How should imaginary numbers be typeset?

Question

I've never had to typeset complex numbers before and I'm finding that I'm uncertain about what best-practices are. My question is really about typesetting just i. (Hence my title referring to imaginary as opposed to complex numbers.)

I would like to be consistent with the textbook which uses a slantstyle. But aside from the choices of the textbook, I'm curious about what others think: Should it be upright? Should it be italic like a variable name?

When I use the default math style, I don't like the appearance, particularly in conjunction with exponents. It looks too crowded and busy to me. Is this just because I'm not used to typesetting for complex numbers? Or, is there some kind of italic correction I could do that would fix things: \/ seems to be ignored in math mode.

Here's my minimal working example:

\documentclass{article}
\usepackage{amsmath}
\pagestyle{empty}
\usepackage[margin=2.25in]{geometry}
\setlength{\parindent}{0pt}
%%
\newcommand{\mi}{\mathrm{i}} %% roman "i"
\newcommand{\di}{i}          %% default math "i"
\begin{document}

    \verb=\mathrm= style: (not consistent with font choice of textbook)
        \begin{align*}
        \mi^0 &= 1     \\
        \mi^1 &= \mi   \\
        \mi^2 &= -1    \\        
        \mi^3 &= -\mi 
        \end{align*}

    Default math style: (better matches the style of the textbook, but already looking crowded.)
        \renewcommand{\di}{i}
        \begin{align*}
        \di^0 &= 1     \\
        \di^1 &= \di   \\
        \di^2 &= -1    \\        
        \di^3 &= -\di 
        \end{align*}
    Whichever choice, the following looks too busy. 
    \begin{align*}
        \mi^n &= \mi^{4\times k + r} = \mi^{4\times k} \times \mi^4 = (\mi^4)^k \times \mi^r = 1^k \mi^r = \mi^r \\
        \di^n &= \di^{4\times k + r} = \di^{4\times k} \times \di^4 = (\di^4)^k \times \di^r = 1^k \di^r = \di^r
    \end{align*}
    And if I change the \verb=\times= to \verb=\cdot= it looks even worse:
    \[
        \di^n = \di^{4\cdot k + r} = \di^{4\cdot k} \cdot \di^4 = (\di^4)^k \cdot \di^r = 1^k \di^r = \di^r
    \]

\end{document}

I know I could completely drop using \times or \cdot but for my particular audience I want to emphasize the multiplication.

I think it's the dot on the $i$ my eye is visually objecting to (in which case there's not much to do about it, I guess).

Answer 1

The possible visual clash of the dot with the exponent can be cured by adding a small kern:

\newcommand{\iu}{{i\mkern1mu}}

Experiment also with smaller kerns and note that the setting depends on the font used, so it can't be a universal recipe. Here's an example: left the kerned version, right the unkerned one.

Some people maintain that mathematicians should conform to ISO standards (see Timtro's answer), but my opinion is that ISO standards should conform to centuries long tradition of mathematical typesetting in the first place. We can look at an article by Sophie Kowalewski published by the Acta Matemathica, one of the journals that set the highest standards for math typesetting. On the first page we see

and on page 89

There is no doubt whatsoever for the meaning of “i” and “d”.

Maybe this is considered too old fashioned. Here is an example from a big publisher, with considerably high standards. It's an excerpt from a paper in “Differential Geometry and its Applications”, volume 26(5) 2008, pages 553–565 (top of page 563). Access is restricted, so I provide a cropped image showing just the important graphic part and no complete text.

Answer 2

I'd always use a math italic i. But conventions vary Many engineering disciplines use j rather than i. Unicode has a specific slot U+2148 (ⅈ) which is a double struck italic i. This is the ⅈ (ⅈ) entity in MathML and HTML5. (The convention started with Mathematica, I can't say I like it much, but it's there if you want an unambiguous notation.)

Answer 3

According to ISO 80000-2:2009, Quantities and units---part2: Mathematical signs and symbols to be used in the natural sciences and technology, the upright i is the correct choice. Quantities which are not variable across time or context (such as immutable constants of nature) are upright while variables, contextual constants, running numbers (dummies), are italic. This is common practice for functions too, and is mostly why we use $\sin(x)$ and not $sin(x)$. (We also wouldn't write $sin(x)$ because the interpreter has no way of knowing the letters sin are not three consecutive variables.)

I sometimes define \newcommand{\iu}{\mathrm{i}\mkern1mu} for my documents. However, it look good without the kern too with LuaLaTeX and unicode-math. In fact, I sometimes find it a bit odd with the kern because there seems to be an imbalance before and after the exponent, which you can see from your 'busy' example.

produced from

%!TEX program = lualatex
%
\documentclass{article}
\usepackage{amsmath}
\pagestyle{empty}
\usepackage[margin=2.25in]{geometry}
\setlength{\parindent}{0pt}
%%
\usepackage[math-style=ISO, partial=upright]{unicode-math}
\usepackage{lualatex-math}

\begin{document}

\newcommand{\iu}{\mathrm{i}\mkern1mu}
   \verb=\mathrm= style, conforming with ISO 80000-2:2009 :
        \begin{align*}
        \iu^0 &= 1     \\
        \iu^1 &= \iu   \\
        \iu^2 &= -1    \\
        \iu^3 &= -\iu
        \end{align*}

\newcommand{\di}{\mathrm{i}}
    Also conforming, but without the manual kern:
        \begin{align*}
        \di^0 &= 1     \\
        \di^1 &= \di   \\
        \di^2 &= -1    \\
        \di^3 &= -\di
        \end{align*}
    Whichever choice, the following looks busy, largely because it is quite busy :)
    \begin{align*}
        \iu^n &= \iu^{4\times k + r} = \iu^{4\times k} \times \iu^4 = (\iu^4)^k \times \iu^r = 1^k \iu^r = \iu^r \\
        \di^n &= \di^{4\times k + r} = \di^{4\times k} \times \di^4 = (\di^4)^k \times \di^r = 1^k \di^r = \di^r
    \end{align*}
    And with the \verb=\times= to \verb=\cdot=:
   \begin{align*}
        \iu^n &= \iu^{4\cdot k + r} = \iu^{4\cdot k} \cdot \iu^4 = (\iu^4)^k \cdot \iu^r = 1^k \iu^r = \iu^r\\
        \di^n &= \di^{4\cdot k + r} = \di^{4\cdot k} \cdot \di^4 = (\di^4)^k \cdot \di^r = 1^k \di^r = \di^r
   \end{align*}

\end{document}

Answer 4

If you're looking for conventions then Elsevier, (a huge publisher) has guidelines: "Preparing Articles in LaTeX", which state (page 11 of PDF, 10 of guidelines) when i is used as an imaginary it is conventionally typeset in a Roman typeface. The example they give has the dot on the i, Roman typeface makes the i upright.

Answer 5

Der Brockhaus Naturwissenschaft und Technik Band 2 (2003) (famous german Encyclopedia on Science) uses upright i

Bergmann Scheafer Lehrbuch der Experimantalphysik (2010) (very famous and rich books on physiks) uses upright i (as fa as i remember)

One possible explanation is, that i is not a variable which usually are typeset in italic. According to a DIN (german industry standard) j for imaginary numbers is allowed in electronic engineering.

Answer 6

Well, based on your comment:

it's the dot on the $i$ my eye is visually objecting to ...

you could use what I have been using which is a dotless i:

Code:

\documentclass{article}
\newcommand*{\I}{\imath}%
\begin{document}
    $\I^2=-1$
\end{document}

Answer 7

To paraphrase Paul J. Nahin who wrote a book on the history of i, it would be no small job to do it well from the Hindus downwards. Here is an Engineers perspective.

In Engineering we have used an i (in all topics, such as thermodynamics, mechanics etc) with the exception of Electrical Engineering, where we use a j and for a good reason as the i is normally used to denote the current. A math italic is the best notation and clear across all disciplines.

As a matter of good practice, use what is most common in your field. If you worried about confusion and want to take the dot out of the i, please don't, as anyone that is confused with such a simple matter is unlikely to spot that the "i" is a dotless "i", not to mention the confusion of Turkish mathematicians. David's suggestion of U+2148 (ⅈ), just looks ugly.