v 6.40.00
19 Dec 2022
updated 21 Dec 2022

Antique Algebra & Higher Arithmetic

Over the years I’ve collected I daren’t say how many books with a view to using them as references when I got the opportunity of putting together a stout volume that would be titled something like “Things I wish I’d known earlier” or “Things I wish had been properly explained at the time”.

In fact several slightly slimmer volumes would have been necessary, separately covering maths, physics and physical chemistry – not textbooks, of course, there are far too many of those anyway, but elaborations upon interesting topics that had only been touched upon, or remedial accounts of important topics that had been rather disgracefully mangled, during my undergraduate and postgraduate education.

The webpages that follow are in the first category – elaborations of algebraic techniques, old or sometimes new, that relate to the theory of numbers in all sorts of different ways. They don’t necessarily dig deep, there’s Wikipedia for the nitty-gritty, but they hopefully bring things out of shadow into sunlight. Many will seem excessively discursive, or maybe just excessive.

Space is also devoted to the Higher Arithmetic, a rather quaint name for the arithmetical topics that are regarded as too esoteric for even senior secondary school but not sufficiently highbrow for university study – continued fractions, for example.

And there are topics which, in my pre-internet era weren’t even covered by the chemical physics literature, such as Gaussian quadrature, numerical simulation and so on – it was considered infra dig to refer to such sordid details, essential though they are to serious computation. I had to work things out for myself, in conjunction with my brilliant postgraduate colleague Robert S Milligan, though now, 50 years on, it’s probably much more professional. And of course now there’s the internet …

The (mostly popular) books listed below contributed to a greater or lesser extent to the stuff I’ve committed to my website – don’t think I’ve read them all cover to cover! Sometimes they contradict one another, and sometimes a popular book can unlock a concept much better than any textbook – as for instance (for me) the significance of Euclid’s Lemma for a really neat proof of the Fundamental Theory of Arithmetic.

Here and there I’ve also quoted from websites as indicated in situ.

But don’t mistake me for an expert in these things. Hopefully, each essay will reach ‘critical clarity’ and enable the interested reader (myself also) to tackle more substantial accounts elsewhere.

In a 2004 interview Martin Gardner (21 Oct 1914 – 22 May 2010), renowned throughout the civilised world, said "I go up to calculus, and beyond that I don't understand any of the papers that are being written. I consider that that was an advantage for the type of column I was doing because I had to understand what I was writing about, and that enabled me to write in such a way that an average reader could understand what I was saying. If you are writing popularly about math, I think it's good not to know too much math.”

I wholeheartedly agree – and trust that what’s sauce for Gardner’s goose is sauce for this elderly gosling too.

You may well be disappointed, albeit only slightly, that there are very few links to the actual essays, the principal reason being that though many of them are substantially complete in MS Word 2003 (!), I’m still uncertain as to how best to present them on-line. There are several possible formats – html, pdf or even jpg, for example, but they each have serious drawbacks.

Vesti la giubba

I must reiterate that this isn’t an attempt at an online textbook (there are plenty of very good ones nowadays) – it’s merely a collection of interesting or useful curiosities that were ignored or glossed over by the textbooks, pedagogues and lecturers during my schooldays and undergraduate era.

There is in fact a cottage industry in maths stuff these days, and the best and most expensive by far is Jan Gullburg's impressive volume. Buy it if you can, steal it if you must.

Number Numbers
Arithmetic Notation and Terminology
New Math
Diophantine equations
Computation Blah
Modern multiplication
Peasant multiplication
Ancient Egyptian multiplication
Calculus Old Math
Fundamental Theorem of Calculus
Algebra Blah
Roots of unity
Roots of minus unity
Minus times minus
Geometry Blah
Probability Newcomb-Benford law
Monty Hall controversy
Curricula Bibliography
The Philosophical Pearl
General Blah
Appropriation All Ideas are Second-Hand
Math and Musical Theater