &$a \cup b$& All numbers in$a$or$b$\\ \hline \end{tabular} A set of which contains all members of another set is a \emph{super set} of the other one which is its \emph{subset}. \section{Introduction to Booleans} Booleans are either true or false. True is represented by$1$and false represented by$0$.\\ There are several logical functions we can use.\\ \begin{tabular}{lll} AND & \&\&, AND,$*$& True when both inputs are true.\\ OR &$\|$, OR,$+\$ & True when either or both inputs are true.\\ NOT & !, ~, ', or NOT & True when input is not true. %Some neat stuff at the wikipedia entry on Logical Conjunction. Read before continuing writing. Also, we've been going very fast: look over work. \end{tabular} %\chapter{Probability} %\chapter{Matricies} %\chapter{Math as a language: Programming} \end{document} [/sourcecode]