A Call for Better Indexes
Indexes. Photo credit: Lendingmemo.
An index is an important component of a book. It provides a view with a much smaller granularity than a table of contents, reveals what is not present as well as what is, and by abstracting concepts can lead the reader to unexpected content.
Most academic books in mathematics are typeset in LaTeX, which has an excellent system for indexing. By inserting \index commands in the source code, and running the MakeIndex program as part of the LaTeXing sequence, an author can iteratively build up an index during the late stages of the writing process, safe in the knowledge that the automatically generated page locators will be correct.
One might expect that the quality of indexes would have improved since the preLaTeX days when indexes had to be generated by hand. But in my view they have not. Most indexes I see have obvious flaws.
Why is this? I think there are two main reasons. First, most authors write at most one or two books during their careers and so indexing is a task that they rarely carry out and therefore have little chance to practice. Second, indexing tends to be left to the last minute, leaving no time to study how to produce a good index. In the days before LaTeX it was much more likely that a professional indexer would be employed, and a professional with some knowledge of the subject area will do a much better job than a hurried author.
I believe it is well worth authors putting a lot of effort into indexing. The benefit is not just a better index but a better book, because the indexing process forces you to think carefully about structure and content. I am working on a book index right now and in the process of choosing entries and inspecting draft indexes I found that some important definitions were missing, spotted improvements to section titles (whose inaccuracy only became apparent when I tried to use them as index entries), and corrected instances where I had spelt words differently at different places. My experience concurs with that of Don Knuth, who said in his article MiniIndexes for Literate Programs, “a little extra time spent on indexing generally leads to significant improvements in the text of any book that is being indexed by its author, who has a chance to see the book in a new light”.
How does one produce a good index? How does one know when one has produced a good index? This post is not the place to try to answer these questions. Some thoughts on how to index are given in Section 13.4 of my Handbook of Writing for the Mathematical Sciences (2nd ed., 1998) and I hope to provide further advice in future blog posts.
But I would like to mention one variable that has a correlation with the quality of an index: its length. Many indexes are simply too small for the content of the book.
The book The Indexing Companion by Glenda Browne and Jon Jermey (CUP, 2007) says that indexes should range from 2% for a “simple book” to 15% for a “complex book”. I think mathematics texts fall towards the “simple” end of the scale, compared with a biography or historical book, for example.
Here is a table showing the relative size of the indexes of some books that I regard as having good indexes.
Book 
Total
pages 
Index
Pages 
Percentage
Index 
Olver et al., NIST Handbook of Mathematical Functions 
951 
65 
6.8 
Horn and Johnson, Matrix Analysis 
643 
36.5 
5.7 
Stewart and Sun, Matrix Perturbation Theory 
365 
15 
4.1 
Press et al., Numerical Recipes 
1235 
41 
3.3 
Knuth, The Art of Computer Programming, Volume 2, 2nd ed 
638 
20 
3.1 
Graham et al, Concrete Mathematics 
657 
20 
3.0 
Boyd and Vandenberghe, Convex Optimization 
716 
16 
2.2 
Strauss, Partial Differential Equations 
454 
9 
2.2 
Trefethen, Approximation Theory and Approximation Practice 
305 
6 
2.0 
My own four books have indexes occupying 2% to 3.7% of the book.
My rule of thumb for a mathematics book is that if your index occupies less than 2% of the book then you should think carefully about whether it needs extending. In particular, ask yourself whether you have indexed items twice when appropriate. For example, “vector space, dimension” and “dimension of vector space” are probably both needed, and likewise “norm, Euclidean” and “Euclidean norm”. Furthermore, you should index synonyms for important concepts, even if they do not appear in the book. For example, if you use the modern term “significand” in floating point arithmetic, you probably need an entry “mantissa, see significand”.
Finally, it is worth emphasizing that readers do care about indexes. The Society of Indexers (based in the UK) produces an excellent journal The Indexer, and all articles over three years old are freely accessible. A regular column Indexes Reviewed collects comments on indexes from book reviews, under the headings “praised”, “censured”, “omitted”, and “obiter dicta”. It makes an interesting read. Here are three notable snippets:
This is a book that clatters around in a dark closet of irrelevancies for 450 pages before it bumps accidentally into its index and stops.
It’s a curious production, without an index. A biography without an index is like a wheelbarrow without handles.
The stupidest index I’ve seen was in the manual for a Kia car. Changing the wheel? Don’t look under C for “changing” or S for “spare wheel” or W for “wheel” or J for “jacking” or T for “tyre” or even F for “flat tyre”. Nope, it was listed under H. For “How to change a wheel.”

Nicholas Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. He is President Elect of SIAM. 