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.
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.
> 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.
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.
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.
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
> 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.
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.
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.
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.
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.
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.
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...
Mind blown, I had no idea about this, thx for sharing
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
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
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...
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.
> 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.
Thought this said Factions at first but still all the interesting.
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!
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.