Egyptian Fractions (2006) | Hacker News Reader

Egyptian Fractions (2006)

blog.plover.com

113 pointsluu24 comments

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+0C7B "౻" TELUGU FRACTION DIGIT THREE FOR ODD POWERS OF FOUR
  (U+0C66 "౦" TELUGU DIGIT ZERO is used for even powers of four too)
   U+0C7C "౼" TELUGU FRACTION DIGIT ONE FOR EVEN POWERS OF FOUR
   U+0C7D "౽" TELUGU FRACTION DIGIT TWO FOR EVEN POWERS OF FOUR
   U+0C7E "౾" TELUGU FRACTION DIGIT THREE FOR EVEN POWERS OF FOUR

Seems complicated at first, but in practice it’s roughly just: circle for zero, and tally marks for one, two and three, alternating vertical and horizontal.

Few Telugu speakers even know about this any more—no one can read even the traditional integers (౦౧౨౩౪౫౬౭౮౯), because 0123456789 have replaced them. (This is the case in most but not all Indian languages. Bengali’s traditional digits are still common, so you can enjoy ৪ being four and ৭ seven.)

A couple of articles and discussions about it:

• https://www.unicode.org/wg2/docs/n3156.pdf is the best public resource I know of (Unicode proposals and related papers are often delightful for information on obscure written stuff, because they had to write down and publish the details to get the characters encoded). One tid-bit: NYSE used a similar decimal/quaternary system until early 2001.

• https://blog.plover.com/math/telugu.html from the same site as the current article, discussed in https://news.ycombinator.com/item?id=14683767 nine years ago.

> One tid-bit: NYSE used a similar decimal/quaternary system until early 2001.

Prior to decimal pricing, us exchanges priced in eighths. An article [1] I found from a source I don't recognize says that there was briefly trading in sixteenths, but I guess I wasn't following the stock markets that closely then and I missed it.

https://tontinecoffeehouse.com/2018/11/05/pricing-in-eighths...

Heh. I've seen the code in the codebase at my previous work that supported not only 1/8 and 1/16 price scales, but all the way up to 1/256... but I don't think I've ever seen a ticker with such price scale outside of unit tests.

> 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 in a mile, base 12 for inches in a foot.

- base 10 for integers and base 14 for fractions (weights in stone and pound). Alternatively, there’s a base 112 when measuring in long tons and stones or 1,016 when using long tons and pounds)

- base 10 for integers and base 4 for fractions (lengths in miles and quarter/half miles)

And that’s ignoring specific units for weighing gold, diamonds, etc.

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 .

Mind blown, I had no idea about this, thx for sharing

I actually made an app a few years ago that converts to the traditional Telugu style of decimals. Its interesting how the fractions in spoken Telugu are based around 1/4ths but few people know about the same applying for the written expression as you mentioned.

https://play.google.com/store/apps/details?id=com.telugu_num...

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.

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 1/7 bar and a 1/28 bar, this is trivially obvious to be fair.

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?

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.

Cool, and something I’ve been curious about (the civil engineering and math involved from earlier representations). I want the logic behind the notation, though. I assume there’s a background system of folding a rope into equilateral pieces that led to this system of fractional math that I would love any YouTube or other recommendations on

We take so much of modern math notation for granted, for centuries people were encumbered by various inefficiencies of what was agreed before them and couldn't be easily broken.

one wonders what notations are we encumbered by even now, without noticing it

There’s several interesting papers on the subject of Egyptian mathematics in the book “From Five Finger to Infinity”.

“The volume of a truncated pyramid in ancient Egyptian papyri”

“Sources of information on Ancient Egyptian mathematics”

“Problems 1-6 of the Rhind mathematical papyrus” (Includes a description of the math for dividing bread rations)

https://archive.org/details/fromfivefingerst0000unse_s0p6

Small error in the last sentence of the 20260617 addendum: in “To do 21×23 they would expand 19=16+4+1” that 19 should be 21.

If this article intrigues, see the chapter on Egyptian mathematics in The Crest of the Peacock: https://www.ms.uky.edu/~sohum/ma330/files/Crest_of_the_peaco...

Thought this said Factions at first but still all the interesting.

> 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!

Very interesting! One thing I don't understand is: doesn't this assume that they could do the calculations to get the coefficients... Using decimal notation? How could they for example know that 18/20 = 9/10? This is straightforward in decimal, but in their notation... Not really? So I am not super convinced this is the actual algorithm they used. Or am I missing something?

they did count in tens, as most civilizations did, though not exactly in "decimal". so 2345 would be either written out as MMCCCXXXXIIIII (but replace the letters with hieroglyphs), or sometimes spelled out phonetically. they had words for twenty, thirty and so on.

so obviously they can factor numbers and they know eighteen is 2*9 and twenty is 2*10 and that they can simplify when dividing 18 by 20, it's just that they don't consider 9/10 a finished result.

I see! Thanks!

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