The AI Journal

An English shopkeeper reckoning change in 1965 could divide a shilling into thirds without leaving a remainder — four pence apiece, exactly — a manoeuvre that the same shopkeeper’s decimalised grandchild, splitting a pound into three, cannot perform without recourse to a third of a penny that does not exist. This is not nostalgia; it is arithmetic. Twelve pence to the shilling meant the shilling divided cleanly by two, three, four and six. A hundred new pence to the pound divides cleanly by two, four, five, ten, twenty, twenty-five and fifty, but not by three, six, seven, eight, nine or eleven — a narrower field of clean division than duodecimal offered, purchased at the cost of exactly the fractions retail trade used constantly and in exchange for fractions banking and engineering used rarely. The received account of the 1971 changeover, that Britain finally shed an antiquated and irrational currency for a modern and sensible one, gets the mathematics backwards. Twelve was never irrational; it was, for the purpose people actually put it to — dividing goods, dividing bills, dividing change among several buyers — the more capable base. What Britain abandoned in 1971 was not bad arithmetic but a system optimised for a kind of arithmetic that fewer and fewer people were still doing by hand.

The case was made, at the time, by someone with no reason to romanticise the old system out of sentiment. Alexander Craig Aitken, the Edinburgh mathematician known for a memory precise enough to recite several hundred digits of pi, published The Case Against Decimalisation in 1962, a year before the government’s own inquiry reported. Aitken’s argument turned entirely on divisibility: ten’s only prime factors are two and five, so any fraction of ten that is not a half or a fifth produces a recurring decimal, while twelve’s factors of two and three give clean division by two, three, four and six, and its square, one hundred and forty-four, divides cleanly by eight, nine, sixteen, eighteen, twenty-four, thirty-six and more besides. For a trade economy built on halves, thirds, quarters, sixths and dozens — parcels sold by the dozen or gross, prices quoted to the nearest farthing, dividends split three or six ways — this was not a curiosity. It was the base doing exactly what a base should do: matching the operations its users performed most often.

The Committee of Inquiry on Decimal Currency, chaired by the Earl of Halsbury and appointed in December 1961 by the Chancellor Selwyn Lloyd, reported in September 1963, and its report is instructive precisely because it did not dispute Aitken’s mathematics. The committee’s terms of reference asked it to advise on the most convenient and practical form a decimal currency might take, and on the timing and cost of the change — not on whether decimalisation was mathematically preferable, a question the government had already settled by fiat before the committee sat. Witold Kula’s history of pre-metric measurement, Measures and Men, supplies the wider pattern this fits into: across early modern Europe, systems of division and grouping that look irregular from a modern vantage point were typically calibrated to the transactions a given trade performed most often, with grain measures, cloth measures and land measures each accreting their own local logic of division rather than sharing one abstract scale. Duodecimal currency was this kind of system — not a relic that had somehow failed to notice the existence of ten, but an adaptation to what shops, markets and bills actually required of it.

What changed by the 1960s was not the mathematics of retail transaction but the identity of the party doing the dividing. Stephen Chrisomalis’s comparative history of numerical notation argues that number systems change not because one base is objectively superior to another in the abstract but because the social, economic and technological context in which people write and manipulate numbers shifts beneath them; a notation optimised for one set of practices becomes a liability once the dominant practice changes. By the early 1960s the dominant practice was migrating from a shop assistant’s mental arithmetic to a till, a ledger machine, and increasingly an electronic calculator, all being built, sourced and exported on a decimal basis because the industrialised world Britain traded with had already gone decimal in its currencies. The Halsbury Report and the parliamentary debates that followed it returned repeatedly to this point: manufacturers of calculating machinery argued that a duodecimal currency, however well suited to a butcher’s mental sums, was becoming a genuine cost to firms trying to mechanise their accounts, and the numismatist Kevin Clancy’s account of the coinage decision records that compatibility with decimal machinery and with the currencies Britain traded against weighed as heavily on the committee as any argument about convenience at the till. Decimalisation was won by the arithmetic of ledgers and export markets, not by a demonstration that ten divided transactions better than twelve did — because it does not.

The strongest objection to this argument is that it overstates how often the disputed fractions actually arose in practice. Most transactions in a shop did not require dividing a bill into thirds or sixths; they required addition, and addition works identically in any base. A reviewer of Aitken’s pamphlet in Nature at the time made exactly this point, judging his historical case for decimalisation’s costs skilful but finding him considerably less persuasive once he moved from diagnosing the problem to prescribing a wholesale return to duodecimal coinage, on the grounds that the everyday advantage of clean thirds was real but marginal against the much larger, recurring cost of teaching every schoolchild and clerk in the country a currency that did not match the base-ten counting system they already used for everything else. This is fair, and it narrows the claim rather than defeating it: duodecimal currency was not obviously worth defending as a going concern in 1963, once transactional convenience is weighed against universal arithmetic instruction and machine compatibility across an entire economy. But narrowing the claim is different from inverting it. The question that survives is not whether Britain was right to decimalise — on balance, given where calculation was heading, it probably was — but why the change is still remembered as the correction of an irrational system rather than as the substitution of one rational system, optimised for hand and till, for another, optimised for machine and export ledger. Andrew Cook’s history of the decision found that even the argument most people assume drove the change, alignment with Europe, barely featured in the committee’s own reasoning; harmonisation with calculating machines and international decimal currencies did the work that harmonisation with the Continent is usually credited with. The old system was not stupid. It was built for a kind of counting that Britain, by 1971, had mostly stopped doing by hand.