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Egyptian Fractions (2006)

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Re: Egyptian Fractions (2006)

#21

Post by mjd »

orthoxerox wrote: post #17

I remember reading a hypothesis that Egyptian fractions were (are?) easier for innumerate people to reason about intuitively. That is, the division of N into M equal parts is easier if everyone gets the same pieces. For example, if I divide three gold bars between seven people naively, some of them get bars that are 3/7 long and some get three small pieces of 1/7 the amount. If instead I give everyone a 1/4 bar, a 1/…

That's an interesting theory but I don't think I find it plausible. Say we're cutting bars like you said. With the obvious strategy I have to cut the three bars into a total of 9 pieces of sizes no less than 1/3 bar each: I cut two of the bars into pieces of 3+3+1 and one bar into pieces of 3+2+2. Then I give five of the people the size-3 bars, and the other two people each get 2+1.

The two people getting the 1/7 + 2/7 pairs can easily verify they are not getting shortchanged, simply by putting their next to one of the 3/7 bars to make sure they add up to the right length.

(Someone dividing 7 sacks of grain among 3 people can do something similar. Maybe they compare two shares of grain on a balance.)

But if you're trying to give everyone a 1/4 bar, a 1/7 bar and a 1/28 bar, sure, it's “trivially obvious to be fair” if you believe you can divide a 1/4 bar into seven exactly equal pieces. But you can't, some will be a little bigger and some will be a little smaller. Seriously, have you ever tried to cut something us unmanageable as a metal bar into seven equal pieces?

Re: Egyptian Fractions (2006)

#22

Post by mjd »

chrismorgan wrote: post #4

We’ve grown used to a full-decimal system, but all kinds of weird stuff has existed in the past. Telugu (a language of southern India) has an interesting traditional numeric system: base ten for integers, and base four for fractions. U+0C78 "౸" TELUGU FRACTION DIGIT ZERO FOR ODD POWERS OF FOUR U+0C79 "౹" TELUGU FRACTION DIGIT ONE FOR ODD POWERS OF FOUR U+0C7A "౺" TELUGU FRACTION DIGIT TWO FOR ODD POWERS OF FOUR U+0C7…

Thanks for the callout to my Telugu fractions article!

If you enjoy ৪ being four and ৭ seven you will probably enjoy the thousand-year-old magic square inscribed at the Parshvanatha temple in Madhya Pradesh.

https://blog.plover.com/math/magic-square-puzzle.html

Shreevatsa R. tells me that the digit symbols are probably Nagari, which predates Devanagari.

Re: Egyptian Fractions (2006)

#23

Post by mjd »

Someone wrote: post #13
chrismorgan wrote: post #4

We’ve grown used to a full-decimal system, but all kinds of weird stuff has existed in the past. Telugu (a language of southern India) has an interesting traditional numeric system: base ten for integers, and base four for fractions. U+0C78 "౸" TELUGU FRACTION DIGIT ZERO FOR ODD POWERS OF FOUR U+0C79 "౹" TELUGU FRACTION DIGIT ONE FOR ODD POWERS OF FOUR U+0C7A "౺" TELUGU FRACTION DIGIT TWO FOR ODD POWERS OF FOUR U+0C7…

> Telugu (a language of southern India) has an interesting traditional numeric system: base ten for integers, and base four for fractions. If they do consistently that is not too bad. Compare that with imperial units, which, depending on the quantity and sometimes its magnitude, uses - base 10 for integers and base 12 for fractions (lengths in feet and inches). Alternatively, base 10 for mikes, base 5,280 for feet’s…

You left out perhaps the most widespread and important example! Many western European cultures, going back I think to Charlemagne, had a monetary system in which a pound was divided into 20 shillings / solidi, and each shilling was divided into 12 pence / denarii. This system persisted in England into the 1970s but it was widespread before the 20th century. For example, in France each livre was divided into 20 sou, and each sou into 12 denier, until 1795.

Leonardo of Pisa's famous book "Liber Abaci" spends a lot of time showing how to do arithmetic on these complicated mixed units, and has an interesting notation for them. If you're interested see https://blog.plover.com/math/liber-abaci-fractions.html .

Re: Egyptian Fractions (2006)

#24

Post by Lucasoato »

fph wrote: post #16
Lucasoato wrote: post #11

> so 6/7=[2,4,14,28]. Whether this is optimal or not is open to argument. It's longer than [2,3,42], but on the other hand the denominators are smaller. Also 6/7 = [2,7,7,14]

That's not an Egyptian fraction decomposition because there are 2 sevens.

> But convention dictated that they could not use the same unit fraction more than once

Damn, I missed this!

Re: Egyptian Fractions (2006)

#27

Post by EdwardCoffin »

mjd wrote: post #21
orthoxerox wrote: post #17

I remember reading a hypothesis that Egyptian fractions were (are?) easier for innumerate people to reason about intuitively. That is, the division of N into M equal parts is easier if everyone gets the same pieces. For example, if I divide three gold bars between seven people naively, some of them get bars that are 3/7 long and some get three small pieces of 1/7 the amount. If instead I give everyone a 1/4 bar, a 1/…

That's an interesting theory but I don't think I find it plausible. Say we're cutting bars like you said. With the obvious strategy I have to cut the three bars into a total of 9 pieces of sizes no less than 1/3 bar each: I cut two of the bars into pieces of 3+3+1 and one bar into pieces of 3+2+2. Then I give five of the people the size-3 bars, and the other two people each get 2+1. The two people getting the 1/7 + 2…

On the other hand, the bars have to be cut no matter which strategy one uses, so this criticism of not being able to cut the bars into exactly equal pieces applies equally to the other strategies.

This Egyptian strategy definitely does have a property of being easier to reason about, and one doesn't have to contend with complaints of say losing out on small amounts of metal around the cuts when one is given three smaller bars that put end-to-end are as long as another, but whose internal mating surfaces don't match up exactly.

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