% This is a LaTeX file. LaTeX is a language which can be used to typeset % documents similar to those one might get from e.g. OpenOffice or Microsoft % Word. The difference is that, while those are "WYSIWYG" ("what you see is % what you get") tools, LaTeX is a markup language. This has advantages and % disadvantages, but one of the major advantages is in writing documents that % involve a lot of symbols or patterns. You are not required to use this file % (or LaTeX in general) to complete your work, but we expect it will help. :) % First and foremost: LaTeX files must be compiled like C++ programs. In order % to compile your LaTeX file, run % % pdflatex LearningLaTeX.tex % % This will produce a file LearningLaTeX.pdf which you may then view with a PDF % viewer: % % evince LearningLaTeX.pdf % % Remember that you must recompile your document if you change the .tex file! % The rest of this document will contain some examples of how to use LaTeX. As % you may have guessed, the '%' character is an end-of-line comment marker, much % like // in C++ or # in Python. Keep reading! % The following command is typically the first in any document. It says that % we want the PDF to be in an article format. It's a sensible default for % simple documents. Commands in LaTeX start with a backslash. Some commands % take arguments, each of which is surrounded by a pair of braces. So, in % LaTeX, % % \documentclass{article} % % is sort of writing this in C++: % % documentclass("article"); \documentclass{article} % Next, we need to bring in common commands, just like we would in Python or % C++. Rather than "import" or "#include", we use the "\usepackage" command. \usepackage{amsmath} % Math notation from the American Mathematical Society. \usepackage{amssymb} % Typographical symbols from the same. % In addition to commands, LaTeX has an idea of "environments". These are sort % of like indentation in Python or the { ... } notation in C++, but they differ % in what they do. The most important environment is called "document" and % everything that appears in your document must be inside of it. We start an % environment with the "begin" command and stop it with the "end" command. % Although it is not necessary, we typically indent the contents of the % environment to make the .tex file easier to read. \begin{document} % Everything we've seen so far has been a comment or a command. If I want % text, all I have to do is put it inside of this document environment. Hello! % Extra spaces in LaTeX don't matter, so that "Hello!" won't be indented. % Instead, it will be in a paragraph. In fact, newlines are treated like % spaces, so the following is on the same line. How are you? % If you want a new paragraph, you insert a blank line. Since the above % line is blank, the following will be in its own paragraph. This is a new paragraph. % Basic text formatting is pretty simple in LaTeX. The \textbf{...} command makes % boldfaced text; the \emph{...} is used to emphasize text; this usually means % making the text italic, unless the surrounding text is already italicized % (in which case, it usually un-italicizes the emphasized text). There is % also \textit{...} which italicizes text always; it does not do anything % different if the surrounding text is already italicized. \textbf{This is bold.} \emph{This is emphasized.} This is plain text. % If you have a good editor, the above might even be rendered to hint at the % font changes. % So far, everything we have written has been in "paragraph mode". It is % also possible to write LaTeX in "math mode". One way of entering math mode % is to write a dollar sign. Another dollar sign leaves math mode. This kind % of math mode is "in-line" and can be written alongside text. Here's something in paragraph mode: 4+2x. Here it is in math mode: $4+2x$. % Math mode is also important because it allows you to use the mathematical % notation from the AMS packages that we named above. For instance, the % command \dots draws a sequence of dots, like an ellipse. Here is an example of using dots in an equation: $1+2+\dots+10 = 55$ % In math mode (but not in paragraph mode), the symbols ^ and _ have special % meaning: they superscript or subscript the very next thing. For instance, % the following produces the letter "x" with a subscripted "i" and a % superscripted "10". Here is an example of subscripts and superscripts: $x_{i}^{10}$ % Notice the use of braces {...}, which indicates that everything in between % the braces should be treated as a one thing, so that everything inside the % braces appears as part of the sub- or superscript. Without them, only the % first symbol would be in the sub- or superscript. In the above example, % the superscript would be '1' instead of '10', and the '0' would appear % outside the superscript: Without braces around the subscript and superscript, only the first symbol appears in the sub- or superscript: $x_i^10$ % To avoid these awkward formatting issues, it's good practice to always % include the braces. % Let's consider using those subscripts for another purpose. We might want % to write a summation; conveniently, this is possible with the \sum command. % Fractions can be written using \frac, which takes two arguments: the % numerator and the denomenator. The following, for instance, is a common % summation: Here is an example of a summation: $\sum_{i=1}^{n} i = 1 + 2 + \dots + n = \frac{n(n+1)}{2}$ % You might notice that the limits on the summation look a little cramped. % That's because LaTeX is trying to squeeze it to fit in-line. We can % instead tell LaTeX to display the math on its own line using \[ and \]. % This is appropriately called "display math mode", and it allows LaTeX to % use more vertical space to make the math look nicer. Here is a summation in display math mode: \[ \sum_{i=1}^{n} i = 1 + 2 + \dots + n = \frac{n(n+1)}{2} \] % Notice that it no longer appears in-line, and latex places the equation on % its own line, with generous vertical space to allow for nicer formatting. % You can also place the \[ and \] on their own line, if you'd like (as % shown in the next example). % For Big-O proofs, we'll also want to be able to write the logical % quantifiers "for all" and "exists" (the upside-down A and the backwards % E). Conveniently, these are accessible via the \forall and \exists % commands. Here is an example using logical quantifiers: \[ \forall x>0, \exists y<0, x+y = 0 \] % We can also get comparison symbols such as "greater than or equal to" if % we know the right name. $x \geq y \leq z$ % Finally, we'll want to be able to write out our proofs in an organized % fashion. We may want to label equations so that we can refer to them in % our text. We can do this using the 'equation' environment: \begin{equation} \label{eq:pythagoras} a^{2} + b^{2} = c^{2} \end{equation} % And we can refer to it in our proofs with \eqref, as shown in the example % below. When compiled, this will automatically insert the equation number. % In this example, ~ tells LaTeX to use a "nonbreaking space", meaning it % won't end the line at the space. This is typically recommended when % referencing equations. Suppose a triangle has a leg of length $x$ and a hypotenuse of length $2x$. Then, by~\eqref{eq:pythagoras}, the other leg has length $\sqrt{(2x)^{2} - x^{2}} = \sqrt{3x^2} = \sqrt{3}\cdot x$. % If we want to show multiple steps simplifying the same equation, we can % use the 'split' environment inside of 'equation'. The whole derivation % will get a single equation number. In a 'split' environment, \\ is used to % start a new line, and & is used to tell LaTeX where to align multiple % lines. % % Also helpful: We can refer to an equation before it appears, as shown % below. As shown in~\eqref{eq:poly-mult} below, $(2x+3)(x^{2} - x)$ is a polynomial of degree three. \begin{equation} \label{eq:poly-mult} \begin{split} (2x + 3)(x^{2} - x) &= 2x(x^{2}) + 3(x^{2}) - 2x(x) - 3(x) \\ &= 2x^{3} + 3x^{2} - 2x^{2} - 3x \\ &= 2x^{3} + x^{2} - 3x \end{split} \end{equation} % If we want the benefits of the 'split' environment without an equation % number, AMS Math also provides the starred environments such as 'equation*': \begin{equation*} \begin{split} (2x + 3)(x^{2} - x) &= 2x(x^{2}) + 3(x^{2}) - 2x(x) - 3(x) \\ &= 2x^{3} + 3x^{2} - 2x^{2} - 3x \\ &= 2x^{3} + x^{2} - 3x \end{split} \end{equation*} % Sometimes we might want to show a multi-step calculation that involves % multiple equations. There is the 'align' environment for this, which works % similarly to 'equation'+'split', except that each line receives a separate % equation number: \begin{align} % As with 'split', 'align' uses \\ to start a new line and & to align % lines. For instance, we can derive the geometric series: S &= \sum_{k=0}^{\infty} ar^{k} \\ S &= a + ar + ar^{2} + ar^{3} + \dots \\ rS &= ar + ar^{2} + ar^{3} + ar^{4} + \dots \\ S - rS &= a \\ (1-r)S &= a \\ S &= \frac{a}{1-r} \end{align} \[ % If you need normal text inside of a math environment, one way to render % it is using the \text command, as shown below. This example also shows % how we can get the Z symbol commonly used for the set of integers % using the \mathbb command from the amssymb package (the same command % with Q and R can give us the symbols for the sets of rationals and % reals, repectively). % We can also use \mid for the vertical bar (this gives better spacing for % set-builder notation than typing |). S = \{ x \in \mathbb{Z} \mid x \text{ is odd} \} \] % Sometimes we might want to typeset a "piecewise-defined function", which % we can do with the 'cases' environment in math mode. \[ f(x) = \begin{cases} x & \text{if } x > 0 \\ 0 & \text{otherwise} \end{cases} \] % LaTeX is a complex language with a lot of community-written libraries and % other features. Conference articles and even entire books are often % written in this format. There's a lot one could learn about this system, % but the above should be a good starting point for CS 41. % % And as an added bonus, you can keep more of your homework on your hard % drive instead of on paper. :) \end{document}