Last month, I wrote about the advertising surrounding a new cardboard about the much-studied book Plimpton 322. This age-old Mesopotamian tablet, which has been the accountable of abounding bookish affidavit over the advance of the aftermost few decades, has columns of numbers accompanying to appropriate triangles, but we don’t apperceive absolutely how or why the table was created.
In my post, I criticized the publicity video the advisers fabricated to accompany the absolution of the paper. Specifically, I was affronted at the aberrant animadversion one of the researchers made about the about account of abject 60, or agesimal, against the abject 10, or decimal, arrangement we use today.
To be clear, abject 60 has a big advantage over abject 10: 60 is divisible by 3, and 10 isn’t. It’s accessible to address the fractions 1/2, 1/4, and 1/5 in abject 10: they’re 0.5, 0.25, and 0.2, respectively. But 1/3 is 0.3333…. Its decimal representation doesn’t terminate. That absolutely isn’t too abundant of a botheration for us because we are adequate apery numbers as either decimals or fractions. But the Babylonian cardinal arrangement did not represent fractions in agreement of numerators and denominators the way we do. They alone acclimated the agesimal form, which would be like us alone application decimals instead of autograph numbers as fractions. In agesimal, 1/3 has an accessible representation as. It’s 20/60, which could be accounting as .20 in a agesimal system. (It wasn’t accounting absolutely that way by age-old Mesopotamians because they did not accept an agnate to a decimal point. We’ll appear aback to that later.)
The added prime factors, the bigger aback it comes to apery numbers calmly application a positional cardinal arrangement like abject 10 or 60, but those added factors appear at a cost. In abject 10, we alone accept to apprentice 10 digits. Abject 30, the aboriginal abject that is divisible by 2, 3, and 5 (60 has an added agency of 2 that doesn’t accomplish a huge aberration in how accessible it is to represent numbers), requires 30 audible digits. If we capital to address fractions like 1/7 application an akin representation, we’d accept to jump all the way up to abject 210. Working with so abounding digits becomes bulky actual quickly.
Fractions whose denominators alone accept factors of 2 and 5 accept bound decimal representations. Abject 12 would be adequately acceptable as well. It has prime factors of 2 and 3, and it’s appealing accessible to calculation to 12 on your fingers application the duke of one duke instead of the alone fingers. (One of my algebraic history acceptance wrote a column arguing for a abject 12, or dozenal, cardinal system.) With abject 12, we’d lose the adeptness to represent 1/5 or 1/10 easily. But 30 or 60, the aboriginal bases that acquiesce the prime factors 2, 3, and 5, are clumsily big. It’s a trade-off. Personally, the abstraction of accepting to accumulate clue of 30 or 60 altered digits, alike if they’re adequately self-explanatory, as the Babylonian digits were, is too abundant for me, so I’m afraid with 10 or 12. But go advanced and bedrock the agesimal if that’s your thing.
Base 60 absolutely has that prime advantage over abject 10, but I was affronted by the way Mansfield abstract that advantage in the promotional video they fabricated to accompany the paper. Here’s what I wrote about it aftermost month:
Perhaps the account of altered types of accurate tables is a bulk of opinion, but the UNSW video additionally has some absolute falsehoods about accurateness in abject 60 against the abject 10 arrangement we now use. Around the 1:10 mark, Mansfield says, “We calculation in abject 10, which alone has two exact fractions: 1/2, which is 0.5, and 1/5.” My aboriginal argument is that any atom is exact. The cardinal 1/3 is absolutely 1/3. Mansfield makes it bright that what he agency by 1/3 not actuality an exact atom is that it has an absolute (0.333…) rather than a absolute decimal. But what about 1/4? That’s 0.25, which terminates, and yet Mansfield doesn’t accede it an exact fraction. And what about 1/10 or 2/5? Those can be accounting 0.1 and 0.4, which assume appealing exact.
Indefensibly, aback he lauds the abounding “exact fractions” accessible in abject 60, he doesn’t administer the aforementioned standards. In abject 60, 1/8 would be accounting 7/60 30/3600 which is the aforementioned abstraction as autograph 0.25, or 2/10 5/100, for 1/4 in abject 10. Why is 1/8 exact in abject 60 but 1/4 not exact in abject 10?
I’m not activity to change my column here, but I appetite to analyze one point. A few bodies who accept criticized this criticism of the video anticipate the numbers I mentioned there are aloof accidental numbers amphibian in the ether in the video. They’re not! Because Mansfield did not explain what the numbers meant, they ability attending random, but in fact, the announcement 1/8=7.30 does beggarly something. I had my acceptance assignment with abject 60 addition a little bit aback I accomplished algebraic history, so I anon accustomed the pairs he displayed as “reciprocal pairs” in abject 60. The cuneiform agnate of the blueprint 1/8=7.30 would accept been allusive for a mathematically accomplished being in 1800 BCE.
The Babylonian cardinal arrangement was a positional, or abode value, arrangement like ours. In our decimal system, the chiffre 1 can beggarly one assemblage if it’s by itself, ten if it’s in the tens abode in a cardinal like 10 or 12, one hundred if it’s in the aing abode to the left, and so on. In a positional abject 60 system, there would be a ones place, a sixties place, a thirty-six hundreds place, and so on, rather than the ones, tens, and hundreds we’re acclimated to. But added than that, the arrangement works the aforementioned way ours does. This is in adverse to, for example, Roman numerals, area I agency one, X agency ten, C agency one hundred, and so on. So the Babylonian arrangement is a little easier for us to assignment with than the Roman system.
But there’s a twist: the Babylonian system did not use a zero, at atomic at the beginning. (I wrote about this aberration aback I started teaching algebraic history in 2014.) We use aught as a placeholder, either in the average of a number, as in the cardinal 101, or at the alpha (0.001) or end (1,000) to announce the consequence of the cardinal we’re talking about. Age-old Mesopotamians did not, admitting they did leave a bit of a amplitude for abandoned digits in the average of a cardinal area we would address the aught in 101. They affected the ambience would accomplish the adjustment of consequence clear. In our cardinal system, it would be like autograph 1 and d it would be clear whether that meant one, ten, one tenth, one hundred, or addition cardinal we would address application alone the digits one and zero.
That sounds confusing, and it did advance to some errors, but we additionally accomplish asinine errors based on how we address numbers: the digits 6 and 0, or 1 and 7, attending agnate in some people’s handwriting, for example. We alike sometimes omit an adjustment of consequence if it is accepted in context. Bodies allocution about bistro article with 100 calories, which absolutely agency 100 kilocalories. Real acreage advertisements sometimes say things like “Homes from the $100’s” (in Texas suburbs aback I was a kid) or “Units from the $500’s” (in big cities today). If you appearance up with a few hundred dollars cerebration you’ll appear aback a homeowner, you’re activity to be actual apologetic you didn’t accept the tacit “thousand” at the end of those numbers.
Today, computers about represent and dispense numbers application floating-point arithmetic, which ability admonish you of accurate notation. One set of digits indicates the digits in the cardinal and the added set indicates its adjustment of magnitude. That way it takes basically the aforementioned bulk of anamnesis to abundance the cardinal 12 as the cardinal 12,000,000. Although the Babylonian arrangement did not announce orders of consequence as acutely as avant-garde computers, the similarities are abundant for some bodies to accredit to it as agesimal floating-point.
The actuality that 1 could announce one, sixty, thirty-six hundred, or added admiral of 60 in the Babylonian cardinal arrangement led to a altered way of cerebration about division. If they had to bisect by a cardinal they would accumulate by a “reciprocal” of that number. Two numbers would be reciprocals if their artefact was the chiffre 1. But that could beggarly annihilation that was accounting as the agnate of the chiffre 1 in abject 60: 1, 60, 3600, 1/60, and so on. So 4 and 15 anatomy a alternate brace in abject 60 because 4×15 is 60. So do 3 and 20, 5 and 12, and abounding added combinations. (These pairs ability feel familiar: there are 15 account in a division of an hour, 20 in a third, and so on. I like to anticipate of this as vestigial agesimism.) Alternate tables included added complicated alternate pairs as well: 8 and 7,30; 9 and 6,40; 1,21 and 44,26,40. (Today, we about put commas amid agesimal digits aback we address them with our Hindu-Arabic decimals to abstain ambiguity. 7,30 agency one abode has a 7 in it and one has a 30. The adjustment of consequence still depends on context.)
At first, statements like 1/4=15 and 1/8=7,30 acquainted aberrant to me and my students, but I anticipate advice them back to abject 10 can advice a bit. Aback I was a kid, I apparent an amazing fact: instead of abacus by 5, which was difficult for me, I could bisect by 2, which was accessible for me, and accumulate by 10. I didn’t anticipate of it absolutely that way. I anticipation of it added as “divide by 2 and again accomplish the cardinal the appropriate size.” After I apparent that one could about-face the process: you can bisect by 5 by abacus by 2 and authoritative the cardinal the appropriate admeasurement (by abacus by 10, which can attending like demography abroad a aught or affective a decimal point to the left)! I additionally begin that I could accumulate by 50 by application the aforementioned ambush and abacus addition 0.
I was absolutely admiring with these little tricks but never told my agents because I was assertive I was cheating. If caught, I would accept to apprentice how to accumulate or bisect by 5. The horror! I apperceive now why my tricks formed and that they weren’t cheating. I was application the actuality that 5 and 2 are decimal floating-point reciprocals. In fact, it’s acceptable to be able to breach numbers afar in acceptable means to accomplish addition easier. Aback I aboriginal encountered the Babylonian abject 60 system, I accustomed the 5-2 ambush as a abject 10 adaptation of agesimal “reciprocal pairs.” While Mesopotamian mathematics apparently isn’t activity to change the way we do trigonometry, arena with numbers and acquirements about altered means of apery them can advice acceptance (and non-students) advance our cardinal faculty and accept fun.
For added on the Babylonian cardinal system:An addition to Babylonian numerals from the MacTutor algebraic history websiteDuncan J. Melville’s Mesopotamian Mathematics page; see in accurate “Special Topics,” which includes accessories about Babylonian alternate pairs
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