How to present mathematical expressions using a language that has so little markup for them? Web authors often need resort to images, but there are more flexible approaches, like MathJax. Moreover, if you need just some special symbols or simple expressions, a lot can be done in HTML, assisted with style sheets (CSS). This document mainly discusses relatively simple mathematical expressions rendered onedimensionally (inline), though possibly with superscripts or subscripts.
The word “mathematical” is used in a rather broad sense here, covering different formalisms and symbols, including the symbols of physics, formal logic, etc. You might wish to take a look at Andreas Prilop’s nice document Mathematical formulas in HTML 4.0, which illustrates well what kinds of symbols and expressions are discussed here. And by looking at the HTML source there, you can see how such things can be done.
This document does not discuss the choice of mathematical notations, only their implementation on web pages. For notations themselves, please refer to my ebook Writing Mathematical Expressions.
You may have wondered why web sites, such as Wikipedia,
use images for presenting mathematical formulas,
even in simple cases like
∇²φ = 0.
Partly the reason is simplicity and uniformity: since some
formulas need to be presented as images, it may sound simpler to
use images for all images—even in cases where
simple HTML markup would do fine:
∇² = <i>φ</i>
or
∇² = <i>φ</i>
Images provide a methodologically uniform approach, but the result is not typographically uniform at all: characters in an image are typically rather different from the same characters in text.
Although HTML markup exists in the sample case, mathematical formulas
are usually much more complicated. HTML lacks markup for mathematical
expressions as structures, and there is no simple way to produce
anything essentially two
Given the fact that HTML was originally developed at CERN, the European Laboratory for Particle Physics, it may sound odd that HTML has rather little markup for mathematical expressions and other special notations used in science. It’s not a surprise that you can’t do math in HTML; after all, it’s a markup language, not a programming language. But how are we expected to write physics reports or even modern biology papers without mathematical notations?
There was once an HTML 3.0 draft, with a section titled HTML Math, suggesting relatively simple markup for some basic mathematics. But it’s all history; the draft expired in 1995. (There was also an earlier idea about HTML+, which would have had a different, more naturallooking math syntax.)
The modern trend is to regard things like mathematical markup as special “applications”, for which specialized languages are used; see W3C’s Math Home Page for information about such trends, including the MathML language. If you ask me, the language hopelessly mixes structure and appearance. In any case, support to MathML in web browsers still isn᾿t wide enough to justify its use on web pages in general.
However, this does not mean that it would be impossible to
write mathematical documents in HTML. There are difficulties and
difficult decisions to be made. For relatively simple mathematical notations,
HTML can be used rather well, especially if you use
handle the toughest parts using
images with adequate alt
texts.
See, for example, Stan Brown’s
articles on math.
Note that there’s the possibility of making an article available both in HTML format
and in some other format, and this might be relatively easy if suitable
tools can be found.
For documents where mathematical notations are needed a lot, other formats than HTML may turn out to be more practical. This might mean for example PDF, PostScript, or TeX format, or perhaps all of them, offered as alternatives.
The benefits of HTML in Web authoring are considerable, however, so it can be a tough decision. This document tries to help in making an informed decision as well as in implementing the HTML way, if that way is taken.
Even if you decide to publish your document in, say, PDF format only, you might still considering making its abstract available in HTML format too. Typically an abstract can be written with just a modest amount of mathematical notations.
There are also JavaScriptbased approaches that use libraries for converting
TeX notations or
TeXlike notations into graphic format. They typically fall back to displaying
such a
notation, like \[P(E) = {n \choose k} p^k (1p)^{ nk} \]
.
The bestknown implementation of the approach is the
MathJax library, but there is also a considerably
smaller and faster library,
jqMath,
which can produce rather amazing results.
For example, using MathJax, the code
$$\sum_{k=1}^Nk(Nk+1)$$
causes the following formula to be displayed:
The approach is simple, from an author’s point of view:
you just insert a fixed script
element in your document, and
you put an inline formula between \(
and
\)
, a display formula
between $$
pairs. In the formula, you use
for example TeX (specifically, AMSTeX)
notations, which is rather simple to learn
for basic mathematical constructs.
Sometimes it is best to present mathematical expressions in
linearized notation.
For example, instead of trying to find a way of presenting
the square root of 2 in the normal mathematical way, you might write
just sqrt(2).
For quotients, you’d use notations like
Note that even in plain text, you can use the multiplication symbol × and do not need to resort to the asterisk (*) of computer jargon. See also other notes on using “normal” characters.
To indicate nesting of mathematical expressions,
linearized notation uses parentheses. It has been common to
use different types of parentheses:
innermost, you use normal
parentheses ();
next level, you use
square brackets []; and outermost you use
curly braces {}, e.g.
[(x + y)^(1/3)]/z.
However, according to the standard on mathematical notations,
ISO 800002, normal parentheses should be used, because other
parentheses have special meanings. Example:
((x + y)^(1/3))/z.
A document containing such an expression should probably contain, near the beginning, a statement like “In this document, the circumflex character ^ is used to indicate exponentiation, i.e. raising to a power.”
Such an approach makes the document accessible to virtually all who have the necessary mathematical prerequisites, if you limit the character repertoire to ISO Latin 1. But naturally it’s a rather simplistic method. You might still consider using it as an alternative format, made available along with some more advanced but less accessible format(s), such as an image.
You can use some software, e.g. something TeX based (see also AMS TeX resources) or a mathematics program with graphic output format option, e.g. Mathematica, to produce visual representations of formulas as images. Often such software gives the result in PostScript format, but for Web use a GIF format is usually better, since it is much more widely supported in browsers by default. Various tools exist for converting from PostScript to GIF format. You could also consider using the Latex2HTML converter which can, among many other things, generate images in GIF format. (There’s another converter, TtH, which uses all kinds of tricks to convert mathematical expressions to HTML without using images. It applies very questionable methods like the Symbol font kludge. You might still get some useful ideas from its output and apply them more reasonably.)
Such an approach is widely used on the Web. See, for example, the wonderful MathWorld pages. Note, by the way, than when you use mathematical terms, say binomial coefficient, on your page but cannot define all of them, it might be a very good idea to link to definitions and descriptions like those on MathWorld. Even if your terms are already known to 95% of readers, you can give the 5% a chance to find it out, and the others might need to brush up their knowledge too, or just check that the term really means what they think it does. See also Eric Weisstein’s Treasure Troves of Science collection, which is impressive, too, not only in content but also in the Web presentation of mathematical formulas. As regards to mathematics proper, e.g. S.O.S. MATHematics also has nice pages on several topics.
img
elementYou would use the
img
element to embed an image into your document, and/or
an a href
element to create link to it.
The latter method is often worth considering,
especially for large formulas.
The reader may
prefer reading the text without distractions and looking at the
formula (image) at the very moment he is prepared to do so. Moreover,
he may prefer looking at it in a separate window (which is separately
adjustable in size and positionable on the screen), or perhaps
print it out (due to better resolution in print than on screen).
As a very simple example, I have used the simple markup
\int_{0}^{\infty}e^{x^2}dx
in TeX and generated
a GIF image file from it, then just embedded the image into an
HTML document using the following markup:
<p><strong>Assignment 42</strong>. Compute
<img src="integral.gif" align="middle" alt=
"the integral of exp(x**2) for x from 0 to infinity"></p>
This looks like the following on your browser:
Assignment 42. Compute
And here’s the same with an intentionally broken reference to an image:
Assignment 42. Compute
Generally, the more complicated formulas you need to present, especially if nonlinear in appearance, the more seriously you need to consider using images for the purpose, despite their drawbacks.
In principle, the
object
element could be used to embed
data in different formats, including images, animations,
and interactive presentations. However, support for object
is limited and buggy. Support for its “brother”, the
iframe
element,
is somewhat less buggy. In special cases, you might consider using
iframe
for large formulas mainly because
iframe
lets you specify an area where the data
is to be presented so that a browser is expected to introduce
scroll bars if needed. You would then need to include the normal
img
element as the content of iframe
,
to provide a fallback for browsers that don’t support
iframe
.
alt
attributeIf you decide to use an image, you still have
the original problem, in a modified form. Due to
accessibility considerations, every img
element must have an alt
attribute that
specifies the textual alternative. This means, in practice, that
you need to write an alternative presentation of the expression
in pure text form, with no HTML markup. On the other hand,
you will do this for “fallback” use only, and you can e.g. write
rather verbose explanations if needed, since the text will normally
not be seen by people who see the image.
The purpose of the alt
attribute is make the text
comprehensible
in a browsing situation where the image is not displayed.
For some notes on the importance and practical use of alt
texts, see my
Guidelines on alt
texts in
img
elements. In the simplest case,
you would use a
simple linearized notation
as alt
text.
A particular problem arises when such alt
texts contain notational features that are not selfevident and
are not otherwise used in the document. For example, suppose that
expressions
contain exponentiation, which is indicated using the usual
twodimensional formatting (an exponent appears raised, perhaps
in smaller font) in image presentations, but a
circumflex character ^ is used in alt
texts.
How can we inform people who need the alt
texts,
without disturbing people who see the images only?
There’s a simple trick: use a “dummy” image, such as
singlepixel transparent GIF, and put the explanations
into its alt
attribute, e.g.
<img src="transp.gif" alt=
"In this document, the circumflex character ^ is used
to indicate exponentiation, i.e. raising to a power.">
And you would put this element somewhere near the start of the
document, before the first occurrence of an alt
text
that uses the convention explained.
If the image has been generated using TeX,
you might also provide a link to the TeX version, since
some people who cannot see the image for some reason might be able
to understand TeX markup, or utilize it indirectly.
Perhaps the image itself could be made into a link.
In fact
the MathWorld
and
Treasure Troves of Science
pages
often include the TeX notation directly as the alt
text;
this is useful but not as useful as giving plain text presentations there
and providing links to TeX versions.
In the document HTML Techniques for Web Content
Accessibility Guidelines 1.0,
section
Markup and style sheets rather than images: The example of math
recommends that if an HTML document has been generated from
a TeX document,
the original TeX (or LaTeX) document too be made available
on the Web, since there are possibilities for
auditory rendition of such versions.
That document also recommends that
if a formula is constructed from several images, a single
alt
attribute should specify a textual alternative to
the entire formula. Apparently, you would put it into one of the
images and use alt=""
for the other images.
Images are widely used for presenting formulas on the Web, and they are quite often the best practical way for large formulas to be presented as “blocks”. For symbols and short formulas in running text, “inline”, similar approach can be less pleasant, since the image does not adapt to text font size and face. Here’s a simple example:
The Greek letter is often used to denote summation, but from the character standards viewpoint, the nary summation symbol is a distinct character, though historically derived from the Greek letter.
If you change
the basic font size from the browser’s menu, you’ll see how
the image remains the same size, instead of being adjusted
according to the font size.
And attempts to use style sheets to scale images
are not always very successful, though browsers tend to scale
downwards relatively well these days;
you could design the image in large size and then use
<img ... height="1em">
to make the browser to scale it
to the font size of the text.
Moreover, line spacing easily gets disturbed.
(See
notes on such issues in
Guide to using special characters in HTML.)
You could also write e.g.
<img src="alpha.gif" alt="α">
,
using a numeric character reference,
and then on some browsers the
document would look better with images off than with images
on. But why not use just α
then?
Within the “normal” set of characters, namely the ISO Latin 1 character repertoire, which you can use pretty safely in HTML authoring, some characters need special attention as regards to use in mathematical notations. (As regards to typing them into your HTML documents, see section Typing characters in my character code tutorial.)
On Macintosh platforms, several browsers have had serious problems with some ISO Latin 1 characters, and these characters include the superscripts 1, 2, and 3 and the vulgar fractions ½ ¼ ¾ as well as the multiplication sign ×. It seems that these problems have finally lost significance: any reasonably modern browser on Mac renders these characters correctly.
Among the basic arithmetic operators, the plus sign + poses no problems. As a minus sign, the hyphenminus character () is commonly used, but using the minus sign character would be more logical. It is usually visually better (longer). Nowadays, it seldom causes character problems. It avoids some line breaking problems that browsers have with hyphenminus: we usually do not want e.g. a unary minus sign to have separated from its operand by a line break. (Browsers often break after hyphenminus, but not minus sign.) For multiplication, there is seldom any reason to use the asterisk *, e.g. a*b (it’s a programming language idiosyncracy), since you can use the multiplication sign, e.g. a × b or perhaps the middle dot, e.g. a · b. When referring to vector operations, you could use × for a cross product and · for a dot product. However, in principle, by the ISO 800002 standard, the correct multiplication dot is not the middle dot (·) but the dot operator (⋅, U+22C5). For fractions and division, the solidus (slash) character / is commonly used, but you might also consider using the division sign ÷ in some cases.
In addition to the abovementioned
use as a multiplication symbol,
the asterisk *
has several uses
as an operator symbol of some kind. Generally such uses are surrogate notations
for various starlike symbols with more specific semantics.
Since the asterisk is displayed in relatively small size and in a superscript
position in most fonts, it does not look good when it should be
binary operator. So if you use it that way, in lack of better characters,
you might consider using markup that suggests
that * be displayed in
a monospace font; the simplest
method is to use
tt
markup, e.g.
<tt>*</tt>
. On the other hand, for use e.g.
as a star denoting
an adjoint matrix, superscript style is better,
and normally achieved (if achievable in a browsing situation)
by not using any font markup for it. When using
“special characters”, there is, in principle,
a rich repertoire of various asterisk and starlike characters available
in Unicode. There is a large number of
characters with “asterisk” in their name, including
“asterisk operator”. Note that there are several Unicode characters with
“star” in their name, including some that are classified as mathematical
(general category: Sm, i.e. Symbol, Math), such as
“star operator”.
Other operators include the less than and
greater than signs (< and >). Since these characters
are essential as tag delimiters in HTML, they should be “escaped”
using the notations <
and
>
when they are to be included into the
document’s textual content, as in a<b.
Similarly the ampersand (&), which might be used e.g.
in logic as an and operator (connective), should be
written as &
.
There is nothing special about the equals sign (=);
it can be typed as such.
But the inequality symbol and the
not equal to, less than or equal to, and
greater than or equal to
symbols are problematic; they do not belong to
ISO Latin 1.
If you have decided to use
“special characters” (i.e. characters outside
ISO Latin 1) despite their problems, then you would use the
numeric references
≠
and
≤
and
≥
(appearance on your browser:
≠ ≤ ≥).
Otherwise you would have to
use some surrogates like =/, <= and >=.
Some authors use =< and => instead of <= and >=, but there’s
anyway the risk that one of the constructs
be mistaken as an implication arrow or another
arrow symbols.
The
not equals sign is the most difficult, since it has
no widely recognized surrogate. Note that
When expressing physical quantities, note the following rules of the SI system of units (see Guide for the Use of the International System of Units (SI) for more information):
em
element
(which is typically presented in italics),
then you should use e.g. something like the following:<em>The measured time was only 42 <span class="unit">s</span>.</em>
.unit { fontstyle: normal; }
4.2 × 10<sup>15</sup>
and, in another usage,
4,2 c5; 10<sup>15</sup>
(renderings: 4.2 × 10^{15} and
4.2 ⋅ 10^{15} but note
the problems with superscripts).
12 °C
(displayed as ±
), as well
as in other contexts, the notation should be such that
“it is completely clear to which unit symbols the numerical
values of the quantities belong”. This means e.g.
Web browsers generally treat any space character as a possible line break point. This is often undesirable in notations like the one discussed here, e.g. between a numeric value and a unit denotation. Thus, some method of avoiding that should be used, e.g. using a nobreak space between them, instead of a normal space. See section Line breaks as problems.
As parentheses you can use normal parentheses
(),
square brackets [],
and braces {}. You might wish to
use font markup
(such as <big>{</big>
)
to make outer parentheses look bigger,
to make the structure visually clearer.
Also note that ISO Latin 1 includes some other more or less mathematical characters beyond those in Ascii: the micro sign µ and the not sign ¬. But beware of the following: assuming that the sharp s ß would be the letter beta; confusing the ordinal masculine indicator º with the degree sign ° or taking either of them as superscript 0; thinking that left guillemet « and right guillemet » would mean ‘much less than’ and ‘much greater than’; or assuming that the letter O with stroke Ø would be an empty set symbol! Don’t be tempted by casual appearances of characters; there are reasons to be strict about the meanings of characters.
In mathematical notations we very often need special symbols, such as Greek letters or a symbol for infinity. In principle, you can use the full Unicode character reportoire in HTML. Unicode Technical Report #25, Unicode Support for Mathematics, says: “Starting with version 3.2, Unicode includes virtually all of the standard characters used in mathematics.”
Previously, browser support had serious flaws when it comes to special characters. The most fundamental problems have almost completely disappeared. But the very important practical limitations in fonts exist.
The problems and solutions are discussed in Guide to using special characters in HTML. The document is newer than the presentation in this document and covers some topics not discussed here.
Support to mathematical characters in fonts varies greatly. Some typical examples:
Unicode  Status in fonts  

±  U+00B1 plusminus sign  Practically universal support 
≤  U+2264 lessthan or equal to  Very good support 
∰  U+2230 volume integral  Limited, but works for most users 
𝑎  U+1D44E mathematical italic small a 
rather limited support; use <i>a</i> instead

In practice, all or most of the mathematical symbols you need are covered by the Lucida Sans Unicode font, which is normally present in Windows systems. It is not of particularly good design, so you might suggest Arial Unicode MS (shipped with Microsoft Office) as a preferred alternative. Moreover, you should include a list of fonts that are commonly available on other platforms (Linux etc.).
For pages that make essential use of mathematical characters that are not covered by fonts normally in use on Windows platforms, it might be a good idea to include a short note about fonts. Perhaps with a few selected characters and with an explanation of what they should look like and a remark: “There are free fonts that contain these characters, such as DejaVu Serif, Quivira, and Symbola.”
You can enter special characters as such
(if the character encoding permits that), or using “escape
notations”. For example, the Greek letter alpha can be written
in HTML source as such (α), as the entity reference
α
, as the hexadecimal character reference
b1;
, or as the decimal character reference
α
. These methods are equivalent in principle,
and they all work on any reasonably modern browser.
HTML 4.0
contains a relatively large set of
named entities like α
for
various symbols,
including Greek letters as used in mathematics.
They are nicely presented in WDG’s document
HTML 4.0 Entities.
(There’s also a more compactly presented list by me:
Character entity references in HTML 4.0.)
However, they are just “named constants”, with definitions that
use numeric character references. Some early browsers
(e.g. Netscape 4) supported them but not the
named entities. Today, this is no longer relevant,
but there’s not much practical reason to use the entities.
Although the entity names are intended to be mnemonic, some of
them are rather cryptic;
using
∑
for the nary summation symbol
would probably work well
(in the abovementioned sense),
but how many readers would intuitively understand what
∋
means?
Note: If the entity is followed by
a space or a line break, the semicolon can be omitted, thereby
making the notation look slightly more natural.
Technically, the semicolon can be omitted if the character
immediately following it is not a letter (A–Z, a–z) or
a digit or one of the following:
._:;
(hyphen, period, underline, colon, semicolon).
HTML5 drafts contain a long list of added entities. They are mostly rather cryptic and practically useless, though browser support was introduced to Firefox near the end of year 2011.
Note that by HTML specifications, you are not limited
to using those numeric character references for which there is
a named entity in HTML 4.0. You can principle use any
&#
number;
As a practical point, note that Microsoft has defined WGL4 (or “PanEuropean”) list of characters, containing some mathematical symbols too. If you use them only, you have relatively good odds of finding a large number of users equipped with fonts that can display the characters. See Using Special Characters from Windows Glyph List 4 (WGL4) in HTML by Alan Wood.
See also the document How to find an &#number; notation for a character, and note that you will probably be interested especially on the following Unicode blocks:
If you only need Greek letters in addition to Ascii characters, then there is the possibility of using an encoding which lets you enter Greek letters directly. You could use ISO 88597, which is relatively widely supported. See section Simpler ways for simpler needs: simple 8bit encodings in the abovementioned document. But note this would give you just Greek letters, not e.g. not equals sign, nabla operator, aleph, etc. ad ∞. Moreover, you would risk losing most of the upper half of ISO Latin 1, due to browser bugs.
There are pitfalls in the area
of characters. Never trust a smiling cat, or what a character
looks like. For example,
in modern character set standards,
the mathematical symbol for
nary summation (∑
,
∑) is distinct from Greek letter capital sigma
(Σ
,
Σ), although it is historically derived from it.
Ditto for nary product symbol (∏
, ∏)
and capital pi (Π
,
Π).
I used to think that someone might still
consider using sigma as a surrogate, assuming it might be better
supported, but
Andreas Prilop kindly
pointed out my mistake, concluding with the following:
“you need only a browser that knows ‘Unicode big numbers expressions’
and Windows 3.1 [or newer] or any MacOS version,
both without any Greek support, to display ∏”.
Another pitfall is that character rendering may vary greatly
across browsers.
For example, the simple rightwards arrow character, which
can be denoted e.g. as →
in HTML, has
very varying shapes, as you can see by using
test page for rightwards arrow.
The variation ranges from an almost dashlike symbol
(with just a tiny arrowhead) to a grotesque version where the arrowhead
dominates (e.g., “” in Calibri).
Moreover, some common fonts like Verdana lack it,
so in text with Verdana as the primary font, the arrows will appear in
whatever font the browser chooses to use as a replacement font.
Therefore, it may be a good idea to suggest a specific, widely
available font where the appearance is acceptable.
You can write e.g.
<span class="arrow">→</span>
in HTML and
.arrow { fontfamily: "Times New Roman"; }
in CSS. In Times New Roman,
the arrow (→)
corresponds to the arrow symbol traditionally used in mathematics,
and it can be used in conjunction with many different fonts.
If the vertical position of the arrow is
unsatisfactory, as it may be when used between capital letters
of the Arial font
(A → B),
you can tune
the rendering
(A → B),
by adding e.g. the CSS rule
.arrow { position: relative; bottom: 0.07em; }
.
The division slash character (U+2215), being defined as a mathematical operator, is in principle more adequate than the common slash character. It belongs to rather many fonts. But the problem is that the appearance in generally unsuitable. Its glyph usually touches the adjacent characters or even strikes through them. Here is an expression using first the common slash, then the division slash:
Doublestruck letters such as ℕ, ℤ, ℚ, ℝ, ℂ are commonly used as symbols of standard sets of numbers in mathematics. Such letters are contained in Unicode, in the Letterlike Symbols block, but their browser support is limited.
It has long been acceptable to use normal letters in bold face instead.
This appears to be even the original notation, and it is preferred in
the international standard on mathematical notations,
ISO
Thus, it is better to use just bold face letters
such as N. You can achieve
this simply by using the b
element in HTML, e.g.
<b>N</b>
.
Similarly, even though Unicode contains characters like
“mathematical bold italic small a” (U+1D482),
it is much safer to use normal letters and simple markup, such as
<b><i>a</i</b>
, to
produce a symbol like a.
This section is not particularly important any more, since most of the special characters discussed here can be used on web pages rather reliably.
For characters that cannot be presented reliably enough but would be needed, different surrogates can be considered. The obvious method is to use words or abbreviations, e.g. “infinity” or “inf” instead of the infinity symbol (∞). In mathematical formulas, words should not be used, so an abbreviation is preferable.
Sometimes one could consider using a character or a sequence of characters in a way which tries to approximate the appearance of a real character which itself cannot be used reliably. There are strong reasons to avoid playing with characters too much, and one should not do things which seriously conflict with the meanings of characters. I’d hesitate using, say, “oo” as a surrogate for the infinity symbol. But there are some notations that one could use, at least if explained in a legend.
In particular, the logical connectives,
corresponding to
“and” and “or”, could rather safely be presented using
A common surrogate for arrow symbols is the use of character pairs like
< or <=, though longer strings like
Note that some of the surrogate notations discussed here may suffer from
line break problems on IE, unless precautions are taken
by using nobr
markup.
Since notations like /\ or > try to imitate the shape of
the real characters they stand for, it can be useful to suggest
some particular font, or a list of fonts in order of precedence,
using the font face
markup in HTML or the
fontfamily
property in CSS.
I have written a test page that shows
text in different fonts which are commonly available in Windows systems.
For example,
In practice, you could use a style sheet like the following:
.logop { fontfamily: "Times New Roman", serif }
.arrow { fontfamily: Tahoma, Symbol, monospace }
.darrow { fontfamily: "Times New Roman", serif }
and HTML markup like the following:
<span class="logop"><nobr>/\<nobr><span>
<span class="arrow"><nobr>></nobr><span>
<span class="darrow">==><span>
This is how they look like on your current browser:
Line breaks are often undesirable inside expressions, but Web browsers generally treat every space as a potential line break position. There are several ways to deal with this:
) instead of a normal space.
nobr
markup around a string that should
be kept together. Example:
<nobr>a + b + c</nobr>
.
nowrap
attribute in a td
(or
th
) element. This is useful if it’s OK to prevent
all line breaks inside a table cell (except line breaks explicitly
requested using br
tags).
whitespace:nowrap
(or maybe whitespace:pre
) in CSS. Typically this means that
you would use something like the following:<span class="q">42 m</span>
.q { whitespace: nowrap; }
It is debatable whether it is more logical to use the CSS way than
nobreak spaces. In a sense, it is a structural property of an expression
like “42 m”
that its two parts belong closely together, so that in any
normal presentation, there should be no line break between them, or any
pause in speech presentation. On the practical side, nobreak spaces surely
work more reliably, partly because browser support for the
whitespace
is still limited, partly due to general
CSS caveats.
Note that the nobr
markup is the only sure cure against IE’s tendency to treat every
hyphen as a potential line break opportunity,
even in a string like ab or a!
But if you have decided to use “special characters”,
then you might
use the real minus sign, −
,
instead of hyphen, and that would avoid
the hyphen problem. See Dashes and hyphens.
There are even more oddities in IE: it may also treat any of
nobr
element is the most effective cure,
as explained in Word division
in IE, but we will discuss
a special case later.
It is customary and recommendable to group digits in long numbers to groups of three digits. But the method of separating the groups depend on cultural conventions and even personal style. This typically means using spaces, commas, periods, or apostrophes as separators.
It is safest to use spaces, since the other alternatives could be misinterpreted. For example, in English 1,005 would mean one thousand and five and 1.005 would mean one and five thousandths; in French, and in several other languages, it’s just the reverse! We need to make some decision concerning the decimal separator, but for integers we can avoid the problem by using spaces: 1 005 is unambiguous. It is true that e.g. in English texts it does not conform to normal English practice, but here we are discussing mathematical texts, where the practice is recommended by standards,
This raises two problems: First, the line breaking problem that we just discussed; but we saw that there are reasonable solutions to it. Second, you might regard spacing between digit groups as typographically excessive, if normal spacing (as between words) is used.
Although there are several
space characters of specific width in Unicode
(in the range U+2000
to U+200B
in the
General
Punctuation block),
using them is not a good idea, as a rule. In practice, they don’t work well,
partly due to limitations in browser support for such “special characters”
(to be discussed in the next section). And in principle,
they are “compatibility
characters” only.
But you can use normal space characters and specify some simple
CSS rules that suggest reduced spacing between “words”,
using the wordspacing
property with a negative value.
That property specifies the spacing to be used in addition to
default interword spacing, so a negative value suggests a reduction
of the spacing.
A “word” is
here any sequence of nonwhitespace characters.
The boring part of the matter is that you typically need to include
extra span
markup just to have some element with
which the rule can be associated.
You can hopefully find some nice program tool for generating the
markup needed, so that you don’t need to type it all by hand.
In my opinion, wordspacing: 0.07em
creates a fairly
nice result for spacing between digit groups.
It means a suggestion to reduce the normal spacing by
7% of the font size, so it naturally adapts to whatever font size
happens to be in use.
This seems to make the digit groups separated visibly but not
disturbingly.
The following demonstrates first a long number without
the effect of such a suggestion, then with that effect, naturally assuming
that your browser supports this part of the CSS specification:
For the latter number, I used the following markup:
<span class="number">123 456 789 000 000 000</span>
and the following style sheet:
.number { wordspacing: 0.07em; whitespace: nowrap; }
The whitespace
rule is unrelated to spacing but a good
idea for other reasons, as explained above.
For related notes, discussing similar problems in text processing, see How to cope with international standards for the thousands separator by William S. Statler.
The “excessive spacing” problem also arises in other contexts in mathematical expressions. It is often regarded as good style to use some spacing e.g. around mathematical operators, but not as big spacing as we get in typical browsing situations if we just use normal spaces.
In highquality typesetting, e.g. when using T_{E}X, spacing is controlled carefully, using advanced tools and techniques. We cannot expect to achieve the same using HTML and CSS, but we can aim at reasonable quality.
An expression like
“a + b”
is usually best written in HTML so that there are spaces
around the operator “+”. This gives more flexibility, since
we can then use wordspacing
in a simple way.
If the spaces were omitted, letterspacing
(which actually affects the spacing between all characters,
not just letters) could be used in simple cases like this, but
things would get much more complicated when variables consist
of several characters (e.g., contain subscripts).
The following example shows the rendering of an expression with several operators, under different style sheets:
unstyled with spaces  a + (b × c) 
wordspacing:0.07em 
a + (b × c) 
wordspacing:0.2em 
a + (b × c) 
unstyled without spaces  a+(b×c) 
Thus, a wordspacing
value like 0.07em creates
an appearance that more or less resembles typical typesetting
of mathematical expressions. A value of 0.2em tends to reduce
spacing so that it almost corresponds to the rendering we would get
if no space characters were used. The effects naturally depend on
the font in use, but these observations apply to typical fonts
like Times New Roman and Arial.
In the old days, one of the most common ways of trying to include Greek letters and
mathematical symbols into Web pages
was to use font face="Symbol"
,
such as writing
<font face="Symbol">c</font>
to get the Greek letter gamma (γ).
You may still find web pages that propagate such usage
as if were clever and useful.
I will not explain here
why that approach is fundamentally wrong; I refer to the excellent
presentations
Using FONT FACE to extend repertoire? by
Alan J. Flavell and
<FONT FACE> considered harmful at the
Alis Babel site.
Briefly, the trick appears to work in many situations,
but that’s because of browser bugs. The invariable
meaning of
<font face="Symbol">c</font>
is the letter c, so any correctly behaving browser will display it
using some physical presentation
(glyph) for that character.
There are several reasonable uses for fontlevel markup that could be applied in mathematical notations. In particular:
b
markup for symbols of vectors
to make them appear in bold face.
But consider it as a presentational suggestion only, making sure that
the text itself indicates the meanings of symbols, and
considering bolding just as extra clue.
i
or
var
markup for variables
and other symbols which are conventionally displayed in
italics, as in f (x) or sin x.
The var
markup may sound more logical for
variables, but it’s questionable whether
var
was meant for mathematical variables;
its original definition (in HTML 2.0) says that it
“indicates a placeholder variable”.
IE 3 oddly displayed var
in monospace font, but current browsers use italics.
See below
for problems that italics may cause.
Note: the constant e (= 2.718…) and
and the imaginary unit i
are often written in italics,
but according to the ISO <b><i>x</i></b>
,
which displays as follows on your browser:
x
small
and big
elements.
But note that if you use sub
and sup
for
subscripts or superscripts,
the characters will by default be smaller than normal,
on many browsers (but not all), so further size reduction
may easily make the text unreadable. But if you like, you could e.g.
make digits after a decimal point appear in smaller font
(as in 42.123 – consider then putting the decimal
point too inside the small
markup).
If you wish to emphasize a formula like
E = mc²,
you can use big
markup or the
fontsize
property in CSS,
in addition to the logical em
or strong
markup
(commonly rendered in italics and in bold, respectively, by default).
Browsers often tend to put characters too close to each other when a character in italics is immediately followed by a nonitalics character, as in f(0). This is more or less an inherent problem with fonts rather than a browser issue. In italic, letters are usually slanted, and this often makes a tall letter hit the next character, if it is upright and tall. For example, a may look reasonable, but T probably looks bad without stylistic tuning.
You might consider using a
nobreak space character between them,
e.g. f (0), using
markup like <i>f</i> (0)
.
Perhaps the best way to deal with the italics issue is to use CSS to
add some empty space after any element rendered
in italics.
You could use either margin or padding property for this.
Using padding is probably better, since some day someone might set a background
color for the element. (The background extends to the padding but not to the
margin.)
If you need the spacing just for an individual expression, you could
use markup like
<i style=
for an expression like f(0).
Or you might even write a general style sheet rule that sets
a right margin for all inline elements that are commonly rendered
in italics. You might explicitly set right margin to zero, since
it is imaginable that some browsers deal with the problem by using
some default right margin, and you don't want a cumulated effect.
Example:
i, em, cite, dfn, var { paddingright: 0.15em; marginright: 0; }
serif  sansserif 

× x  × x 
U ∪  U ∪ 
ε ∈  ε ∈ 
a a  a a 
o o  o o 
Fonts with serifs are usually better than sansserif fonts for mathematical texts. This may sound strange, because mathematical operators, like the multiplication sign ×, typically have no serifs. They have rather similar design in all fonts. But there is usually a considerable difference between serif and sansserif design for letters.
It is important to distinguish mathematical symbols from each other and from letters and other characters. The serifs and the varying stroke width in serif fonts often help this. Moreover, serif fonts typically make a better distinction between upright and italic style.
This usually rules out Arial, the most commonly used font on web pages.
In most browsers, the default font is Times New Roman, which is a good
serif font for printed
matter but problematic on screen due to the much smaller resolution. Suitable
serif fonts that work both on screen and on paper include Cambria,
Georgia, Palatino Linotype, and Bookman Old Style.
Just remember to list down a few of them in your
fontfamily
, in order of your preference,
since none of them them is universally installed on computers.
Hint: after writing a CSS rule like
body { fontfamily: Cambria, Georgia, Palatino Linotype, Bookman Old Style, serif; }
use Firefox with
Web Developer Extension to view your page, then select
“Edit CSS” in its “CSS” menu.
Modify the style sheet by removing the first font in the list, look at the page,
remove the next font, etc. This way you can quickly test the page on
all the fonts you suggest, without editing the page itself.
In good typography, we avoid mixing fonts in text. We can use normal, italic, and bold versions of a font, but not fonts of different design in the same paragraph or other block of text. However, in mathematical texts, it is often more or less necessary to mix fonts, taking letters and other common characters from one font and mathematical symbols from another.
The main reason is that most fonts have a limited character repertoire, as described in section Using special characters. When you need to pick up special characters, you cannot be too picky. In particular, a large number of mathematical symbols can be found in commonly available sansserif fonts like Lucida Sans Unicode and Arial Unicode MS but not in common serif fonts.
For example, the
nabla
operator ∇ (U+2207) is present in several fonts,
but not in any serif font commonly available on Windows systems.
Thus, to write the expression
∇f you should use markup
like
<font class=nabla>∇</font><i>f</i>
together with a CSS rule like the following:
.nabla { fontfamily: Arial Unicode MS, DejaVu Serif, Linux Libertine, Lucida Sans Unicode; }
While it has been conventional to some extent to use different characters for nested parentheses, using ( ), [ ], { }, and then angle brackets, such practice is not endorsed by standards. Quoting ISO 800002:
It is recommended to use only parentheses for grouping, since brackets and braces often have a specific meaning in particular fields. Parentheses can be nested without ambiguity.
Thus, angle brackets should only be used in specialized meanings,
such as L² inner product of functions, or maybe for an arithmetic mean if the primary notation (line over) is not applicable.
Instead of usage like [(a + b)/c]², normal
parentheses should be used:
If angle brackets are used in math, then they should, according to the standard, be MATHEMATICAL LEFT ANGLE BRACKET U+27E8 and MATHEMATICAL RIGHT ANGLE BRACKET U+27E9. The HTML entities
⟨
and
⟩
denote other characters, U+2329 and U+232A. While
they “work” more often than the correct characters,
“working” here means just getting displayed in some odd font.
It is not appropriate to use the less than sign (<) and the greater than sign (>) as angle brackets. In mathematical texts, their usage should be limited to the relational meanings (though the relation could of course be other than the common ordering relation). It should never be a matter of glyph preference which character you use, though in this imperfect world, violations of this principle are sometimes understandable and foregiveable.
The Unicode standard says that the use of U+2329 and U+0232A as mathematical brackets is “strongly discouraged, because of their canonical equivalence to CJK angle brackets. This canonical equivalence is likely to result in unintended spacing problems if these characters are used in mathematical formulae.” In practice, when you use these characters, they will most probably be picked up from a font designed for ChineseJapaneseKorean (CJK) “ideographs”, therefore designed to fit into a largish square, typically causing typographic mismatch. On the other hand, font support to the correct mathematical angle brackets is still rather limited, so avoid them unless the contents absolutely needs them.
There’s a special of oddity
with the entity references for angle brackets.
Even though HTML specifications clearly define
⟨
and
⟩
as
〈
and
〉
, i.e. as denoting U+2329 and U+232A, most browsers treat them as denoting U+27E8 and U+27E9.
Strangely enough, in this issue, IE seems to be the only browser that works by the specifications.
HTML5 drafts have silently changed the meanings
of
⟨
and
⟩
to correspond to the behavior of most browsers.
Conclusion: Avoid entity references. By using character references, you will at least know which character will be used, even though you
still have all the font problems.
This section deals with fractional numbers; the section Towards twodimensionality deals with more complicated fractional expressions.
For fractions like 6/7, the common linearized notation is usually the best, especially within running text. It is not typographically good, but it is robust, and people are accustomed to such simple presentations of fractions on web pages.
In
twodimensional display formulas,
even fractions can be shown using a horizontal line, with
a number above and below it, but inside text, it’s hardly feasible—
Is 1 / 3 or ⅓ better?
The technique used for the first presentation of the fraction in the above example is based on heavy font size reduction of digits and using bolding to keep them fairly legible. This technique may fail, among other things, if the browser has a minimum setting for font size.
If the only fractions in your document are
vulgar fractions
½ ¼ ¾,
you might consider using the
ISO
Latin 1 characters for them, e.g. as entity references
½
,
¼
,
¾
. But if you need other
fractions too, this is not a good idea, since it would be odd
if different fractions had essentially different appearances.
In Unicode, there are a few more fraction characters, for 1/3, 2/3, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/8, 3/8, 5/8, 7/8, in the Number Forms block, but it would still be a limited repertoire. Moreover, although they are covered by many fonts, the font support is far from universal. The newest fraction characters, namely those for 1/7, 1/9, 1/10, and 0/3, have very limited font support (only a few fonts, none of which is shipped with any operating system or popular software).
You might use a linear notation with
sup
markup for the numerator
and the
sub
markup for the denominator.
The main problem is then that an expression like
^{5}/_{8} tends to cause uneven line spacing, due to
the poor quality of implementation of superscript and subscript style
in most browsers. It is therefore better to use CSS to reduce font size
and change vertical position.
You might also consider using the fraction slash (U+2044) character which could, according to the Unicode standard, solve the problem for numeric fractions in an elegant way. The fraction slash character is often more slanted than the normal slash (solidus) character. This is intended to correspond to special rendering where the numbers around it are in reduced size and vertically positioned in a manner that reflects a traditional way of writing fractions. However, browsers are not required to do such things.
As of this writing (2021), desktop browsers generally use such fraction rendering. This means that when they encounter two digit sequences separated by a fraction slash character, they construct a representation of a fraction from the string. This takes place automatically. The quality varies. Android browsers seem to display the construct just linearly,
Thus, if you have e.g.
5⁄8
in your HTML code,
some browsers will render
it as a formatted fraction, whereas others will display just
a normal 5 and a normal 8 separated by a fraction slash.
In the latter case, the result may look worse than
a simple 5/8 notation, due to the slanted appearance of the
fraction slash character: 5⁄8.
In both cases, the results depend on the font, and as usual,
the result can be messy if the
basic font used for the text, and hence for the digits, does
not contain the fraction slash character, which thus needs
to be taken from some fallback font.
Here is the rendering on your current browser:
5⁄8.
Some fonts (currently, mostly the socalled Microsoft C fonts like Cambria) contain information for constructing fractions using special shapes and positioning of digits and the slash. Using socalled OpenType features, such construction can be asked for.
On web pages,
contain superscript variants of glyphs, typically
for digits, lowercase letters a–z, and a few operators.
it has become possible to utilize OpenType features using the CSS
property
fontfeaturesettings
.
Browser support is becoming more widespread.
Using this approach, the fraction is written in simple linear
notation but wrapped in an element for which the OpenType feature
"frac"
is requested for.
OpenType also defines the feature "afrc"
for alternative
fraction format (typically, with horizontal line, not sloped fraction slash).
It is however supported in even fewer fonts than
"frac"
.
It is easy to create fractions using MathJax, with the
\frac
command. However, it creates a fraction with
numerator and denominator stacked, with horizontal line between them.
Such a presentation is usually OK in display formulas, but less so
in text. About tuning, see the Q/A pages
LaTeX force slash fraction notation
and
How do I typeset arbitrary fractions like the standard symbol for .5 = ½?
In MathML, a fraction can be described as a special
case of a fractional expression, using the
mfrac
element. The code is verbose, but a more
serious problem is that not all browsers support MathML,
especially when embedded in an HTML document.
On nonsupporting browsers, the code degrades to a rendering like
“5 8” instead of “5/8”.
To demonstrate what the different approaches yield on your current browser with its current settings, here is a table of different presentations for 5/8:
Approach  Notation in HTML document  Appearance 

Linear notation  5/8  5/8 text 
Special character  ⅝  ⅝ text 
sup and sub 
<sup>5</sup>/<sub>8</sub>  ^{5}/_{8} text 
Fraction slash  5⁄8  5⁄8 text 
Fraction slash and CSS 
<font class=num>5</font>⁄<font
 5⁄8 text 
OpenType "frac" 
<span class=frac>5/8</span>
 5/8 text 
MathJax 
\(\frac{5}{8}\)
 \(\frac{5}{8}\) text 
MathML 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mfrac bevelled="true">
<mrow> <mn>5</mn> </mrow>
<mn>8</mn>
</mfrac>
</math>
 $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{1ex}{$8$}\right.$ text 
The style sheet used is the following:
.num, .denom { fontsize: 70%; }
.num { position: relative; bottom: 0.5ex; left: 0.2em; }
.denom { position: relative; left: 0.05em; }
.ofrac {
mozfontfeaturesettings: "frac";
webkitfontfeaturesettings: "frac";
msfontfeaturesettings: "frac";
fontfeaturesettings: "frac";
}
There can be
line breaking problems
with the “/” character as well as the fraction slash, though
currently on minority browsers only.
To stay on the safe side, you could use markup like
<span class="frac">
for fractions and use the style sheet rule
.frac {
.
To underline something, you could use the
u
element in HTML. However, underlining is widely
taken as indicating a link on the Web.
Links want to be links, and you should
avoid doing anything that might make something look like a link
if it isn’t. But if underlining a symbol is part of an established
tradition in some field, go ahead and use the u
element.
It would be less logical to use the CSS declaration
, since here
underlining is not just a suggestion on rendering style but an essential
feature of content.
As an alternative, you could use the combining low line character (U+0332) after the symbol to be underlined. However, this character appears in a few fonts only and does not necessarily produce any better rendering. A simple test (with an underlined character and a symbol with combining low line): x, x̲.
It is common to use overlining
in mathematics, e.g. to indicate an average.
Somewhat illogically, HTML has no markup for overline.
In a
style
sheet you could suggest overlining, using the declaration
textdecoration: overline
.
However, as the property name says, it’s assumed to be decoration, not
part of the content proper, and in any case
style sheets are for suggestions only;
you should expect style sheets to get ignored fairly often.
Presentationally, note that the overline appears rather high above
the symbol.
Overlining something like
x
might be adequate if the context or explicit explanations make
it clear what is meant even if the overline does not appear. For casual
overlining a single symbol, you could use an embedded style sheet as follows:
<b style="textdecoration:overline"><i>x</i></b>
.
If overlining is essential, consider using the combining overline character (U+0305) after the symbol to be overlined. There are risks with fonts, but in most browsing situations, this method works. The rendering varies but is generally much better than in the CSS methods, as the overline is close to the letter. A simple test, first with a CSSoverlined letter, then a letter with combining overline: x, x̅.
For radicals (expressions of roots), it is customary in typeset mathematics to use a vinculum it more evident what belongs “under the root”. The vinculum is a horizontal line that joins with the radical sign, and the joining is difficult to arrange without using specialized software that draws math expressions.
You might consider the following options:
You might suggest overlining to make to produce a sort of vinculum:
√(a² + b²)
This
uses simple markup where the expression in parentheses is
enclosed between
<span
and </span>
.
In this context, the relatively high vertical placement of the
overline does not disturb. It is even desirable, and perhaps an even higher placement would be desirable.
Things may get distorted on many browsers
if there are exponents written using sup
under the root.
In our example, the parentheses are redundant
when overlining is applied.
I have experimented with tricks which would put the parentheses
inside span
elements with style sheets suggesting
the suppression (display:none
) of them in presentation.
Instead of textdecoration: overline
, you might set
a top border for the radicand, e.g.
.radic { bordertop: solid 1px }
. This seems to produce
reasonable presentation even when there are exponents
and subscripts (sup
and sub
elements)
in the radicand, as the following example illustrates:
What about roots other than the square root? There are
Unicode characters for the cubic root and fourth root symbol,
though they are less widely supported than the square root symbol.
For a general root, you might put the radical index
right before the radical symbol, in superscript style. Besides, you
could use CSS to suggest reduced spacing between those characters.
This would mean HTML markup like the following:
<span class="radic"><sup><var>n</var></sup>√</span><span class="radicand"><var>x</var></span>
with CSS like the following:
.radic {letterspacing:0.15em; }
.radicand {textdecoration:overline; }
This looks like the following in your current browsing
situation: ^{n}√x.
HTML tables
are intended for presenting data which is tabular.
We will not discuss here the tabulation of numeric data in general, since
the basic HTML constructs are simple and the fine tuning,
using attributes in HTML markup and/or style sheets, is beyond our scope.
But it needs to be noted that numeric data should normally be
rightaligned, which is not the default alignment in HTML tables,
so you often need the
align="right"
attribute (in td
or
tr
elements). It would often be desirable to align
numeric data on the decimal point, but this is in practice not possible
in the way defined in the HTML 4.0 specification. Instead,
some tricks are needed, such as using a monospace font and
right padding with
nobreak spaces so that the items in a column have the same
number of characters to the right of the decimal point.
For a matrix, the conventional notation
in mathematics is to use large parentheses around. This would
be rather difficult in HTML and would work well for very small
matrices only (cf. to methods discussed in the
Towards twodimensionality section below).
It’s probably best to use a different presentation
which makes matrices have an appearance which is suitably
distinctive, such as a special but not too striking
background color for cells. You might
write, say, <table class="matrix">
for any
table which presents a matrix, and include a style sheet rule like
table.matrix td { background: #fda none; color:#000; }
This results in something like the following on your browser
(when a cell spacing of 4 pixels and centering of cell contents
has been suggested too):
A = 

In the example above, the table representing a matrix has been embedded into an outer table so that we have been able to associate a symbol with the table. Similar techniques can be used e.g. when you wish to present the sum of tables; you would write an outer onerow table which contains the matrices in its cells and a plus sign in a cell of its own between them. The following example illustrates this:

+ 

= 

A superscript can be presented in an HTML document in several ways:
sup
markup.
It has essential problems in typographic quality, however.
Browsers implement it using reducedsize font in a raised position,
causing too small stroke width and too large size, as well as
uneven line spacing.
fontfeaturesettings
.
Browser support is becoming more widespread.
The following table illustrates the approaches. The last may show just normal characters instead of superscripts, if your computer lacks the Cambria font or your browser does not support access to superscript variants at the font level.
For subscripts, the situation is rather similar. However, there is a more limited set of subscript characters than superscript characters.
The HTML language has the
sub
and sup
elements for subscripts and
superscripts. But they should primarily be regarded as
stylistic suggestions only, rather than as
essential parts of the notation.
(See my notes on the intended use
of sub
and sup
.)
Naturally they can be valuable for “styling” math, too. To quote the
descriptions of
sub
and sup
in
WDG’s
HTML 4.0
Reference, replacing their markup examples with their appearance on
your browser:
Since SUB is inherently presentational, it should not be relied upon to express a given meaning. However, it can be useful for chemical formulas and mathematical indices, where the subscript presentation is helpful but not required. For example:
 Chemical formulas include H_{2}O (water) and C_{21}H_{27}NO (methadone).
 Let x = x_{1} + x_{2} + ... + x_{n}.
Since SUP is inherently presentational, it should not be relied upon to express a given meaning. However, it can be useful for mathematical exponents where the context implies the meaning of the exponent, as well as other cases where superscript presentation is helpful but not required. For example:
 The rent is due on the 1^{st} of each month.
 An example of a quadratic polynomial is 3x^{2} + 5x  7.
However, especially when superscripting is used to express exponentiation, superscripting is essential, and there need not be any contextual hints. It really makes a difference if 10^{9}, which is intended to mean 10 to the power 9, actually gets displayed as 109. The same applies to using superscripts e.g. in denoting the transpose of A by A^{T} (i.e., A immediately followed by T in superscript style).
Superscripts are often used for footnote references in print media. But in mathematical texts, such practice is best avoided in order to remove any risk of confusing such usage with exponentiation or other mathematical superscripting. Besides, such footnote references don’t work well on Web pages in general, as explained in the document Footnotes (or endnotes) on Web pages.
Twodimensionality in formulas will be discussed later, but here we mention some possibility of using superscripts and subscripts to simulate notations that should really be written above and below a symbol. The following example (which also uses special characters) shows the markup for an infinite sum and the resulting appearance on your browser.
∑<sub><var>i</var>=0</sub><sup>∞</sup> <var>x<sub>i</sub></var> 
∑_{i=0}^{∞} x_{i} 
That means summation of x_{i} (with i as subscript) from i=0 to infinity. In browsing situations where the infinity symbol is correctly displayed, the main problem on most browsers is that the infinity symbol does not appear above i=0 but to the right of it (in superscript style though). In the worst case, the reader might misunderstand the upper limit as an exponent of the lower limit! So it is perhaps better not to use a superscript at all but put the limits into a subscript, e.g. as i=0,…,∞ (which makes use of the horizontal ellipsis character; midline horizontal ellipsis would be better, but it’s less widely supported) giving the following appearance: ∑_{i=0,…,∞} x_{i}.
The infinity symbol ∞ might appear in fairly small size. In general, special symbols easily become unreadable when font size is reduced, so you might consider setting the font size larger than normal. In the above example with HTML markup for a formula and the formula itself side by side, the font size for the formula has been set to 125%.
The presentation of a summation expression could be tuned in different ways, some of which will be discussed in the sequel. But generally they lead to rather complicated constructs, and the complexity may cause problems on different browsers, current and future. However, some simple superscript positioning problems can be addressed relatively easily. Let us take the example of just positioning a simple onecharacter superscript above a onecharacter subscript.
In chemistry, sometimes both subscripts and superscripts
are used, for example in formulas for ions.
Consider the formula
NO<sub>3</sub><sup>−</sup>
where letter O has both subscript “3” and
superscript “–”.
The latter is the minus sign, and the
Ascii hyphen is a particularly poor surrogate here due to its shortness; such
problems were discussed in the section on
special characters.
It seems that the stylistically preferred
notation for ions has the superscript in the same horizontal
position as the subscript.
See, for example, the Ions
page in Eric Weisstein’s World of Chemistry,
which uses images to create such appearance.
The markup mentioned above by default creates an
appearance where the superscript is on the right of the subscript
(O_{3}^{−}).
Changing the order of the superscript and
subscript would not help much. But we can try to affect the horizontal
placement by using a negative margin. Since, in general, the notation
for an ion always has a superscript and may or may not have
a subscript, it seems practical to put the sup
element first and move the subscript to the left. This would mean
markup like
<span class="ions">NO<sup>−</sup><sub>3</sub></span>
(or maybe with div
instead of span
)
and a style sheet like the following:
.ions { lineheight: 1.8; }
.ions sub { marginleft: 1ex; }
and it would result in the following on your browser:
O^{−}_{3}.
Creating good appearance for variables with both
a subscript and a superscript is rather challenging: finetuning is
needed, and the rendering will still greatly depend on the font used.
Beware that widths of characters vary by font, so a horizontal shift
created by marginleft
or some other method
might be adequate for some fonts and poor for others.
Moreover, it cannot as such be used in conjunction with the
method of making line spacing even that will be described later
in this document.
The marginleft
property effectively shifts the
subscript to the left. The lineheight
ions
contains a lone
superscript, you would need to take extra measures, since the CSS code
above postulates that a sup
element appears immediately
after a sub
element.
From the HTML perspective, the basic problem
in situations like this
in that the sub
and sup
elements
have been defined in a rather presentationoriented manner rather than
structurally. When you write
<sub>i</sub>
, you're saying that
“i”
is a
subscript but not what it is associated with. In a case like
“a_{i}^{2}”,
“i”
is a
subscript for “a” whereas
“2”
is a superscript (exponent) for the expression
consisting of “a”
with subscript “i”.
There is no way to express this
structural relationship in HTML. Using extra parentheses, like
As you may have noticed
on many web pages,
subscripts and superscripts tend to mess up line spacing.
For example, a superscript
expression like A^{T} makes the line have more than normal
vertical spacing above it.
The reason is that subscripts and superscripts may increase the vertical
space needed for a line, and browsers quite naturally increase
height of a line box (making it larger than the value of the
lineheight
property).
The simple solution to this problem is to use a style sheet
that positions subscripts and superscripts vertically using relative
positioning, instead of the verticalalign
property. This prevents the effect that makes some lines higher
than others in the same paragraph.
The method is described in more detail in the document How to prevent uneven linespacing when subscripts or superscripts are used on web pages.
It is usually best to avoid setting the font size of sub
and sup
elements. The reason is that IE has a longstanding
bug (which is still present in IE 11):
It looks like IE (all versions till IE9) multiplies the font size of the <sub> and <sup> and their descendants with some variable coefficient (sth between 0.6 – 0.8 depending on the fontsize).
Even though it might seem suitable to set the font size to achiever similar sizing
across browsers, it has just the opposite effect. If you don’t set it,
browsers generally apply a size reduction (by about 80%)
fairly consistently. But if you set
fontsize
on sub
or
sup
,
IE will interpret it differently from other browsers.
If you really need to set the font size of subscripts or superscripts, you have a few options, like the following:
span
markup
(with class
attributes) instead of
sub
and sup
,
and set verticalalign
and fontsize
in CSS.
sub
and sup
but use JavaScript to
convert them to span
elements.
Demo page:
sup size fix.
Most browsers render
superscripts properly, or at least tolerably well. But they
often fail to handle
nested superscripts (or subscripts) well. And that means that
exponentiation in an exponentiation may get lost in graphic
presentation, too.
For example, some old versions of Internet Explorer render
a<sup>b<sup>c</sup></sup>
the same way as
a<sup>bc</sup>
.
However, modern browsers, including reasonably new versions of IE,
honor nested superscripts in rendering,
Here is test for your
browser: a^{bc}
(the letter c should be a superscript of b).
The preceding paragraph may
illustrate the problem that nested superscripts easily cause problems by
(almost) hitting the preceding line.
small font.
One approach (perhaps observable in this paragraph) is
to set the
There are various ways to avoid the problems with superscripts by using other notations:
The superscript and subscript problems can also be seen as special cases of a wider problem: how to present mathematical expressions in the conventional twodimensional format?
In mathematics, it is common to number equations and put the number on the right of an equation, in parentheses or brackets, as follows:
(a + b)^{2} = a^{2} + 2ab + b^{2}  (42) 

e^{x} ≈ 1 + x + x^{2}/2 + x^{3}/6 + x^{4}/24 + x^{5}/120 + x^{6}/720 + x^{7}/5040 + x^{8}/40320 + x^{9}/362880 + x^{10}/3628800  (43) 

Several approaches have been proposed to achieve such layout using just CSS and no presentational markup. However, the CSS methods (whether based on floating or on positioning) seem to suffer from various problems on current browsers. Some methods work well if the equation fits on one line but lead to confusion when it is divided over two or more lines. Since we wish to create pages that adapt to varying rendering widths (“fluid design”), a simple table is the practical solution:
<table class="eq" summary="Equation and its number." width="100%"> <tr> <td>the equation</td> <th align="right" valign="bottom">(number)</th> </tr> </table>
If you regard such a table as a deprecated “layout table”,
consider its markup, specifically the summary
attribute and
the use of th
(table header cell) element for the equation number,
which logically acts as a header for the row (the equation).
You can get rid of the presentational attributes width
,
align
and valign
by using
corresponding
CSS properties. If you like prevent the equation number from appearing in
bold, you should either replace (somewhat illogically) the
th
element by a td
element, or use the
fontweight
property in CSS.
As a whole, this could mean the following style sheet:
.eq { width: 100%; } .eq th { textalign: right; verticalalign: bottom; fontweight: normal; }
The following equation has been formatted with such a style sheet (and its HTML markup is “pure”):
(p + q)(r + s) = (p + q)r + (p + q)s = pr + qr + ps + qs  (44) 

For simplicity, let us first assume that we wish to present the
expression x divided by
a − b
in the conventional twodimensional format.
In this trivial case, the linearization
x/(a − b) would do just fine,
but in more complicated cases, twodimensionality would greatly
improve the clarity.
Using an image is one possibility, and if the
linearized version isn’t too complicated and you include it as the
textual alt
ernative, it might work fine. But let’s
see some other possibilities.
One might present the expression as twodimensional
preformatted text and include it using the
pre
element. This would be rather simple in our trivial case:
x  a  b
In more complicated cases, you could use a sort of Ascii art like the following:
b / f(x)   dx / 1 + x a
Several mathematical programs can format expressions that way
for you, and you could just cut and paste them.
Note that some markup, such as i
for italics,
is allowed within pre
elements.
And you need not be limited to Ascii; you could even use the
special characters outside ISO Latin 1,
in principle, though with special problems. Example (where the
integral sign may or may not
display correctly):
b f(x) ∫  dx 1 + x a
In any case, the visual quality of this method cannot be very impressive. Moreover, it creates accessibility problems, since it’s gibberish unless seen in the exact preformatted way.
An HTML
table might be used to make an expression appear
twodimensionally. Such an approach could be seen as
avoidable “use tables for mere layout”, though in a sense a table
construct would reflect the structure of data, e.g. in
<table cellspacing="0" cellpadding="0">
<tr><td align="center"><i>x</i></td></tr>
<tr><td valign="middle"><img src="1px.gif" alt="divided by"
width="100%" height="1"></td></tr>
<tr><td align="center"><i>a</i> − <i>b</i></td></tr>
</table>
width
attribute to a horizontal line.
The “table” displays as follows:x 
a − b 
Reasonable appearance might be achieved that way,
with perhaps tolerably graceful degradation in textonly media:
a simple character cell browser like Lynx
would basically display it as
x
divided by
a  b
which might be understandable.
But designing and writing suitable table markup would be rather awkward
for nontrivial expressions.
When presenting a display fraction using a table,
it would be simpler and more natural to have two rows only,
using a bottom border for the first cell. The border would create
a suitable horizontal line. However, in textonly presentation
and in nonvisual presentation, the data would appear as
x
a  b
which can be difficult
to interpret. This could be partly addressed
by using a summary
attribute for the table, e.g.
<table summary=
.
Let’s see what we could do with the integral above. Using a bit contrived table markup, we might get something like the following:
b  
∫  f (x)  dx 
1 + x  
a 
In any simple text presentation, it would look rather awful, though. You might consider providing a separate link to an alternative presentation, for such reasons.
For a rather common case of a definition that is most naturally presented in two lines, this approach might work tolerably, for simple definitions:
δ_{ij} =  { 
Returning to simple examples, let us consider how we might
present (a − b)/x
twodisplay
property in CSS (cf.
ideas about removing redundant parentheses above).
We would start from the simple linear notation
(a−b)/x. We would put the parentheses and the slash each
inside a span
element containing that character only,
and we would also use span
for the numerator and denominator.
Then, using class names assigned to the span
elements,
we would suggest in CSS the suppression of the display of those
characters, presenting both the numerator and the denominator as a block.
To create a horizontal division line, one might use
underlining of the numerator,
but using a bottom border is simpler and gives a better rendering.
This means markup like the following:
<span class="fracexp"><span class="nom"><span class="lin">(</span><i>a</i> −
<i>b</i><span class="lin">)</span></span><span
class="lin">/</span>
<span class="den"><i>x</i></span></span>
with the following style sheet:
.lin { display: none; }
.fracexp { display: inlineblock; }
.den, .nom { display: block; }
.nom { borderbottom: solid 1px; }
This is what it looks like on your browser:
(a − b)/ xAnd it degrades to (a − b)/x in nonCSS browsing situations. If you think that such graceful degradation is not needed, you can use simpler markup, with just one element for the fractional expression containing two elements, one for the nominator and one for the denominator.
In HTML5,
the canvas
element lets you write text
to specific positions, by pixels.
The writing operations are based on JavaScript functions
standardized in HTML5, and the code needed to draw
an equation is relatively bulky. In special cases,
this may be feasible,
especially if you also use canvas
for other
purposes (like an animation).
We could consider other approaches too, such as using positioning with style sheets instead of tables. But it probably suffices to conclude with the following note:
In HTML, a rich set of mathematical symbols and some other basic notations can be used, but currently with accessibility problems that users need to solve. Twodimensional presentation of expressions via HTML markup is trickery and handcraft, now and in the foreseeable future. Thus, quite often it is better to use JavaScriptbased tools or images for such purposes.