Typesetting Mathematics. Dr. C. F. Rocca Jr.

Transcription

1 Typesetting Mathematics Dr. C. F. Rocca Jr. September 16, 2010

2 2 Typesetting Mathematics General Notes On Typing Up Math: What follows is largely based on my own experience and observations. Another good source for information on mathematical writing can be found at [9] or at the more extensive [6]. When you are typing a piece of mathematics there are two ways to enter an equation, inline or displayed. An inline equation is typed right along with the text of a sentence as if it were part of the text. This is what you want to do for smaller less significant equations for example when you assume f(x) = x 2 4x + 3 you type this inline. Now if you want to show something more significant or more lengthy then you want to display it centered on its own line. For example if we factor our quadratic f(x) = x 2 4x + 3 = (x 1)(x 3) or use the quadratic formula x = 4 ± ( 4) = 4 ± 4 2 = 1 or 3. When you including a graphic you can choose to have it displayed centered on a line of its own or to have text written beside it. Which you choose may depend on what you are doing with the graphic. In this example the graphic needs to be large so that there is room for the students to work so it makes sense for it to be on its own. (1) Using the given graph of a function identify where the derivative of the function is zero, positive, and negative and then graph the derivative on the same plot.

3 C.F. Rocca Jr. 3 If however you only need to make reference to the image you may decide to make it smaller and place the text beside the image like in this problem. (2) Answer the following questions about the given graph. At which points does a limit exist? 2. is the function defined? 3. is the function continuous? 4. is the function differentiable? However even in this situation it is simpler and perhaps easier to read if you still just center the image on its own line. Lists, both bulleted and numbered, occur a lot in mathematical writing. They can be used to help strive for clarity if you are making a list of points

4 4 Typesetting Mathematics or to enumerate a number of different exercises or different answers for a multiple choice question. Depending on the use you need to decide if you want a bulleted list or enumerated and if you want a single column or multiple columns. It is my personal opinion that you should find a reasonable system that you are comfortable with and that is clear and then stick with that system. Clear consistency is better that inconsistency even if it is not a formally correct format. When choosing between bulleted lists versus numbered lists I normally ask myself if the order really matters. If, as with the objectives I have listed above, the order is secondary to the information being conveyed and if the information could be written out in a comma separated list then I choose to use bullets. However, for a list of exercises on a worksheet, quiz, or test order is crucial and so a numbered list is appropriate. And, lastly, you should only split lists into columns in particular cases such as when: you are listing multiple short answers to a single question; you are listing multiple short questions where the work will be done elsewhere; you are listing multiple short questions with similarly short answers. Finally you should keep in mind that mathematics is a language of its own and as such has certain conventions and special notations that have evolved to help make communication clear and consistent. For example when discussing number systems you should use the conventional symbols N, Z, Q, R, and C for the natural, integer, rational, real, and complex numbers. Further there are conventions like the following that re not at all hard and fast rules but are generally accepted as a sort of default, a, b, c, d are constants, x, y, z are variables, i, j, m, n are integers, f, g, h, s, v are function names, e is reserved for the base of the natural log, r is mostly used for radius or rate, t is a variable generally reserved for time, p, q are normally used for probabilities or primes depending on context,

5 C.F. Rocca Jr. 5 o is not often used since it is to easily confused with 0, k, l, u, w get used for various purposes though l should be avoided if there is a chance of confusion with 1, α, β, γ, θ, φ are frequently, though not exclusively, used for angles, and π is reserved for the constant equal to the ratio of a circles circumference to its diameter. In the next three sections we will look at the technical nuts and bolts of accomplishing some of what we have been discussing here. In particular we will for Microsoft Word, Open Office, and L A TEXlook at how we can input equations, images, lists, columns, and special symbols.

6 6 Typesetting Mathematics L A TEX Latex is a programing language designed from the ground up for typesetting mathematics and science. The basics of Latex are relatively straight forward but to really learn it can take time. While it does not have the nice point and click interface of word processors it does produce a crisper more professional looking document especially with mathematics. Also, when you put something some place, like an image, it stays where you put it and if you number a list in a particular way it stays numbered in that way. In other words you get a great looking document and you are in control of the layout. Add to this that Latex is free and it starts to be worth the extra work to learn how to use it. In the next few sections we will look at just a few of the things you can do in latex and how to do them. Keep in mind that Latex is a programing language and syntax is important. Each document should have the basic structure \documentclass[12pt]{report} [HEADER] \begin{document} [BODY] \end{document} The header is where we can put formatting commands and commands to tell Latex what we need and the body is where most of our text will go. So let s look at some of what we can do. Entering Equations: Inline equations are entered by placing the equation between a set of dollar signs like so: Let us begin by assuming that ɛ > 0 and that f(x) = x 2. Let us begin by assuming that $\epsilon>0$ and that $f(x)=x^2$.

7 C.F. Rocca Jr. 7 Notice that to typeset the greek letter ɛ we just type out its name with a slash before it, this is true of all the greek letters. Furthermore if we want a capital greek letter such as Γ instead of lowercase γ you type the name capitalized, $\Gamma$. Also notice that to get the exponent for the x 2 we used a carrot just as you would on a calculator. Many of the mathematical symbols or expressions you might want to use follow similarly logical patterns. Let s look now at an example of typesetting a displayed equation which is done similarly to inline equations except it is typed in between a \[ and \]. According to the Fundamental Theorem of Arithmetic if f(x) is continuous on the interval [a, b] and a x b then F (x) = x a f(t)dt is an antiderivative of f(x). That is According to the Fundamental Theorem of Arithmetic if $f(x)$ is continuous on the interval $[a,b]$ and $a\leq x\leq b$ then \[F(x)=\int_a^x f(t) dt\] is an antiderivative of $f(x)$. That is \[\frac{df}{dx}=f(x).\] df dx = f(x). Finally, it is sometimes necessary to have an entire series of equalities stacked on top of each other like when we are solving for a variable or proving an equality. Consider the following example from trigonometry. sin(3θ) sin(θ) = sin(2θ) cos(θ) + cos(2θ) sin(θ) sin(θ) = 2 sin(θ) cos 2 (θ) + cos(2θ) sin(θ) sin(θ) = cos(2θ) sin(θ) + sin(θ)(2 cos 2 (θ) 1) = cos(2θ) sin(θ) + cos(2θ) sin(θ) = 2 cos(2θ) sin(θ) Which we type like this: \begin{eqnarray*} \sin(3\theta)-\sin(\theta)

8 8 Typesetting Mathematics &=&\sin(2\theta)\cos(\theta)+\cos(2\theta)\sin(\theta)-\sin(\theta)\\ &=&2\sin(\theta)\cos^2(\theta)+\cos(2\theta)\sin(\theta)-\sin(\theta)\\ &=&\cos(2\theta)\sin(\theta)+\sin(\theta)(2\cos^2(\theta)-1)\\ &=&\cos(2\theta)\sin(\theta)+\cos(2\theta)\sin(\theta)\\ &=&2\cos(2\theta)\sin(\theta)\\ \end{eqnarray*} In what is above the &=& tells Latex where to line things up and the \\ tells it where to end the line. The * after the word eqnarray turns off line numbering, with out it our equations would have looked like this: sin(3θ) sin(θ) = sin(2θ) cos(θ) + cos(2θ) sin(θ) sin(θ) (1) = 2 sin(θ) cos 2 (θ) + cos(2θ) sin(θ) sin(θ) (2) = cos(2θ) sin(θ) + sin(θ)(2 cos 2 (θ) 1) (3) = cos(2θ) sin(θ) + cos(2θ) sin(θ) (4) = 2 cos(2θ) sin(θ) (5) (6) Including Images: Now suppose that we wish to include a graphic to help illustrate a point such as We can approximate the value of 1 2 x 2 + 3dx by using either right boxes or left boxes as we see on this graph

9 C.F. Rocca Jr. 9 Then that is typeset in Latex as We can approximate the value of \[\int_{-2}^1 x^2+3 dx\] by using either right boxes or left boxes as we see on this graph \begin{center} \resizebox{3in}{!}{\includegraphics{../images/riemannsum6boxes2.jpg}} \end{center} Notice that the lower limit of integration 2 is inside a set of braces, if we did not do this then only the negative symbol would have gone into the subscript. Also in the command to resize the picture the 3in adjusts the width of the picture while the! tells Latex to adjust the height so that the picture is not distorted. Making Lists: Now suppose that you need to give students, or anyone, a list of steps of directions. If the order of the steps matter then we might want to write:

10 10 Typesetting Mathematics Getting Dressed 1. Pick out clothes. \begin{center} Getting Dressed \end{center} 2. Put on under clothes. 3. Put on your pants and shirt. 4. Put on your socks and belt. 5. Put on your shoes. \begin{enumerate} \item Pick out clothes. \item Put on under clothes. \item Put on your pants and shirt. \item Put on your socks and belt. \item Put on your shoes. \end{enumerate} Now if the order of the list does not matter then you may instead want to use a bulleted list like this: Milk Shopping List Orange juice Bread (Seven Grain) Cheese Eggs \begin{center} Shopping List \end{center} \begin{itemize} \item Milk \item Orange juice \item Bread (Seven Grain) \item Cheese \item Eggs \end{itemize} If you are making up a multiple choice test then you may end up with a list inside a list (one for the questions and one for the possible answers) and you might want to put the possible answers into multiple columns. This is accomplished like so: 1. This theorem Theorem: If a, p Z +, gcd(a, p) = 1, and p is prime then a p 1 1 (mod p). is known as (a) Wile s Theorem (b) Fermat s Little Theorem (c) Euler s Theorem (d) Zimmermann s Theorem

11 C.F. Rocca Jr. 11 Which is typeset as: \begin{enumerate} \item This theorem \textbf{theorem:} \textit{if $a,p\in\mathbb{z^+}$, $gcd(a,p)=1$, and $p$ is prime then $a^{p-1}\equiv 1\pmod{p}$.} is known as \begin{multicols}{2} \begin{enumerate} \item Wile s Theorem \item Fermat s Little Theorem \item Euler s Theorem \item Zimmermann s Theorem \end{enumerate} \end{multicols} \end{enumerate} There are a few things that we should point out in this example. In this example, as in some of the previous examples, we use a \begin{...} and \end{...} in order to create particular structures like columns or lists, these are all called environments. In particular enumerate is used for numbered lists, itemize is used for bulleted list, and multicols for multiple columns. For the multiple columns there is an additional {2} after the begin command which tells Latex how many columns to use. Also, to use multiple columns in this way we need to put \usepackage{multicol} in the header of the document to let Latex know what we want to do. Finally, notice that with the nested list environments Latex automatically switched from numbering with numbers to numbering with letters so that we did not have to do it ourselves. If we had wanted to enumerate a list item in a special way we would type \item[widget] where widget was what we wanted in front of the list item. Creating Tables: Sometimes you ll need tables to either present information or for students to fill in as part of a worksheet. Is accomplished in Latex using the tabular

12 12 Typesetting Mathematics environment. Here is an example. Type Alcohol Cal/12 oz Carbs Porter 5.6% g I.P.A. 6.9% g Light 4.2% g Stout 5.8% g Which is typeset like this. \begin{tabular}{l c c c} Type & Alcohol & Cal/12 oz & Carbs \\ \hline Porter & 5.6\% & 209 & 18.4g \\ I.P.A. & 6.9\% & 231 & 20g \\ Light & 4.2\% & 110 & 6.6g \\ Stout & 5.8\% & 225 & 22.3g \\ \end{tabular} There are a number of important components here. First after the \begin{tabular} we see {l c c c} this sequence of letters and vertical lines (called pipes) tells Latex how many columns you want, one for each letter, how to align them, l for left, c for center, and r for right, and finally where to draw vertical lines between columns, wherever there is a pipe between the letters. Next, the & indicates when in a row we are done typing material for a column, while the \\ marks the end of each row. Finally, after any row where we want a horizontal line we type \hline. Other: Symbols and Hyperlinks: In the example where we wanted multiple columns where we wanted multiple columns for our list we had to place \usepackage{multicol} in the header of the document, what this did was load a special set of commands for formating columns. Such sets of commands are called packages. There are many different packages in Latex for a variety of purposes, let s look at a couple that can be used for symbols and one that can be used for creating hyperlinks. Many, if not most, of the symbols we use in mathematics are built in so that we don t need a special package.

13 C.F. Rocca Jr. 13 For example all the greek letters α, β, γ, π, δ... are built in as are such symbols as ±,,,,,,,, and many others. For example all the greek letters $\alpha,\beta,\gamma,\pi,\delta \ldots$ are built in as are such symbols as $\pm, \div, \leq, \geq, \equiv, \Rightarrow, \Leftarrow, \forall,\angle$ and many others. But here are some other packages that you might consider using and what symbols they can produce (for a more comprehensive list see [15]): amsfonts loaded with \usepackage{amsfonts} N $\mathbb{n}$ Z $\mathbb{z}$ Q $\mathbb{q}$ R $\mathbb{r}$ C $\mathbb{c}$ ifsym loaded with \usepackage[misc]{ifsym} \Cube{1} \Cube{2} :\StrokeOne ::\StrokeTwo \Cube{3} \Cube{4} :::\StrokeThree ::::\StrokeFour \Cube{5} \Cube{6} ;\StrokeFive hieroglf loaded with \usepackage{hieroglf} \textpmhg{\hone} 2 \textpmhg{\hten} 3 \textpmhg{\hhundred} 4 \textpmhg{\hthousand} 5 \textpmhg{\hxthousand} 6 \textpmhg{\hcthousand} 7 \textpmhg{\hmillion} We can hyperlink elements of a document such as the table of contents, indices, and bibliographies and place hyperlinks into a document using the hyperref package (see [5]) which is loaded by placing \usepackage[linkcolor=blue,citecolor=blue,anchorcolor=blue, urlcolor=blue,colorlinks=true]{hyperref}

14 14 Typesetting Mathematics in the header (note that it should all be on one line). Then much of the hyperlinking is done automatically while we can place url s in the document with the command \href{ Rocca s Home Page} Which produces Dr. Rocca s Home Page. Or with the command \url{ Which would give us What we have seen here is just a taste of what L A TEX can do for more detail you should look at [7] and [12].

15 C.F. Rocca Jr. 15 Microsoft Word Microsoft Word, and Microsoft Office in general, is exceptionally common. It is unlikely that you want have cause to use it at some point in you career. It is a general purpose word processor though and as such does not do as nice a job as Latex which was designed specifically for writing up mathematics. Also it is not cheap to purchase and so that is a mark against it when compared to both Latex and Open Office which we will look at later. Entering Equations: To insert an equation (or just about anything) in Word you need to go to the Insert Tab next to the Home Tab. Then on the far left you can click on the menu for entering an equation and choose either on of the predefined equations or a new equation. Once you start editing your equation you will see at the top of the screen you get a new Equation Tools Design Tab. which includes palettes with all of the most commonly used symbols and operations. Most of what you want to enter you will need to find in the appropriate palette but some really basic operations such as exponents and fractions can

16 16 Typesetting Mathematics be entered by hand as x^2 and x/y. When you enter work by hand it will not initially look pretty to get it to you click on the down arrow in the bottom right corner of the box containing your equation which produces a drop down menu where you can select a professional look for your equation. That drop down menu is also where you can choose between having an inline or displayed equation. In order to get a stacked equation like so you will need to insert a matrix that has 3 columns and as many rows as you think you will need for you equation. The middle column in each row will then have an equal sign while the right and left columns will contain the right hand and left hand equations. The empty little boxes that you see in the above figure are there when you are working on your document but will not be there when you print the final version. Including Images, Creating Tables, Symbols, & Hyperlinks: Equations are not the only reason to visit the insert tab, it is also the place to go for images (or pictures), tables, symbols and hyperlinks. When including images you can choose to insert a piece of clip art by clicking the Clip Art button or a picture from a file by clicking the Picture

17 C.F. Rocca Jr. 17 button and then in each case choosing the desired file. Alternately, if you are viewing a graphic in another program you can use copy and paste to enter it into Word. However you insert the image you will likely want to adjust it in some way once you have it in Word. When you click on the image you can adjust it using options given to you along the top in the menu bar or by right clicking and choosing the option to look at the images properties in which case a new window opens where yo can adjust the image as you please. Note that if you just need to adjust the size of the images you can click and drag the little squares that appear around the image after you selected it, but if you do not want it to get distorted then you need to hold down the shift key as you do so. To insert a table you, predictably, click on the Table button and selecting what size table you want. If you are unsure of what size table you need that is fine. After inserting a table you can add more rows by highlighting a row above or below where you want the new row, right clicking on the row you highlighted and choosing Insert Rows. Inserting new columns works the same way. For symbols you place the cursor at the spot in the document where you want the symbol, press the Symbol button on the insert tab then choose the symbol you want. There are a lot of symbols so you should make note of where to find the symbols you most often need. Also, if there is one symbol you use repeatedly within a document then using copy and paste to copy it from one spot to the next will be most convenient. To insert hyperlinks you should type in the text you wish to have be your link, highlight this with the cursor, and then press the Hyperlink button. A new window will now pop up where you can enter what you want to link to. If you are linking to a web site it is easiest to copy and pasted the web address from your web browser. You can also link to other files on your hard drive by entering there location or browsing for them in the window that pops up. Making Lists & Columns: Let s briefly touch on making lists and making columns. These are not made from the insert tab. For lists you need to go to the Home tab

18 18 Typesetting Mathematics and for columns you click on the Page Layout tab When you make a list you can choose from either a bulleted list or numbered list, each with a variety of styles. To start a list move your cursor to a new line where you want the list and then click the little icon in the paragraph pane on the home tab which looks like the sort of list you want. Then you just type/enter what you want for the first item. When you are done with one line in your list hit enter and word will give you another line automatically. When you are done hit enter to get a new line then click on the list icon to exit list mode. To get a list within a list, like a outline or a set of possible answers to a multiple choice question,

19 C.F. Rocca Jr. 19 then you start a new line and click the Indent button word will indent the list and change the numbering or symbol as needed. To place your list in columns as in the above example, or to place any text in columns for that matter, you highlight the text you want in columns and then on the Page Layout tab select the columns button and choose the number of columns to break it into. As with the introduction to L A TEX above this introduction to Word of course does not demonstrate all of its capabilities. Hopefully though this will give you enough information to get you started with some basics. Really learning how much Word or any program can do for you takes time and practice.

20 20 Typesetting Mathematics Open Office The last method we will look at for typing up mathematics is a program called Open Office [14] which is a complete freeware office suite much like Microsoft Office. Like Microsoft s office suite Open Office has a word processor, spread sheet, presentation program (like Power Point), a drawing program, and database program. Unlike Microsoft s software Open Office is free, but it is also a little rougher around the edges, though very similar. Entering Equations: To enter equations in Open Office you need to insert them as you do in Word. Unlike the most recent version of Word though Open Office uses menus and tool bars like the older versions of Word and like Word of Mac. To insert an equation you need to choose Insert Object Formula

21 C.F. Rocca Jr. 21 This creates a box into which the formula will be placed, gives a specific tool bar an symbol palette for creating formulas, and opens an editing window at the bottom of the screen where you edit the formula. Because the equation is in a box like that it is treated like an image and so you can not really typeset proper inline equations they are all displayed. However since many if not most equations should be displayed for the sake of legibility this is not a huge loss. What is also nice it that there are palettes with commands so that you don t have to remember them but if you do remember them you can just type them in at the bottom in the editing window. Including Images, Lists, Tables, Symbols, & Links: All the other features that we have been discussing are accomplished in Open Office in much the same way that they are accomplished in Word once you know where to look. Special Characters and Images are inserted using items off the Insert menu

22 22 Typesetting Mathematics like we did for inserting a formula. Note that you can, as in Word, also insert images using copy and paste. For lists, tables, and links the directions are identical to those for Word. Even the icons on the menu bar are almost identical to those used in Word. And, lastly columns are accomplished in much the same way by choosing columns off the Page Layout menu.

23 C.F. Rocca Jr. 23 Note that all three programs do allow you to save/export your documents as pdf s which is convenient if you wish to make an electronic copy available to students. For Latex, or recent versions of Latex, pdf is the default format, for Word you have the option of saving as pdf, and for Open Office you can choose to export as a pdf.

24 24 Typesetting Mathematics So, there we have three ways to typeset mathematics each with its own advantages and disadvantages. Latex produces a very attractive and professional finished product. Additionally you retain control over pretty much every element of the document you are creating and so if you format things in a particular way they stay formatted that way. However Latex is a programing language that can be hard to learn so to use it you need to be willing to work a little. Word is a highly polished program with a nice user interface if you can remember where all the buttons are located. Word also handles a great deal of the formatting for you whether you ant it to or not. For example list numbering and its attempts to automate the process for you can be particularly frustrating and unstable. Also, the general look and feel of the equations in Word, while nice, is not as crisp as what you get in Latex and it can get frustrating to have to spend your time hunting through menu after menu to find a symbol that you need. Open Office has many of the advantages of Word with respect to having an easy user interface though its equations are even less crisp. One of the big advantages that Open Office is that it is free and effectively as functional as Word which is not. Another advantage, at least from my perspective, is that after you have used a symbol once while entering a formula you can learn the text command so the next time you can just type it in without having to sift through menus. As we move forward we will see that a lot of these comments will not only apply to other types of software. That is we will see that basic software does not produce as nice a product or have as much versatility as more advanced software and commercial software is more polished than none commercial but can be very expensive.

25 25 Word Open Office Latex Equations Insert Symbols Equation Insert Object Formula then View Formula Elements Type either \[ or $ to open a math environment type your equation then either \] or $ to close the equation Symbols Insert Symbols More Symbols Insert Special Characters Load the appropriate package, if a special one is required, and type the command Images Insert Picture Insert Picture From File Place \usepackage{graphics} in your header then enter the picture where you want it with \resizebox{width}{!} - {\includegraphics{image}} Lists Columns Tables Hyperlinks Home Press the Bullets or the Numbering Icon First highlight the portion of the document you want in columns then go to Page Layout Columns Place the cursor where you want the table then click the table icon, on the insert tab, and choose the dimensions of your table. Highlight the text that will become the hyperlink, click on the insert tab, then on the hyperlink icon, and then follow directions Format Bullets and Numbering First highlight the portion of the document you want in columns then go to Format Columns Place the cursor where you want the table then click the table icon and choose the dimensions of your table. Highlight the text that will become the hyperlink, then on the hyperlink icon, and then follow directions Enclose the list in either and itemize or enumerate environment with a \item before each list item Place \usepackage{multicol} in your header then place what you want in columns in between \begin{multicols}{# of Cols} and \end{multicols} Use the tabular environment see page 11 for details Use the hyperref package see page 13 for details PDF s Menu Icon Save As PDF or XPS File Export as PDF Typically compiles directly as a PDF

26 Bibliography [1] Al Cuoco, Mathematical Connections: A Companion for Teachers and Others. Mathematical Association of America, Newton MA, [2] GeoGebra, June [3] The Geometer s Sketchpad Resource Center, dynamicgeometry.com/, June [4] Michael Goossens, Frank Mittelbach, Alexander Samarin, The L A TEX Companion. Addison-Wesley, Boston MA, [5] Hypertext marks in L A TEX : a manual for hyperref, org/applications/hyperref/ftp/doc/manual.html, June [6] Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts, Mathematical Writing, pdf, June [7] Leslie Lamport, L A TEX A document Preparation System: User s Guide and Reference Manual. Addison-Wesley, Boston MA, [8] L A TEX project: L A TEX - A Document Preparation System, June [9] Kevin P. Lee, Mathematical Writing: Tips and Guidelines,,http: //ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/, June [10] Math Software for Engineers, Educators & Students Maplesoft, June [11] MiKTEX Project Page, June

27 C.F. Rocca Jr. 27 [12] The Not So Short Introduction to L A TEX2ε, tex-archive/info/lshort/english/lshort.pdf, June [13] Office - Microsoft Office, June [14] OpenOffice.org - The Free and Open Productivity Suite, openoffice.org/, June [15] Scott Pakin, The Comprehensive L A TEX Symbol List http: // symbols-letter.pdf, June [16] Sage: Open Source Mathematics Software, org/, June [17] Texmaker (free cross-platform latex editor), net/texmaker/, June [18] TEXShop, June [19] Wolfram Alpha - Computational Knowledge Engine, wolframalpha.com/, June 2010.