He also presents some historical and international context, looking at ancient Egypt and Rome and at China and Japan, and takes the time to dwell on interesting details. (The Romans wrote four as IIII and not IV, for example. That and the other "subtractive" representations became popular in the 13th century, for shortening dates on inscriptions, but were never used in calculations, for obvious reasons.)
Lockhart starts with the different motivations for representing numbers and some of the possible approaches. As well as the common five-barred gate tally system, he imagines a Hand people, who count with thumps and claps, a Banana people, whose language has base four numbers, and a Tree people, who have a symbolic base seven system. He introduces the Egyptian and Roman "marked value" systems and the Chinese and Japanese systems, with separate symbols for the first nine numbers and the use of a soroban or bead abacus.
What are the goals of a number representation scheme and what trade-offs are involved? "Our criteria for written systems should be ease of perception, ease of reading and writing, and a sensible grouping size." The "bold and original approach to arithmetic devised by the Hindu mathematicians of the sixth century AD" involved the introduction of zero and a full place-value system; this enabled the methods we use now for addition and subtraction.
"we have arrived at a point in our brief pseudohistory of arithmetic where things are starting to feel a bit more familiar... The problem with familiarity is not so much that it breeds contempt but that it breeds loss of perspective. Having grown up with the Hindu-Arabic decimal place-value system and being constantly inundated with these particular symbols and digit sequences, as well as our names for them, it is easy to lose sight of the big picture and to allow convention (not to mention schooling) to substitute for understanding."
The major European improvement on the Hindu-Arabic decimal place-value system was the metric system. Units of measurement as such are outside Lockhart's scope, but he touches on the non-decimal number systems we have retained for time and angle measurements, the old British pounds-shillings-pence currency system, and conventions for decimal points.
Turning to multiplication, Lockhart looks at how it works in the different number systems he has presented, real and invented, and their advantages and disadvantages. This forces the reader to consider the underlying mathematics and to separate that from the particulars of specific methods. And he presents division as necessary both abstractly, as an inverse to multiplication, and for the practical problem of sharing.
A kind of digression, a chapter "Machines" gives a brief history of calculating machines and an argument for using them to avoid boring computations. Lockhart then argues that "the arithmetic of fractions is not really of any practical concern" and
"they are in no way necessary in the practical everyday world. Even if you live in a backward country like the United States that still uses outdated imperial units, you can still just take out a calculator and do all of the necessary computations. Put everything in the same decimal format, and let the machine do its thing. Then use as many digits of this approximation as are appropriate."
But he looks at some of the elegant, beautiful and amusing properties of fractions, and at how the different ways of representing them work for comparison, approximation, and arithmetical operations.
Lockhart concludes his tour of arithmetic with the introduction of negative numbers, which serves as an example of the mathematical concept of extension and allows the construction of what mathematicians call the rational number field, "the proper setting for arithmetic". And he ends with a glimpse at some simple combinatorics, at what we can do if the things we are trying to count have some structure, some pattern, order or symmetry.
Arithmetic is not aimed at children at all — its vocabulary and syntax are formal, though not academic — but with a tiny bit of massaging I used the first chapters to teach a six year old that there are different ways of representing numbers, and how to count in base four. It helped that her age in "Banana language" was ba na na. And we are continuing to work through it, with the simple problems sprinkled throughout a big attraction: How would you design a tabula abacus for the Banana people?
A more obvious audience would be primary school teachers. Arithmetic is not a replacement for traditional teaching materials — scaling what I'm doing with one child to a classroom seems sadly impractical — but it offers a broader perspective on the standard fare of the curriculum and perhaps some ideas for making it engaging. More generally, anyone curious about numbers and their representation and manipulation, and willing to rethink what may seem like obvious concepts, may find revisiting arithmetic surprisingly fun.