Lesson #4: Mathematcs

Download the lesson example and open it up in your development environment. The text here is complementary to the source code, and this lesson will mostly be a description of how the source code works. It's probably worth compiling the source code just to get a notion of what the result of each bit of the source code looks like.

The main reason professionals in math-heavy fields like LaTeX is
that it handles equations and mathematical formatting so very
well. And surely you've been impatient to get to the good stuff! The
big idea is that LaTeX has some things which it'll only do in *text
mode*, like small caps, and some things which it only does
in *math mode*, like fractions. It also handles ordinary text
in math and text modes differently; in math mode, every letter is
assumed to be a variable, and so it's typeset in italics and spaced a
little bit differently than letters in a word would be.

To switch between modes in a paragraph, you just include a dollar
sign character to switch to math mode, and another dollar sign to
switch back. So writing something like `We know that if $x>3$, then
$2x>6$.` will format right, with the right symbols, styling on the
variable, and spacing. If you wrote this without the dollar signs, it
would compile fine but not look right. Lines 17–23 of the lesson
example exhibit some simple examples of this "inline math mode".

If you want "start math mode" and "end math mode" to look
different, you can use `\(` to start writing math
and `\)` to end it. But using `$` for both the beginning
and end is pretty standard.

If you want to put an equation (or other bit of mathematics), all
on a line by itself, either for emphasis or because it's large and
wants to be alone, you can do it in any of three different ways. You
can enclose the math to be displayed in two *pairs* of dollar
signs, or can bracket it with `\[` and `\]`, or even put
it in an environment called `displaymath`. All three of these
are shown in the lesson example: double dollar signs are used on line
20, backslashed brackets on line 49, and the displaymath environment
on line 53.

But to do anything really interesting in math mode (aside from
spacing variables to look like variables) you need commands. Some of
those commands are used to produce mathematical symbols, just like we
needed commands to produce unusual symbols in text. There are a huge
number of those, because there are a huge number of mathematical
symbols you might want to use, so there's `\times` for a
cross-multiplication and `\cdot` for a multiplication dot,
and `\subseteq` for a nonproper-subset symbol, and `\ge`
for a greater-than-or-equal-to sign, and so forth. A good list of the
built-in symbols made for math mode appears in Tables 48, 50, 88, and
122 of
the Comprehensive
LaTeX symbol list. Don't worry, you probably won't need all of
them at once, and most of the time the names make some sense.

One problem with the way math mode formats text is that it's not
right for a lot of the common functions in math. Logarithms, sines,
and the like should use upright letters typeset as words, not italic
individual variables. If you write `$sin(x)$`, it doesn't
actually look right; the letters are italicized and spaced
wrong. Compare that with `$\sin(x)$`, which has the sine
function typeset upright. These are broadly known to LaTeX as
"log-like" symbols, and a complete list of them is in Table 181 of the
above-mentioned symbol list. Contrast how these two are typeset on
line 42 of the example source.

Defining new log-like symbols is possible but a bit difficult at
this stage, but in a later lesson we'll see how to define, say, a
command `\arccsc`.

Math is often more than just horizontally laid-out text, though. Mathematical expressions often have superscripts, subscripts, fractions, radicals, matrices, and so forth. For now, we'll look just at the first four in that list.

To typeset a fraction in math mode, use the
command `\frac{`*numerator*`}{`*denominator*`}`. The
numerator and denominator can be as complicated as you like and can
even include fractions in their own right! See examples of this usage
on line 47 and 49.

To typeset something with a superscript in math mode, use the
command *base*`^{`*exponent*`}`. For
subscripts, use the
command *variable*`_{`*index*`}`. Note that
these don't include backslashes but are just single-character commands
in their own right. Recall that a literal underscore would be coded
as `\_`. A literal caret would be `\^{}`
(because `\^` is actually an accent, so you have to tell LaTeX
to put it over nothing). Examples of subscripts and superscripts are
in lines 27 and 28 of the sample source; these examples also show that
subscripts and superscripts can themselves be superscripted or
subscripted.

A square root is written
as `\sqrt{`*parameter*`}`; as with other commands,
the parameter itself can be complicated, and the square root will
resize to accomodate it. If you want radicals other than square roots,
this command has an optional
parameter: `\sqrt[`*n*`]{`*parameter*`}`
will make a radical with a small number to its left; for
instance, `\sqrt[3]{5}` is the standard way to depict the cube
root of 5. A simple square root is exhibited in line 53 of the source
example.

You can, and some users do, omit braces when the terms in question
are a single character long. Thus `\frac25` is as good a way of
saying two fifths as `\frac{2}{5}` is, and
likewise `\frac{x}{y}` could be written as `\frac xy`
(note that space, because `\fracxy` would just be a single
invalid command). Likewise `x^2` works just as well
as `x^{2}`. The danger of this practice is that if you get into
the habit of simplifying this way, you're almost certain to at some
point do it with a parameter that's more than one character long. Once
you think of `x^4` and `x^7` as natural ways to
write *x* raised to the fourth and seventh power, it's easy to
fall into the trap of writing the twelfth power as `x^12`,
which won't look right. That would be parsed as just superscripting
the 1; the correct way to write it is `x^{12}` (see lines 30
and 31 for further examples of this problem).

We have two different versions of math mode, and they have somewhat distinct typesetting priorities. Inline math wants to keep your paragraph formatting uninterrupted, so its priority is to use as little vertical space as possible, whereas display math is already breaking out of the paragraph structure, and so it can be as tall as it needs to be without making the line of text it's in look strange.

Consequently, some expressions typeset differently in display mode than they do in inline mode. One easy example is that fractions are a lot taller in display mode: numerators and denominators are full-size, unlike the half-size numerals used in inline mode. Contrast the fractions in line 47 of the source example with the derivative (typeset as a fraction) in line 49, or the fraction in line 61.

Another family of expressions which are typeset differently in
display and inline modes are those symbols which conventionally have
additional text above and/or below them. Such symbols include the
integral (`\int`), big-sigma sum (`\sum`), and limit
(`\lim`), and the additional text is passed to LaTeX as a
subscripts and/or superscripts. Inline these are treated like
conventional subscripts and superscripts, but in display math, where
vertical space is not at a premium, these elements are horizontally
aligned with the symbol. Contrast the way the range on the sum is
typeset in the examples in line 59 and 61 of the source code example.

The `equation` environment functions like
the `displaymath` environment (or either of its shorthand
forms), except that, in addition, each `equation` receives a
sequential number for future reference. Lines 60–63 of the
source example are such an environment, so when this bit of math is
typeset it receives a flush-right label "(1)". Like any other label,
this one can be referred to by using the `\label` command and a
later `\ref`, as seen on line 69.

Most simple math errors come from failing to turn math mode on or off appropriately. The error messages mostly say exactly what needs to be done, but not always in the right place.

: You have mismatched (or absent) math-mode signs, so you're not in math made somewhere where you want to use a fraction, or a square root, or some other uniquely mathematical command. Note that the actual site of your error is probably some ways before the place where LaTeX ran into trouble; it's an earlier math-mode expression which may have a missing dollar-sign.`Missing $ inserted`: You have failed to end a display-math equation, probably somewhere before where the error occurred.`Display math should end with $$`: This one is much rarer, but it denotes an attempt to begin a display-math environment when already in one, or an attempt to end one when you aren't. This usually means your display-math symbol-pairs are mismatched somehow.`Bad math environment delimiter`: This is a catchall for all sorts of unparseable commands in math mode. For instance, it arises when a math expression ends with a fraction missing its denominator. Best practices are just to look very closely at the equation immidiately preceding it. Often there is more going on that just a missing close-bracket.`Missing } inserted`: This is a similar catchall to the above, although it usually comes of soem sort of unhappy interaction between math mode and environments. Check that all math expressions are closed properly.`Missing \endgroup inserted`

Write a document which compiles to make a PDF looking like this.

Return to the tutorial table of contents.