How to best typeset "special" constants (complex unit i, Euler's number e,...)?

Question

I'm trying to come up with a typographically appealing way to express "special" numbers such as the complex unit i = sqrt(-1) or Euler's number e. It has to be such that it cannot be confused with regular numbers (such as the running index i, for example), and would ideally work for serif as well as sans-serif fonts.

I was briefly thinking of typesetting these numbers in bold, but found that bold face is often used to indicate vector quantities.

[Edit: Johannes Küster suggests to use upright for constants (about 14 minutes into his talk).]

I've seen double-stroked small letters too, and must say this isn't without appeal. I have no idea how to consistently typeset those, though. The (outdated?) bbm package seems to provide at least some basic functionality.

What do you use to represent special numbers, why do use it, and how to you typeset your solution?

Answer 1

Consistency is the primary goal. So the first task is to know what "special numbers" we need and define commands for them:

\newcommand{\euler}{e}
\newcommand{\ramuno}{i}

(ramuno was how some Italian mathematicians of the 16th century called the quantity that squared gives –1; then Euler started using i).

The mathematical typography tradition usually didn't have a special treatment of these symbols. See, for example, n. 359 in Gauss's Disquisitiones Arithmeticae, where the equivalent of

$\cos\frac{\lambda kP}{e} + i\sin\frac{\lambda kP}{e}$

is found (the edition I consulted is from the Werke by the Königlichen Gesellschaft der Wissenschaften in Göttingen, vol. 1, 1863, page 450). There's no doubt what this i is denoting.

http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235993352&IDDOC=137206

However, in recent times, under the influence of physics and applied mathematics, people started to denote "constants" with upright letters. There's even an ISO regulation about this, which is compulsory in some fields where uniformity among papers and books is very important.

In pure mathematics there's essentially no rule. Do as you like or how your field is used to. Using special names for the special numbers allow you to change the appearance of your document just by changing the definition.

If you feel that there may be confusion between the "imaginary unit" (no worse name could be chosen for it) and an index (for summations, for instance), you have three strategies:

1. use a special denotation for the imaginary unit;

2. don't use i as an index;

3. forget about it and let the reader know from the context.

Strategy 2 is used by Graham, Knuth and Patashnik in their "Concrete Mathematics". Strategy 3 is very common in math textbooks.

Answer 2

Unicode has special glyphs for these symbols: 0x2148 for imaginary i, 0x2149 for imaginary j, 0x2107 for Euler's constant, etc (although on most fonts they look ugly).

If you are using a unicode aware engine and a opentype math font, you can just type these directly or use the corresponding macro for them (ConTeXt uses \imaginaryi, \imaginaryj, \Eulerconst, etc.; I don't know what names unicode-math package for LaTeX uses).

EDIT: I don't know whether \Eulerconst refers to "Euler's number" e or not. As pointed out by Caramdir, it not used frequently.

Answer 3

Whenever I'm unsure about how I'll eventually want something to look, I'll create a \newcommand so that I can hope to control things globally.

In this particular case, I might define something like

\newcommand{\myspecial}[1]{\mathrm{#1}}

which could be used, for example, as

$\myspecial{e}$

(I wouldn't necessarily use \mathrm, it's just to demonstrate the idea).

Another reason I find this approach useful is that I can then easily grep the file to find all occurrences, something like

egrep 'myspecial' myfile.tex

Answer 4

How about reserving Sans Serif type for special numbers? Usually the main font is some kind of Roman, so displaying the imaginary unit in a non-italic Sans Serif would make a difference.

So, for example

\newcommand{\imagi}{\mathsf{i}} \newcommand{\myexp}{\mathsf{e}}

produce something like what there is in the screenshot

I hope the letters are big enough to tell the difference

Answer 5

Just use roman (\mathrm) for all non-variables. In particular, this includes standard operators, functions, and constants (such as $\mathrm{lim, sup; exp, ln; e, i}$) and can be extended to any mathematical object that has a unique fixed meaning, without needing any parameters to be specified, throughout a particular work in which it is defined.

Answer 6

This is more of an extended comment —

If your reader or audience quickly glances at your article/slide, will they be able to tell an i (the imaginary unit) from an i (a running index)? I would contend not. They may be able to avoid getting confused by considering the context or paying close attention to the typeface, but ideally they shouldn't have to.

Working in quantum informatics, this is a notational issue which arises all the time: we often use complex numbers and combinatorics simultaneously. Sometimes people make no distinction at all between indices and the imaginary unit, which is obviously not great. Occasionally someone decides to use \iota for the imaginary unit, which is not a real improvement over using just i. People who don't often write on the subject and are therefore less accustomed to using complex numbers will sometimes represent the imaginary unit explicitly as \sqrt{-1} (which looks terrible, especially if they occur several places in equations including exponents, please don't do this if you use the imaginary unit more than a couple of times).

After one time encountering the expression e^{2\pi iijk/N} in someone else's work, I made the choice to never, ever to use i as a running index if there is even the slightest chance of me using complex numbers in my article, notes, or slides. That goes for hand-written work as well. My indices start with either h or j, usually the latter. I've found that this is a good policy.

I similarly try to avoid using e for anything but Euler's constant, despite often writing about the edges of graphs — though at least I have \exp to use as a respectable alternative when I exponentiate. I'm not quite as adamant about that, as e is larger and easier to spot variation of typeface for, and because I can keep my uses of e as a non-constant well contained in practise.

Having said that, I make the imaginary unit and Euler's constant appear however I find most aesthetically pleasing. For me that is \mathrm e and plain-old i.

Answer 7

Sorry to chime in so many years later. I often like to use a different upright font for them. Here, for example, is Hermann Zapf’s Euler as a companion for (a clone of) his Palatino:

\documentclass[varwidth, preview]{standalone}
\usepackage{mathtools}
\usepackage{unicode-math}
\defaultfontfeatures{Scale=MatchLowercase}
\setmainfont{TeX Gyre Pagella}[
  Scale=1.0,
  Ligatures={Common, Discretionary, TeX}]
\setmathfont{Asana Math}
\setmathfont[range={up/{Latin,latin,Greek,greek},
                    bfup/{Latin,latin,Greek,greek},
                    cal, bfcal, frak, bffrak},
              script-features={},
              sscript-features={}
            ]{Neo Euler}
\newcommand\upi{\symup{i}}
\newcommand\upe{\symup{e}}
\begin{document}
\begin{align}
  \upe^{\upi x} &= \cos{x} + \upi \sin{x} \
  \upe^{\upi \uppi} + 1 &= 0
\end{align}
\end{document}

This can be done just as easily in PDFTeX, by loading the family zeur as a math alphabet.

Euler is also a good match for Optima, by the same designer. This is also a good use for the Upright Italic style of Computer Modern.

Answer 8

First off, since they are scalar constants (and not vectors) they should not be boldface. Furthermore, numeric constants such as values 0, 1, 2, .. are typeset upright, and not slanted like variables. Sans-serif styles seem to be reserved for tensor quantities in isomath when not in component form.

So use \mathrm{e} and \mathrm{i} or {\rm e} and {\rm i}

Example:

_{PS. Example isomath pdf output}