Typesetting Text. Spring 2013 TEX-Bit #1 Math 406

Transcription

1 Spring 2013 TEX-Bit #1 Math 406 Typesetting Text Whitespace characters, such as blank or tab, are treated uniformly as space by L A TEX. Several consecutive whitespace characters are treated as one space. Whitespace at the start of a line is generally ignored, and a single line break is treated as whitespace. An empty line between two lines of text defines the end of a paragraph. Several empty lines are treated the same as one empty line. 1 From the above paragraphs note the way that quotes are done. You do not use the double quotes Like this you use two graves quotes at the beginning and two single quotes at the end like this. Notice the difference at the beginning, with the double quotes you get 99 and 99 but doing it with the single quotes you get 66 and 99 as we would expect. The same is true for single quotes, note the difference between quote and quote. Also, the characters # $ % ˆ & { } and \have a special meaning so to typeset these you use \# \$ \% \^{} \& \_ \{ \} \~{} \textbackslash respectively. Notice a couple things at this point. First L A TEX automatically formats the lines so that the spacing between the words is at its best. Another feature that it has is that it will automatically hyphenate a word if it will make the line look better. L A TEX always tries to produce the best line breaks possible. If it cannot find a way to break the lines in a manner that meets its high standards, it lets one line stick out on the right of the paragraph. L A TEX then complains ( overfull hbox ) while processing the input file. This happens most often when L A TEX cannot find a suitable place to hyphenate a word. 2 You can force a line break but it is not advisable. In most cases L A TEX will break a line at a better point than you will. As you have probably noticed all L A TEX commands (i.e. special things) begin with a \. So changing font styles can be done with bold, italic, typewriter and Small Caps. You may not have noticed a special feature that is done automatically and makes the document easier to read, these are called Ligatures. Notice the f and the i in final, the two f s in shelfful and the two f s and i in traffic. Forcing these to display like Word you would have final, shelfful and traffic. One last bit of information for now. When you make an ellipsis, those three dots you do not use three dots you use the command \ldots, notice the difference... verses... 1 Taken from The Not So Short Introduction to L A TEX 2ε 2 Although L A TEX gives you a warning when that happens (Overfull \hbox) and displays the offending line, such lines are not always easy to find. If you use the option draft in the \documentclass command, these lines will be marked with a thick black line on the right margin. Taken from The Not So Short Introduction to L A TEX 2ε

2 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit001.tex \documentclass[12pt]{article} \usepackage[pdftex]{graphicx} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} Tuesday, July 09, :56 PM \pagestyle{empty} \parskip=5pt \begin{document} \noindent Spring 2013 \hfill {\Huge\textbf{\TeX{}-Bit \#1}} \hfill Math 406 \vspace{20pt} \noindent\textsc{\large Typesetting Text} \vspace{10pt} % Comments are any line after a percnet % sign. This does not show up in the document. ``Whitespace'' characters, such as blank or tab, are treated uniformly as ``space'' by \LaTeX{}. Several consecutive whitespace characters are treated as one space. Whitespace at the start of a line is generally ignored, and a single line break is treated as whitespace. An empty line between two lines of text defines the end of a paragraph. Several empty lines are treated the same as one empty line.\footnote{taken from \textit{the Not So Short Introduction to \LaTeXe{}}} From the above paragraphs note the way that quotes are done. You do not use the double quotes " Like this" you use two graves quotes at the beginning and two single quotes at the end ``like this''. Notice the difference at the beginning, with the double quotes you get 99 and 99 but doing it with the single quotes you get 66 and 99 as we would expect. The same is true for single quotes, note the difference between 'quote' and `quote'. Also, the characters \# \$ \% \^{} \& \_ \{ \} \~{} and \textbackslash have a special meaning so to typeset these you use \verb \# \$ \% \^{} \& \_ \{ \} \~{} \textbackslash respectively. Notice a couple things at this point. First \LaTeX{} automatically formats the lines so that the spacing between the words is at its best. Another feature that it has is that it will automatically hyphenate a word if it will make the line look better. -1-

3 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit001.tex Tuesday, July 09, :56 PM \LaTeX{} always tries to produce the best line breaks possible. If it cannot find a way to break the lines in a manner that meets its high standards, it lets one line stick out on the right of the paragraph. \LaTeX{} then complains (``overfull hbox'') while processing the input file. This happens most often when \LaTeX{} cannot find a suitable place to hyphenate a word.\footnote{ Although \LaTeX{} gives you a warning when that happens (\texttt{overfull \textbackslash hbox}) and displays the offending line, such lines are not always easy to find. If you use the option \texttt{draft} in the \texttt{\textbackslash documentclass} command, these lines will be marked with a thick black line on the right margin. Taken from \textit{the Not So Short Introduction to \LaTeXe{}}} You can force a line break \\ but it is not advisable. In most cases \LaTeX{} will break a line at a better point than you will. As you have probably noticed all \LaTeX{} commands (i.e. special things) begin with a \textbackslash. So changing font styles can be done with \textbf{bold}, \textit{italic}, \texttt{typewriter} and \textsc{ Small Caps}. You may not have noticed a special feature that is done automatically and makes the document easier to read, these are called \textit{ligatures}. Notice the f and the i in final, the two f's in shelfful and the two f's and i in traffic. Forcing these to display like Word you would have f\mbox{}inal, shelf\mbox{}ful and traf\mbox{}f\mbox{} ic. One last bit of information for now. When you make an ellipsis, those three dots you do not use three dots you use the command \texttt{\textbackslash ldots}, notice the difference \ldots verses... \end{document} -2-

4 Spring 2013 TEX-Bit #2 Math 406 Typesetting Text Lists and a few other things. There are three basic types of lists, enumerations, itemization and descriptions. Enumerations are numbering systems 1, 2, 3,... or a, b, c,..., etc. Itemizations are bulleted lists and descriptions are of course descriptions. These can be combined in any manner. Creating then is simple and the same syntax for each type. Lists are an example of an environment. An environment in L A TEX is any block of text (or math) that is between a \begin{---} and an \end{---} where what is between the curly brackets is the type of environment. So an enumeration environment starts with a \begin{enumerate} and ends with an \end{enumerate}. Each item in the list is preceded by a \item. The same is true for an itemize environment. The description environment is only slightly different in that for each item you put the thing being described in square brackets for example, \item [First] This is the first item. The words inside the square brackets are automatically bold faced. 1. Item number Item number Item number 3. The first bullet. The second bullet. The third bullet. First This is the first item. Second This is the second item. Third This is the third item. You can combine these in any order and to any reasonable depth. enumeration inside an enumeration, For example, an 1. Item number 1. (a) Item number 1. (b) Item number 2. (c) Item number 3. 1

5 2. Item number Item number 3. Bullets and enumerations together, 1. Item number Item number Item number 3. The first bullet. The second bullet. The third bullet. Bullets inside bullets inside bullets. The first bullet. The second bullet. The first bullet. The first bullet. The second bullet. The third bullet. The second bullet. The third bullet. The third bullet. Finally a really long one, combining all three types. 1. Item number 1. The first bullet. The second bullet. The third bullet. 2. Item number 2. (a) Item number 1. (b) Item number 2. i. Item number 1. ii. Item number 2. 2

6 The first bullet. The second bullet. The third bullet. A. Item number 1. B. Item number 2. C. Item number 3. iii. Item number 3. (c) Item number Item number 3. First This is the first item. (a) Item number 1. (b) Item number 2. (c) Item number 3. Second This is the second item. The first bullet. The second bullet. The third bullet. Third This is the third item. If you looked through the mark-up of this document you will notice a couple other things. First, you can create a horizontal line in the document by using the \hrule command, which of course stands for horizontal rule. You will also notice several uses of \verb and some vertical line characters. The verb stands for verbatim and the syntax is \verb followed by a particular character, everything between the two occurrences of that character is set in typewriter font and the L A TEX commands are ignored. There is also a verbatim environment for multiple lines of text. This is text in the verbatim environment note that it keeps spaces and does not reformat as it would if the text were outside the environment. Also note in the first paragraph of the typeset document that there are two hyphenated words at the end of lines but in the original paragraph there were no hyphenations in those words. As we pointed out before, if L A TEX needs to hyphenate a word to make the word spacing look good it will do so automatically. 3

7 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit002.tex \documentclass[12pt]{article} \usepackage[pdftex]{graphicx} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} Tuesday, July 09, :58 PM %\pagestyle{empty} \parskip=5pt \begin{document} \noindent Spring 2013 \hfill {\Huge\textbf{\TeX{}-Bit \#2}} \hfill Math 406 \vspace{20pt} \noindent\textsc{\large Typesetting Text --- Lists and a few other things.} \vspace{10pt} There are three basic types of lists, enumerations, itemization and descriptions. Enumerations are numbering systems 1, 2, 3, \ldots or a, b, c, \ldots, etc. Itemizations are bulleted lists and descriptions are of course descriptions. These can be combined in any manner. Creating then is simple and the same syntax for each type. Lists are an example of an environment. An environment in \LaTeX{} is any block of text (or math) that is between a \verb \begin{---} and an \verb \end{---} where what is between the curly brackets is the type of environment. So an enumeration environment starts with a \verb \begin{enumerate} and ends with an \verb \end{enumerate}. Each item in the list is preceded by a \verb \item. The same is true for an \verb itemize environment. The \verb description environment is only slightly different in that for each item you put the thing being described in square brackets for example, \verb \item [First] This is the first item. The words inside the square brackets are automatically bold faced. \begin{enumerate} \item Item number 1. \item Item number 2. \item Item number 3. \end{enumerate} \begin{itemize} \item The first bullet. \item The second bullet. \item The third bullet. \end{itemize} \begin{description} \item [First] This is the first item. \item [Second] This is the second item. \item [Third] This is the third item. \end{description} You can combine these in any order and to any reasonable depth. For example, an enumeration -1-

8 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit002.tex inside an enumeration, Tuesday, July 09, :58 PM \begin{enumerate} \item Item number 1. \begin{enumerate} \item Item number 1. \item Item number 2. \item Item number 3. \end{enumerate} \item Item number 2. \item Item number 3. \end{enumerate} \hrule Bullets and enumerations together, \begin{enumerate} \item Item number 1. \item Item number 2. \item Item number 3. \begin{itemize} \item The first bullet. \item The second bullet. \item The third bullet. \end{itemize} \end{enumerate} \hrule Bullets inside bullets inside bullets. \begin{itemize} \item The first bullet. \item The second bullet. \begin{itemize} \item The first bullet. \begin{itemize} \item The first bullet. \item The second bullet. \item The third bullet. \end{itemize} \item The second bullet. \item The third bullet. \end{itemize} \item The third bullet. \end{itemize} \hrule Finally a really long one, combining all three types. -2-

9 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit002.tex \begin{enumerate} \item Item number 1. \begin{itemize} \item The first bullet. \item The second bullet. \item The third bullet. \end{itemize} \item Item number 2. \begin{enumerate} \item Item number 1. \item Item number 2. \begin{enumerate} \item Item number 1. \item Item number 2. \begin{itemize} \item The first bullet. \item The second bullet. \item The third bullet. \begin{enumerate} \item Item number 1. \item Item number 2. \item Item number 3. \end{enumerate} \end{itemize} \item Item number 3. \end{enumerate} \item Item number 3. \end{enumerate} \item Item number 3. \begin{description} \item [First] This is the first item. \begin{enumerate} \item Item number 1. \item Item number 2. \item Item number 3. \end{enumerate} \item [Second] This is the second item. \begin{itemize} \item The first bullet. \item The second bullet. \item The third bullet. \end{itemize} \item [Third] This is the third item. \end{description} \end{enumerate} Tuesday, July 09, :58 PM If you looked through the mark-up of this document you will notice a couple other things. First, you can create a horizontal line in the document by using the \verb \hrule command, which of course stands for horizontal rule. You will also notice several uses of \verb \verb and some vertical line characters. The verb stands for verbatim and the syntax is \verb \verb followed by a particular character, everything between the two occurrences of that character is set in typewriter font and the \LaTeX{} commands are ignored. There is also a verbatim environment for multiple lines of text. -3-

10 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit002.tex Tuesday, July 09, :58 PM \begin{verbatim} This is text in the verbatim environment note that it keeps spaces and does not reformat as it would if the text were outside the environment. \end{verbatim} Also note in the first paragraph of the typeset document that there are two hyphenated words at the end of lines but in the original paragraph there were no hyphenations in those words. As we pointed out before, if \LaTeX{} needs to hyphenate a word to make the word spacing look good it will do so automatically. \end{document} -4-

11 Spring 2013 TEX-Bit #3 Math 406 A Little Bit of Math The real reason for using L A TEX is for typesetting mathematics. Although I use it for just about everything its real strength is in the presentation of mathematics. L A TEX has two different modes, text mode and math mode. In the first two TEX-Bits we were always in text mode. Math mode is when you are between two dollar signs $ $ or you are between \[ and \]. When you are between two dollar signs you are in in-line mode and when you are between \[ and \] you are in display mode. In-line mode is for formulas that are inside a paragraph like, y = x 2 3x + 2 and display mode will center the formula and place it in its own vertical space. For example, y = x 2 3x + 2 There are other differences between the two types of math modes, for example, take the definition of the derivative f f(x+h) f(x) (x) = lim h 0 is the in-line and this is the displayed, h f (x) = lim h 0 f(x + h) f(x) h I am not going to go into all of the options for mathematical formula in L A TEX since I really do not have time to write a book but we will look at a few examples of things you will use frequently and I point out some mistakes I tend to see with new TEX users. For a good background on typesetting mathematics that will probably cover 99% of all you will need read chapter 3 of The Not So Short Introduction to L A TEX 2ε. Exponents are done with a ^ symbol, as with most calculation devices. x 2, x 3, x xx. Subscripts are done with the underscore, x 2, x 3, x a1. These can, of course, be combined, x x x 2 n. Fractions use the \frac command. With it the stuff in the first set of curly brackets is in the numerator and the stuff in the second set of curly brackets is in the denominator. Fractions in displayed equations are much larger than those in the in-line equations. For example, x2 +x 1 x 2 +x+1 verses x 2 + x 1 x 2 + x + 1 So if you are doing compound fractions you will probably want to make it a displayed equation for readability, notice x 1 x verses x 2 x 7 +1 x 1 x x x 7 Also notice with the in-line version, L A TEX only made the character so small. Instead of making them smaller, to the point where they would be impossible to read it moved the line so that the in-line equation would fit. 1

12 Roots are equally simple, use the syntax \sqrt{---} where the thing inside the curly brackets is what you are taking the square root of. For example, 2, x, and 4x 2 6x + 1. To do roots that are not square roots just put [ ] with something inside right after the \sqrt. For example, 3 2, 7 x, and t 4x2 6x + 1. As with just about anything in L A TEX you can combine these structures, x 1 3 x x 5 2 x L A TEX has an extremely extensive list of symbols. The Not So Short Introduction to L A TEX 2ε has tables of mathematical symbols that go on for several pages and for the complete list take a look at The Comprehensive L A TEX Symbol List by Scott Pakin. This contains 330 tables of symbols. L A TEX even has a complete list of symbols for the Linear A language, which is an ancient and so far undecipherable language. Of course, there is a complete set of symbols for Greek, α, β, Γ and so on. If you are using the TeXnicCenter to do your editing you will find under Math in the main menu listings of many symbols and mathematical constructs, so for most of your work you will not need to memorize or look up commands. One more idea for this TEX-Bit, parentheses. Notice the following output, ( 1 x )5 This does not look very good, to say the least. It would be better if the parentheses were larger so that the 1 was contained in them. To do this we replace ( ) with \left( \right). x We get, ( ) 5 1 x It does add some length to expressions but the result is very readable, no matter how complex the expression is. For example, ( ( 3 4) 2 ) 5 1 x 2 +1 x 3 2 x 1 x+7 Before we end this TEX-Bit I want to show you a common error that is made by those new to L A TEX, too many dollar signs. You do not need to put dollar signs around every symbol, just every expression. In math mode you should put an entire formula or expression inside one set of dollar signs or one set of displayed equation delimiters. For example, x A verses x A or y = x 2 3x + 2 verses y = x 2 3x + 2. In these examples the second form is preferred. In some cases it is difficult to tell the difference but there are spacing differences between the two and frankly the second forms are easier to write and read. 2

13 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit003.tex \documentclass[12pt]{article} \usepackage[pdftex]{graphicx} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} Tuesday, July 09, :58 PM %\pagestyle{empty} \parskip=5pt \begin{document} \noindent Spring 2013 \hfill {\Huge\textbf{\TeX{}-Bit \#3}} \hfill Math 406 \vspace{20pt} \noindent\textsc{\large A Little Bit of Math} \vspace{10pt} The real reason for using \LaTeX{} is for typesetting mathematics. Although I use it for just about everything its real strength is in the presentation of mathematics. \LaTeX{} has two different modes, text mode and math mode. In the first two \TeX{}-Bits we were always in text mode. Math mode is when you are between two dollar signs \verb $ $ or you are between \verb \[ and \verb \]. When you are between two dollar signs you are in in-line mode and when you are between \verb \[ and \verb \] you are in display mode. In-line mode is for formulas that are inside a paragraph like, $y = x^2-3x+2$ and display mode will center the formula and place it in its own vertical space. For example, \[ y = x^2-3x+2 \] There are other differences between the two types of math modes, for example, take the definition of the derivative $f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ is the inline and this is the displayed, \[ f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \] I am not going to go into all of the options for mathematical formula in \LaTeX{} since I really do not have time to write a book but we will look at a few examples of things you will use frequently and I point out some mistakes I tend to see with new \TeX{} users. For a good background on typesetting mathematics that will probably cover 99\% of all you will need read chapter 3 of \textit{the Not So Short Introduction to \LaTeXe{}}. Exponents are done with a \verb ^ symbol, as with most calculation devices. $x^2$, $x^3$, $x^{x^x}$. Subscripts are done with the underscore, $x_2$, $x_3$, $x_{a_1}$. These can, of course, be combined, $x_1^2 + x_2^2+\cdots + x_n^2$. Fractions use the \verb \frac command. With it the stuff in the first set of curly brackets is in the numerator and the stuff in the second set of curly brackets is in the denominator. Fractions in displayed equations are much larger than those in the in-line equations. For example, $\frac{x^2+x-1}{x^2 +x+1}$ verses \[ -1-

14 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit003.tex Tuesday, July 09, :58 PM \frac{x^2+x-1}{x^2+x+1} \] So if you are doing compound fractions you will probably want to make it a displayed equation for readability, notice $\frac{x-\frac{1}{x}}{\frac{x^2}{x-7}+1}$ verses \[ \frac{x-\frac{1}{x}}{\frac{x^2}{x-7}+1} \] Also notice with the in-line version, \LaTeX{} only made the character so small. Instead of making them smaller, to the point where they would be impossible to read it moved the line so that the in-line equation would fit. Roots are equally simple, use the syntax \verb \sqrt{---} where the thing inside the curly brackets is what you are taking the square root of. For example, $\sqrt{2}$, $\sqrt{x}$, and $\sqrt{4x^2-6x+1}$. To do roots that are not square roots just put \verb [ ] with something inside right after the \verb \sqrt. For example, $\sqrt[3]{2}$, $ \sqrt[7]{x}$, and $\sqrt[t]{4x^2-6x+1}$. As with just about anything in \LaTeX{} you can combine these structures, \[ \frac{x-\sqrt{\frac{1}{\sqrt{\sqrt[3]{x}}}}}{\frac{x^2}{\sqrt[5]{x-7}}+1} \] \LaTeX{} has an extremely extensive list of symbols. \textit{the Not So Short Introduction to \LaTeXe{}} has tables of mathematical symbols that go on for several pages and for the complete list take a look at \textit{the Comprehensive \LaTeX{} Symbol List} by Scott Pakin. This contains 330 tables of symbols. \LaTeX{} even has a complete list of symbols for the Linear A language, which is an ancient and so far undecipherable language. Of course, there is a complete set of symbols for Greek, $\alpha$, $\beta$, $\Gamma$ and so on. If you are using the TeXnicCenter to do your editing you will find under Math in the main menu listings of many symbols and mathematical constructs, so for most of your work you will not need to memorize or look up commands. One more idea for this \TeX{}-Bit, parentheses. Notice the following output, \[ (\frac{1}{x})^5 \] This does not look very good, to say the least. It would be better if the parentheses were larger so that the $\frac{1}{x}$ was contained in them. To do this we replace \verb ( ) with \verb \left( \right). We get, \[ \left( \frac{1}{x} \right)^5 \] It does add some length to expressions but the result is very readable, no matter how complex the expression is. For example, \[ \left( \frac{\left( \frac{1}{\sqrt{x^2+1}} \right)^5}{\frac{x^3- \frac{2}{x-1}}{x+7}} \right)^{\left(\frac{3}{4}\right)^2} \] Before we end this \TeX{}-Bit I want to show you a common error that is made by those new to \LaTeX{}, too many dollar signs. You do not need to put dollar signs around every symbol, just every expression. In math mode you should put an entire formula or expression inside one set of dollar signs or one set of displayed equation delimiters. For example, $x$ $\in$ $A$ verses $x \in A$ or -2-

15 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit003.tex Tuesday, July 09, :58 PM $y$ $=$ $x^2-3x+2$ verses $y = x^2-3x+2$. In these examples the second form is preferred. In some cases it is difficult to tell the difference but there are spacing differences between the two and frankly the second forms are easier to write and read. \end{document} -3-

16 Spring 2013 TEX-Bit #4 Math 406 A Little More Math Arrays and Equation Arrays An equation array is a lit of equations in which all of the equal signs line up. These are environments so they start with \begin{eqnarray} and ends with \end{eqnarray}. There are two types, the eqnarray which numbers all of the equations and eqnarray* which does not number the equations. Also, if you use eqnarray you can keep L A TEX from numbering an equation by using \nonumber after the equation. The equal signs that are to be lined up are put between two and symbols (&) and we must end every line except the last with \\ so, \begin{eqnarray*} - x_1 + 6x_2 + 16x_3 & = & 1 \\ 11 x_1-22x_2-44x_3 & = & -3 \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray*} produces x 1 + 6x x 3 = 1 11x 1 22x 2 44x 3 = 3 22x x x 3 = 8 and \begin{eqnarray} - x_1 + 6x_2 + 16x_3 & = & 1 \\ 11 x_1-22x_2-44x_3 & = & -3 \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray} produces x 1 + 6x x 3 = 1 (1) 11x 1 22x 2 44x 3 = 3 (2) 22x x x 3 = 8 (3) Finally, an example of using \nonumber after an equation, 1

17 \begin{eqnarray} - x_1 + 6x_2 + 16x_3 & = & 1 \nonumber \\ 11 x_1-22x_2-44x_3 & = & -3 \nonumber \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray} produces x 1 + 6x x 3 = 1 11x 1 22x 2 44x 3 = 3 22x x x 3 = 8 (4) One thing that is a bit out of the ordinary with the equation arrays is that they are not placed inside a displayed equation. so we do not need to put \[ and \] around them as we do with other displayed mathematics. The next few examples simply show some more equation arrays. The first also uses the \displaystyle command. When you are in the inline math mode and you want to force L A TEX to typeset the equation like it does in display math mode simply put \displaystyle as the first thing inside the $ $. Note in the next example how the limits are displayed. The inline mode would put the x 0 as a subscript to the limit but since we used \displaystyle the x 0 is placed under the limit. sin(x) cos(x) 1 Using the fact that lim = 1, lim = 0 and the definition of the derivative, x 0 x x 0 x prove that d (sin(x)) = cos(x). dx Proof. f (x) = f(x + h) f(x) lim h 0 h = sin(x + h) sin(x) lim h 0 h = lim h 0 sin(x) cos(h) + cos(x) sin(h) sin(x) h sin(x) cos(h) sin(x) cos(x) sin(h) = lim + lim h 0 h h 0 h cos(h) 1 = sin(x) lim h 0 = cos(x) h + cos(x) lim h 0 sin(h) h 2

18 The next example was an exam question on one of my Calculus I exams a few years ago. lim x2 + 2x ( x 2 3x = lim x2 + 2x ) x2 + 2x + x x 2 3x 2 3x x x x2 + 2x + x 2 3x (x 2 + 2x) (x 2 3x) = lim x x2 + 2x + x 2 3x 5x x2 + 2x + x 2 3x = lim x 5 = lim x 1 + 2x x 1 x 1 x = 5 2 In all of the above examples we used equal signs between the & signed but this is not necessary, any symbol will work. For example, given that a and b are non-negative numbers we have, a 2 + b 2 a 2 + 2ab + b 2 = (a + b) 2 When we work with matrices we use a different type of array environment, the array environment. These do have to be inside the display math delimiters. We also need to tell it how many columns we intend to have and how we want to align the columns. Consider the following example, \[ A = \left[ \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right] \] This produces the following matrix, A = Notice in the code that after \begin{array} we have {rrr}. This tells L A TEX that we are going to use three columns and each will be aligned on the right. Notice in the resulting 3

19 matrix the three columns line up on the right. Changing {rrr} to {ccc} will center the columns as in the example below. A = Changing {rrr} to {ccc} will center the columns as in the example below. A = The alignment for any column is independent of the alignment of any other column, so changing the alignment to {clr} is valid and produces, A = As you noticed we did a left and right commands to place the brackets around the array, without them the array simply looks like, We can also use other brackets, such as parentheses or vertical bars in the case of a determinant. We added a few other things in the example below as well. Notice the use of \qquad, \quad and \qquad are two spacing commands you can use in math mode that put some space between items. The \qquad is twice the width of the \quad. These come in handy when you want to put two different but related items in the same displayed math area, like we did below. Another new thing was the \mbox{...}. The \mbox{...} is used to write text inside math mode. Without the \mbox{...} the so we have would be treated as mathematics, italicized and no spaces. A = so we have A = Without \mbox{...} the result does not look very good, A = sowehave A = = 14 = 14 4

20 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit004.tex \documentclass[12pt]{article} \usepackage[pdftex]{graphicx} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} Tuesday, July 09, :58 PM \usepackage{amsthm} %\pagestyle{empty} \parskip=5pt \begin{document} \noindent Spring 2013 \hfill {\Huge\textbf{\TeX{}-Bit \#4}} \hfill Math 406 \vspace{20pt} \noindent\textsc{\large A Little More Math --- Arrays and Equation Arrays} \vspace{10pt} An equation array is a lit of equations in which all of the equal signs line up. These are environments so they start with \verb \begin{eqnarray} and ends with \verb \end{eqnarray}. There are two types, the \verb eqnarray which numbers all of the equations and \verb eqnarray* which does not number the equations. Also, if you use \verb eqnarray you can keep \LaTeX{} from numbering an equation by using \verb \nonumber after the equation. The equal signs that are to be lined up are put between two and symbols (\&) and we must end every line except the last with \verb \\ so, \begin{verbatim} \begin{eqnarray*} - x_1 + 6x_2 + 16x_3 & = & 1 \\ 11 x_1-22x_2-44x_3 & = & -3 \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray*} \end{verbatim} \noindent produces \begin{eqnarray*} - x_1 + 6x_2 + 16x_3 & = & 1 \\ 11 x_1-22x_2-44x_3 & = & -3 \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray*} \noindent and \begin{verbatim} \begin{eqnarray} - x_1 + 6x_2 + 16x_3 & = & 1 \\ 11 x_1-22x_2-44x_3 & = & -3 \\ -22 x_1 + 55x_ x_3 & = & 8-1-

21 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit004.tex \end{eqnarray} \end{verbatim} Tuesday, July 09, :58 PM \noindent produces \begin{eqnarray} - x_1 + 6x_2 + 16x_3 & = & 1 \\ 11 x_1-22x_2-44x_3 & = & -3 \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray} Finally, an example of using \verb \nonumber after an equation, \begin{verbatim} \begin{eqnarray} - x_1 + 6x_2 + 16x_3 & = & 1 \nonumber \\ 11 x_1-22x_2-44x_3 & = & -3 \nonumber \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray} \end{verbatim} \noindent produces \begin{eqnarray} - x_1 + 6x_2 + 16x_3 & = & 1 \nonumber \\ 11 x_1-22x_2-44x_3 & = & -3 \nonumber \\ -22 x_1 + 55x_ x_3 & = & 8 \end{eqnarray} One thing that is a bit out of the ordinary with the equation arrays is that they are not placed inside a displayed equation. so we do not need to put \verb \[ and \verb \] around them as we do with other displayed mathematics. The next few examples simply show some more equation arrays. The first also uses the \verb \displaystyle command. When you are in the inline math mode and you want to force \LaTeX{} to typeset the equation like it does in display math mode simply put \verb \displaystyle as the first thing inside the \$ \$. Note in the next example how the limits are displayed. The inline mode would put the $x \rightarrow 0$ as a subscript to the limit but since we used \verb \displaystyle the $x \rightarrow 0$ is placed under the limit. \noindent Using the fact that $\displaystyle \lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$, $\displaystyle \lim_{x \rightarrow 0} \frac{\cos(x)-1}{x} = 0$ and the definition of the derivative, prove that $\frac{d}{dx}(\sin(x)) = \cos(x)$. \begin{proof} \begin{eqnarray*} f'(x) & = & \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ & = & \lim_{h \rightarrow 0} \frac{\sin(x+h)-\sin(x)}{h} \\ & = & \lim_{h \rightarrow 0} \frac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}{h} \\ & = & \lim_{h \rightarrow 0} \frac{\sin(x)\cos(h)-\sin(x)}{h} + \lim_{h \rightarrow 0} \frac{ \cos(x)\sin(h)}{h} \\ -2-

22 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit004.tex Tuesday, July 09, :58 PM & = & \sin(x)\lim_{h \rightarrow 0} \frac{\cos(h)-1}{h} +\cos(x) \lim_{h \rightarrow 0} \frac{ \sin(h)}{h} \\ & = & \cos(x) \end{eqnarray*} \end{proof} The next example was an exam question on one of my Calculus I exams a few years ago. \begin{eqnarray*} \lim_{x \rightarrow \infty} \sqrt{x^2+2x}-\sqrt{x^2-3x} & = & \lim_{x \rightarrow \infty} \left( \sqrt{x^2+2x}-\sqrt{x^2-3x} \right) \cdot \frac{\sqrt{x ^2+2x}+\sqrt{x^2-3x}}{\sqrt{x^2+2x}+\sqrt{x^2-3x}} \\ & = & \lim_{x \rightarrow \infty} \frac{(x^2+2x)-(x^2-3x)}{\sqrt{x^2+2x}+\sqrt{x^2-3x}} \\ & = & \lim_{x \rightarrow \infty} \frac{5x}{\sqrt{x^2+2x}+\sqrt{x^2-3x}} \cdot \frac{\frac{1}{x }}{\frac{1}{x}} \\ & = & \lim_{x \rightarrow \infty} \frac{5}{\sqrt{1+\frac{2}{x}}+\sqrt{1-\frac{3}{x}}} \\ & = & \frac{5}{2} \\ \end{eqnarray*} In all of the above examples we used equal signs between the \& signed but this is not necessary, any symbol will work. For example, given that $a$ and $b$ are non-negative numbers we have, \begin{eqnarray*} a^2 + b^2 & \leq & a^2 + 2ab + b^2 \\ & = & (a+b)^2 \end{eqnarray*} When we work with matrices we use a different type of array environment, the array environment. These do have to be inside the display math delimiters. We also need to tell it how many columns we intend to have and how we want to align the columns. Consider the following example, \begin{verbatim} \[ A = \left[ \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right] \] \end{verbatim} \indent This produces the following matrix, \[ A = \left[ \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right] \] -3-

23 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit004.tex Tuesday, July 09, :58 PM Notice in the code that after \verb \begin{array} we have \verb {rrr}. This tells \LaTeX{} that we are going to use three columns and each will be aligned on the right. Notice in the resulting matrix the three columns line up on the right. Changing \verb {rrr} to \verb {ccc} will center the columns as in the example below. \[ A = \left[ \begin{array}{ccc} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right] \] \noindent Changing \verb {rrr} to \verb {ccc} will center the columns as in the example below. \[ A = \left[ \begin{array}{lll} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right] \] \noindent The alignment for any column is independent of the alignment of any other column, so changing the alignment to \verb {clr} is valid and produces, \[ A = \left[ \begin{array}{clr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right] \] As you noticed we did a left and right commands to place the brackets around the array, without them the array simply looks like, \[ \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \] We can also use other brackets, such as parentheses or vertical bars in the case of a determinant. We added a few other things in the example below as well. Notice the use of \verb \qquad, \verb \quad and \verb \qquad are two spacing commands you can use in math mode that put some space between items. The \verb \qquad is twice the width of the \verb \quad. These -4-

24 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit004.tex Tuesday, July 09, :58 PM come in handy when you want to put two different but related items in the same displayed math area, like we did below. Another new thing was the \verb \mbox{...}. The \verb \mbox{...} is used to write text inside math mode. Without the \verb \mbox{...} the ``so we have'' would be treated as mathematics, italicized and no spaces. \[ A = \left( \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right) \qquad \mbox{so we have} \qquad A = \left \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right = 14 \] \noindent Without \verb \mbox{...} the result does not look very good, \[ A = \left( \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right) \qquad so we have \qquad A = \left \begin{array}{rrr} 2 & 1 & 0 \\ -2 & 2 & 2 \\ 1 & -3 & 0 \\ \end{array} \right = 14 \] \end{document} -5-

25 Spring 2013 TEX-Bit #5 Math 406 A Little More Math Theorems As you can see we added the amsthm package and did some definitions in the preamble that set up Theorems, Lemmas, Propositions, Corollaries and Examples. These are environments that will automatically number themselves when you compose the document. We also made definitions, remarks and notations. Definition 1. A Rational number is a real number that can be written as a where a and b b are integers and b 0. Definition 2. An Irrational number is a real number that is not rational. Remark 1. Another equivalent way to define an Irrational number is one that cannot be written as a where a and b are integers. b Theorem 1. The 2 is irrational. Proof. By way of contradiction we will assume that 2 is rational. Then there must exist integers a and b such that 2 = a. Without loss of generality we may also assume that the b fraction a is in lowest terms, thus there is no common factor between a and b. Squaring b both sides of the equation 2 = a a2 we get 2 = and thus 2b 2 = a 2. So a 2 must be even b b 2 which implies that a is even, so 2 is a factor of a and we can write a = 2t for some t. Then 2b 2 = a 2 = 4t 2, giving b 2 = 2t 2. So b 2 must be even which implies that b is even, so 2 is a factor of b. But this implies that both a and b have a common factor of 2 which contradicts the assumption that a is in lowest terms. Therefore, 2 is irrational. b Corollary 1. The p is irrational for any prime number p. Proof. The proof is similar to the proof of theorem 1 and is left as an exercise. The next proof is my favorite proof. Theorem 2. There exists two irrational numbers a and b such that a b is a rational number. Proof. By theorem 1 we know that 2 is irrational. So either 2 2 is rational or it is irrational. If it is rational we are done and if it is irrational then ( 2 2 ) 2 = = 2 2 = 2 Remark 2. The reason I like theorem 2 is because it is a constructive proof and constructs the actual numbers that satisfy the theorem but you never know what those number actually are. 1

26 Definition 3. An irrational number is one that cannot be written as a where a and b are b integers. Theorem 3. The 2 is irrational. Proof. By way of contradiction we will assume that 2 is rational. Then there must exist integers a and b such that 2 = a. Without loss of generality we may also assume that the b fraction a is in lowest terms, thus there is no common factor between a and b. Squaring b both sides of the equation 2 = a a2 we get 2 = and thus 2b 2 = a 2. So a 2 must be even b b 2 which implies that a is even, so 2 is a factor of a and we can write a = 2t for some t. Then 2b 2 = a 2 = 4t 2, giving b 2 = 2t 2. So b 2 must be even which implies that b is even, so 2 is a factor of b. But this implies that both a and b have a common factor of 2 which contradicts the assumption that a is in lowest terms. Therefore, 2 is irrational. b Corollary 2. The p is irrational for any prime number p. Proof. The proof is similar to the proof of theorem 3 and is left as an exercise. The next proof is my favorite proof, again. Theorem 4. There exists two irrational numbers a and b such that a b is a rational number. Proof. By theorem 3 we know that 2 is irrational. So either 2 2 is rational or it is irrational. If it is rational we are done and if it is irrational then ( 2 ) 2 2 = = 2 = 2 Remark 3. Did you notice the use of labels and references in the theorems above? Remark 4. Labels and references are very convenient. Think about this, what if you changed the order of some of the theorems? The theorem numbers would change when you recompose the document and you would have to go back and change the numbering in the proofs. This is what you would have to do on the first page of this handout. But with the labels and references the renumbering happens automatically. Remark 5. When you use of labels and references you need to compose the document twice. The first time sets the theorem numbers and the second time puts them into the references. Notation. Notice that the Notation environment does not do any numbering. That is because of the * that we used after the \newtheorem command. 2

27 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit005.tex \documentclass[12pt]{article} \usepackage[pdftex]{graphicx} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsmath} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} Tuesday, July 09, :58 PM \usepackage{amsthm} %\newtheorem{thm}{theorem}[section] \newtheorem{thm}{theorem} \newtheorem{cor}{corollary} \newtheorem{lem}{lemma} \newtheorem{prop}{proposition} \newtheorem{ex}{example} \theoremstyle{definition} \newtheorem{defn}{definition} \theoremstyle{remark} \newtheorem{rem}{remark} \newtheorem*{notation}{notation} %\pagestyle{empty} \parskip=5pt \begin{document} \noindent Spring 2013 \hfill {\Huge\textbf{\TeX{}-Bit \#5}} \hfill Math 406 \vspace{20pt} \noindent\textsc{\large A Little More Math --- Theorems} \vspace{10pt} As you can see we added the \texttt{amsthm} package and did some definitions in the preamble that set up Theorems, Lemmas, Propositions, Corollaries and Examples. These are environments that will automatically number themselves when you compose the document. We also made definitions, remarks and notations. \begin{defn} A Rational number is a real number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. \end{defn} \begin{defn} An Irrational number is a real number that is not rational. \end{defn} \begin{rem} Another equivalent way to define an Irrational number is one that cannot be written as $\frac{a }{b}$ where $a$ and $b$ are integers. -1-

28 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit005.tex \end{rem} Tuesday, July 09, :58 PM \begin{thm} The $\sqrt{2}$ is irrational. \end{thm} \begin{proof} By way of contradiction we will assume that $\sqrt{2}$ is rational. Then there must exist integers $a$ and $b$ such that $\sqrt{2} = \frac{a}{b}$. Without loss of generality we may also assume that the fraction $\frac{a}{b}$ is in lowest terms, thus there is no common factor between $a$ and $b$. Squaring both sides of the equation $\sqrt{2} = \frac{a}{b}$ we get $2 = \frac{a^2}{b^2}$ and thus $2b^2 = a^2$. So $a^2$ must be even which implies that $a$ is even, so 2 is a factor of $a$ and we can write $a = 2t$ for some $t$. Then $2b^2 = a^2 = 4t^2$, giving $b^2 = 2t^2$. So $b^2$ must be even which implies that $b$ is even, so 2 is a factor of $ b$. But this implies that both $a$ and $b$ have a common factor of 2 which contradicts the assumption that $\frac{a}{b}$ is in lowest terms. Therefore, $\sqrt{2}$ is irrational. \end{proof} \begin{cor} The $\sqrt{p}$ is irrational for any prime number $p$. \end{cor} \begin{proof} The proof is similar to the proof of theorem 1 and is left as an exercise. \end{proof} The next proof is my favorite proof. \begin{thm} There exists two irrational numbers $a$ and $b$ such that $a^b$ is a rational number. \end{thm} \begin{proof} By theorem 1 we know that $\sqrt{2}$ is irrational. So either $\sqrt{2}^{\sqrt{2}}$ is rational or it is irrational. If it is rational we are done and if it is irrational then \[ \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = \sqrt{2}^{\sqrt{2}\sqrt{2}} = \sqrt{2}^2 = 2 \] \end{proof} \begin{rem} The reason I like theorem 2 is because it is a constructive proof and constructs the actual numbers that satisfy the theorem but you never know what those number actually are. \end{rem} \newpage \begin{defn} An irrational number is one that cannot be written as $\frac{a}{b}$ where $a$ and $b$ are integers. \end{defn} \begin{thm} -2-

29 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit005.tex \label{sqrt2irr} The $\sqrt{2}$ is irrational. \end{thm} Tuesday, July 09, :58 PM \begin{proof} By way of contradiction we will assume that $\sqrt{2}$ is rational. Then there must exist integers $a$ and $b$ such that $\sqrt{2} = \frac{a}{b}$. Without loss of generality we may also assume that the fraction $\frac{a}{b}$ is in lowest terms, thus there is no common factor between $a$ and $b$. Squaring both sides of the equation $\sqrt{2} = \frac{a}{b}$ we get $2 = \frac{a^2}{b^2}$ and thus $2b^2 = a^2$. So $a^2$ must be even which implies that $a$ is even, so 2 is a factor of $a$ and we can write $a = 2t$ for some $t$. Then $2b^2 = a^2 = 4t^2$, giving $b^2 = 2t^2$. So $b^2$ must be even which implies that $b$ is even, so 2 is a factor of $ b$. But this implies that both $a$ and $b$ have a common factor of 2 which contradicts the assumption that $\frac{a}{b}$ is in lowest terms. Therefore, $\sqrt{2}$ is irrational. \end{proof} \begin{cor} The $\sqrt{p}$ is irrational for any prime number $p$. \end{cor} \begin{proof} The proof is similar to the proof of theorem \ref{sqrt2irr} and is left as an exercise. \end{proof} The next proof is my favorite proof, again. \begin{thm} \label{irrtoirrisrat} There exists two irrational numbers $a$ and $b$ such that $a^b$ is a rational number. \end{thm} \begin{proof} By theorem \ref{sqrt2irr} we know that $\sqrt{2}$ is irrational. So either $\sqrt{2}^{\sqrt{2}}$ is rational or it is irrational. If it is rational we are done and if it is irrational then \[ \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} = \sqrt{2}^{\sqrt{2}\sqrt{2}} = \sqrt{2}^2 = 2 \] \end{proof} \begin{rem} Did you notice the use of labels and references in the theorems above? \end{rem} \begin{rem} Labels and references are very convenient. Think about this, what if you changed the order of some of the theorems? The theorem numbers would change when you recompose the document and you would have to go back and change the numbering in the proofs. This is what you would have to do on the first page of this handout. But with the labels and references the renumbering happens automatically. \end{rem} \begin{rem} When you use of labels and references you need to compose the document twice. The first time -3-

30 C:\Users\despickler\LaTeX\Classes\Math406\TeXBit005.tex sets the theorem numbers and the second time puts them into the references. \end{rem} Tuesday, July 09, :58 PM \begin{notation} Notice that the Notation environment does not do any numbering. That is because of the * that we used after the \verb \newtheorem command. \end{notation} \end{document} -4-

31 Spring 2013 TEX-Bit #6 Math 406 Including Graphics Including graphics into L A TEX is not as easy as including graphics into other word processing systems, like Word. Although it requires a little doing, the graphics that are produced are of the highest quality. One thing you will notice about L A TEX graphics is that the image is well preserved when it is scaled up or down. This is not always the case with Word. There are several types of graphics files than can be loaded into a L A TEX document. The most common formats used today are jpeg and png. The type of graphics you load into the document will depend on what you are showing. If the image is a photograph then a jpeg is probably the best. On the other hand, if you are showing a chart or a diagram you would probably want to save the image as a png. To include a graphics file in your L A TEX document make sure that you have the \usepackage[pdftex]{graphicx} in the preamble of your document. Also make sure that the image file you are inserting is in the same folder as the tex document. Now to include a graphics file you just need the \includegraphics command. For example the code \begin{center} \includegraphics[scale=0.9]{texbit6pic001.png} \end{center} produces. 1

32 In the command \includegraphics[scale=0.9]{texbit6pic001.png} the graphics file is texbit6pic001.png the [scale=0.9] is an option that scales the image by 0.9, that is 90% of its original size and maintains the images aspect ratio. The same images stored as a jpg file would be loaded in with \begin{center} \includegraphics[scale=0.4]{texbit6pic001.jpg} \end{center} and produces. Note that the scale factor is different, different image files store sizes differently so the same image under two different formats may require different scales. There are many other options that can go inside the [ ] but scaling is probably the most common. L A TEX has a very active following and several communities devoted to the language. TUG (TEX Users Group) is one of them. If you have a question on how to formulate something in L A TEX, chances are the same question has already been asked and a quick Google search will bring up numerous forums with answers. I have found a lot of nifty tricks and packages this way that I never knew existed. 2