The Universe of Discourse

Base-4 fractions in Telugu

Rik Signes brought to my attention that since version 5.1 Unicode has contained the following excitingly-named characters:

    0C78 ౸ TELUGU FRACTION DIGIT ZERO FOR ODD POWERS OF FOUR
    0C79 ౹ TELUGU FRACTION DIGIT ONE FOR ODD POWERS OF FOUR
    0C7A ౺ TELUGU FRACTION DIGIT TWO FOR ODD POWERS OF FOUR
    0C7B ౻ TELUGU FRACTION DIGIT THREE FOR ODD POWERS OF FOUR

    0C7C ౼ TELUGU FRACTION DIGIT ONE FOR EVEN POWERS OF FOUR
    0C7D ౽ TELUGU FRACTION DIGIT TWO FOR EVEN POWERS OF FOUR
    0C7E ౾ TELUGU FRACTION DIGIT THREE FOR EVEN POWERS OF FOUR

I looked into this a little and found out what they are for. It makes a lot of sense! The details were provided by “Telugu Measures and Arithmetic Marks” by Nāgārjuna Venna.

Telugu is the third-most widely spoken language in India, spoken mostly in the southeast part of the country. Traditional Telugu units of measurement are often divided into four or eight subunits. For example, the tūmu is divided into four kuṁcamulu, the kuṁcamulu, into four mānikalu, and the mānikalu into four sōlalu.

These days they mainly use liters like everyone else. But the traditional measurements are mostly divided into fours, so amounts are written with a base-10 integer part and a base-4 fractional part. The characters above are the base-4 fractional digits.

To make the point clearer, I hope, let's imagine that we are using the Telugu system, but with the familar western-style symbols 0123456789 instead of the Telugu digits ౦౧౨౩౪౫౬౭౮౯. (The Telugu had theirs first of course.) And let's use 0-=Z as our base-four fractional digits, analogous to Telugu ౦౼౽౾. (As in Telugu, we'll use the same zero symbol for both the integer and the fractional parts.) Then to write the number of gallons (7.4805195) in a cubic foot, we say

7.-Z=Z0

which is 7 gallons plus one (-) quart plus three (Z) cups plus two (=) quarter-cups plus three (Z) tablespoons plus zero (0) drams, a total of 7660 drams almost exactly. Or we could just round off to 7.=, seven and a half gallons.

(For the benefit of readers who might be a bit rusty on the details of these traditional European measurements, I should mention that there are four drams in a tablespoon, four tablespoons in a quarter cup, four quarter cups in a cup, four cups in a quart, and four quarts in a gallon, so 4⁵ = 1024 drams in a gallon and 7.4805195·4⁵ = 7660.052 drams in a cubic foot. Note also that these are volume (fluid) drams, not mass drams, which are different.)

We can omit the decimal point (as the Telegu did) and write

7-Z=Z0

and it is still clear where the integer part leaves off and the fraction begins, because we are using special symbols for the fractional part. But no, this isn't quite enough, because if we wrote 20ZZ= it might not be clear whether we meant 20.ZZ= or 2.0ZZ=.

So the system has an elaboration. In the odd positions, we don't use the 0-=Z symbols; we use Q|HN instead. And we don't write 7-Z=Z0, we write

7|ZHZQ

This is always unambiguous: 20.ZZ= is actually written 20NZH and 2.0ZZ= is written 2QZN=, quite different.

This is all fanciful in English, but Telugu actually did this. Instead of 0-=Z they had ౦౼౽౾ as I mentioned before. And instead of Q|HN they had ౸౹౺౻. So if the Telugu were trying to write 7.4805195, where we had 7|ZHZQ they might have written ౭౹౾౺౾౸. Like us, they then appended an abbreviation for the unit of measurement. Instead of “gal.” for gallon they might have put ఘ (letter “gha”), so ౭౹౾౺౾౸ఘ. It's all reasonably straightforward, and also quite sensible. If you have ౭౹౾౺ tūmu, you can read off instantly that there are ౺ (two) sōlalu left over, just as you can see that $7.43 has three pennies left over.

Notice that both sets of Telugu fraction digits are easy to remember: the digits for 3 have either three horizonal strokes ౾ or three vertical strokes ౻, and the others similarly.

I have an idea that the alternating vertical-horizontal system might have served as an error-detection mechanism: if a digit is omitted, you notice right away because the next symbol is wrong.

I find this delightful. A few years back I read all of The Number Concept: Its Origin and Development (1931) by Levi Leonard Conant, hoping to learn something really weird, and I was somewhat disappointed. Conant spends most of his book describing the number words and number systems used by dozens of cultures and almost all of them are based on ten, and a few on five or twenty. (“Any number system which passes the limit 10 is reasonably sure to have either a quinary, a decimal, or a vigesimal structure.”) But he does not mention Telugu!

[ Addendum 20200821: Bengali currency amounts do something similar, but the sections alternate between base-10 and base-4 numerals! ]

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