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’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.
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.
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.
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?
> 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.
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.
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[ 2.5 ms ] story [ 37.5 ms ] threadTelugu (a language of southern India) has an interesting traditional numeric system: base ten for integers, and base four for fractions.
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.
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.
Also 6/7 = [2,7,7,14]
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.
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?
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.
“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