Split Number systems (Split from: 60 years of silence - so far)

[steve_bank]

I remember some from a Chinese class.

yi,er,san 1,2,3

Description. Chinese numerals The Arabic numeral system used today in China was introduced to China by the Europeans in the early 17th century. But the Chinese character-based number systems are still in use. The financial numerals are used only when writing an amount on a form for remitting money at a bank.Aug 1, 2020

chinese numbers - Google Search

 

[steve_bank]

Hey mods, a split to math forum 'Numers And Numer Systems'?

 

[lpetrich]

Don’t forget elephant trunks. They are pretty good at grasping, and of course elephants are also quite intelligent.

Yes, joining octopuses and New World monkeys.

What base woud elephants count in? They have no digits, but a perfectly serviceable grasping organ.

2 (front legs), 3 (front legs, trunk), 4 (all legs), 5 (all legs, trunk)

 

[Copernicus]

Luckily, the numerical symbolic representation of decimal numbers is universal across languages, but the words used to represent decimal numbers can be very complex, partly because of the historical legacy of different base counting systems used to give names to numbers.

Universal??? I have actually used 3 different encodings.

For Indian/Arabic numerals? I don't know what you are referring to, but the (originally Indian) use of integers gave rise to the current numerical semiotic system that is used in pretty much every language that I've come across. The more complicated issues have to do with the naming conventions used for integers.

In addition to the 0123456789 there was something used in Iran that was the standard system--when I was there 50 years ago prices were expressed in it. Presumably everything else but we only learned to read the numbers. My memory is it is Arabic-looking but I doubt I would even recognize it by now. I believe this might be the digits but since I don't recall them I can't swear to it: ۰۱۲۳۴۵۶۷۸۹

I hate the rubric "Arabic", because Muslims got the system from Hindus, who had used it for centuries before Islam even became a thing. I don't know whether they also used the Vedic Sanskrit symbology in Persian Avestan literature.

China also has an old system--you can function perfectly well without it but it does show up. There's a set of Mahjong tiles around here that use it--no way to play the game if you can't read them. (There are other tiles with Chinese characters but they only form sets, not runs, and thus you can simply compare them without knowing what they say.) I've seen it occasionally in China--think of how we often have Roman numerals in copyright dates etc. By now I've forgotten some of the digits and I never knew the zero as it doesn't appear in the game. Some work with google turns up: 〇一二三四五六七八九十

OK, but China's symbolic writing system stayed largely confined to East Asia. Not as efficient as the Brahmic quasi-alphabetic scripts that eventually got exported to the East.

And I'm ignoring Roman numerals because they aren't really digit encoding. Digging up the Chinese characters also reveals an ancient Greek system that isn't rendering on my system so I don't want to try to paste it in.

I'm not at all sure what your system is or what you are trying to say here. Are you talking about high level programmatic representation of numbers, as opposed to binary machine representations?

 

[steve_bank]

Arabs are credited with publishing the first general text in algebra.

Math and science from Arabs and Persians flowed into Europe as the Mid East declined. Newton used Persian astronomical data.

To be fair.

 

[Copernicus]

Arabs are credited with publishing the first general text in algebra.

Math and science from Arabs and Persians flowed into Europe as the Mid East declined. Newton used Persian astronomical data.

To be fair.

It doesn't matter. The Hindus invented that numbering system centuries before the birth of Muhammed. They got it from India and exported it to Europe, which had been using alphabetic letters and combinations of letters to represent numbers.

 

[Loren Pechtel]

Luckily, the numerical symbolic representation of decimal numbers is universal across languages, but the words used to represent decimal numbers can be very complex, partly because of the historical legacy of different base counting systems used to give names to numbers.

Universal??? I have actually used 3 different encodings.

For Indian/Arabic numerals? I don't know what you are referring to, but the (originally Indian) use of integers gave rise to the current numerical semiotic system that is used in pretty much every language that I've come across. The more complicated issues have to do with the naming conventions used for integers.

I was talking about systems for writing numbers.

In addition to the 0123456789 there was something used in Iran that was the standard system--when I was there 50 years ago prices were expressed in it. Presumably everything else but we only learned to read the numbers. My memory is it is Arabic-looking but I doubt I would even recognize it by now. I believe this might be the digits but since I don't recall them I can't swear to it: ۰۱۲۳۴۵۶۷۸۹

I hate the rubric "Arabic", because Muslims got the system from Hindus, who had used it for centuries before Islam even became a thing. I don't know whether they also used the Vedic Sanskrit symbology in Persian Avestan literature.

I said "Arabic-looking"--my vague memories was that it struck me as Arabic writing. Whether it actually is or not I do not know. What I do know was that in 1975 Iran that's how prices were displayed.

China also has an old system--you can function perfectly well without it but it does show up. There's a set of Mahjong tiles around here that use it--no way to play the game if you can't read them. (There are other tiles with Chinese characters but they only form sets, not runs, and thus you can simply compare them without knowing what they say.) I've seen it occasionally in China--think of how we often have Roman numerals in copyright dates etc. By now I've forgotten some of the digits and I never knew the zero as it doesn't appear in the game. Some work with google turns up: 〇一二三四五六七八九十

OK, but China's symbolic writing system stayed largely confined to East Asia. Not as efficient as the Brahmic quasi-alphabetic scripts that eventually got exported to the East.

That's still over a billion people. And I said it's not widely used now, our digits have mostly driven it out.

And I'm ignoring Roman numerals because they aren't really digit encoding. Digging up the Chinese characters also reveals an ancient Greek system that isn't rendering on my system so I don't want to try to paste it in.

I'm not at all sure what your system is or what you are trying to say here. Are you talking about high level programmatic representation of numbers, as opposed to binary machine representations?

I found what the Unicode code points are supposed to be but my system has nothing that will actually display them.

 

[Swammerdami]

And I'm ignoring Roman numerals because they aren't really digit encoding. Digging up the Chinese characters also reveals an ancient Greek system that isn't rendering on my system so I don't want to try to paste it in.

The ancient Greeks used 27 number symbols denoting 1, 2, 3, ..., 9, 10, 20, ... 90, 100, 200, ... 900. These symbols were simply the letters of the Greek alphabet. The Greek alphabet has only 24 letters, so three antique Greek letters were also used. (Perhaps these three were the symbols that didn't render for Loren.) IIRC we were shown this by our high school math teacher who had started a pseudo-frat whose name used the three obsolete Greek letters.

Three of the original Phoenician letters dropped out of use before the alphabet took its classical shape: the letter Ϻ (san), which had been in competition with Σ (sigma) denoting the same phoneme /s/; the letter Ϙ (qoppa), which was redundant with Κ (kappa) for /k/, and Ϝ (digamma), whose sound value /w/ dropped out of the spoken language before or during the classical period.

After all these decades, "qoppa" and "digamma" ring bells. The pseudo frat name was something like Digamma Qoppa Sanpi.

Worth mentioning is the Mayan numbering system, which had a full-grown base-20 place value system using three symbols: 1 (dot), 5 (dash), Zero (ornate curves). Decoding Mayan script was a major achievement -- apparently no natives were found who could still read it at the time of the Conquistadores. It was the numeral system that got decoding started IIRC. Naturally there was redundancy in the stone inscriptions recording the bios (with birth, coronation and death years, etc.) of their Kings.

 

[Swammerdami]

On the topic of rendering numbers on computers, don't forget systems used for compression.

Simple and popular is the Golomb-Rice code. I'll illustrate it by encoding 53 using the Golomb-Rice code with parameter 8 = 2³.

53 = 6*8 + 5.​

53 = 1111110 101​

This is 6 written in unary (with a 0 appended as delimiter) followed by the remainder 5 in 3 bits of binary. Using the Golomb-Rice code with parameter 16, the rendered token ends with 4 bits of binary:

53 = 3*16 + 5.​

53 = 1110 0101​

The Golomb-Rice provides optimal compaction when the non-negative integers to be encoded follow an exponential distribution of known parameter.

Related to that code is the Elias gamma code, useful for a certain power-law distribution, or for an exponential of unknown parameter. This code is used to encode positive integers, zero disallowed. An elegant way to implement (or define by example) the Elias gamma code is that the first 1 bit will always be in the middle of the code word. Here are the first several integers encoded with that code

1 = 1​

2 = 010​

3 = 011​

4 = 00100​

5 = 00101​

6 = 00110​

7 = 00111​

8 = 0001000​

9 = 0001001​

Ordinary binary! except for the leading zeros.

The Golomb-Rice code above used the following unary encoding of the non-negative integers:

0 = 0​

1 = 10​

2 = 110​

3 = 1110​

4 = 11110​

Variants are possible, e.g.

0 = 1​

1 = 00​

2 = 010​

3 = 0110​

4 = 01110​

This variant can be used for robustness in the presence of errors. Note that in this encoding every code word is a palindrome. Thus the decoding is the same in either direction.

This same idea adapts to the Elias gamma code also! Simply interleave the significand bits into the palindromic bits.

 

[Swammerdami]

Speaking of Unicode, the prescription for rendering UTF-16 as UTF-8 has a structure reminiscent of the "unary" code already mentioned. This rendering has several elegances, e.g. correct ordering, automatic sync.

0 xxxxxxx = Ascii-7 character
10 xxxxxx = continuation byte
110 xxxxx = one of 2^11 UTF-16 codes^*; followed by one continuation byte
1110 xxxx = one of 2^16 UTF-16 codes^*; followed by two continuation bytes
11110 xxx = one of (up to) 2^21 extended ("alternate plane"?) UTF codes; followed by three continuation bytes
11111 xxx = ?? ; used in UTF-8 BOM prefix

^* - This includes illegal redundant codes. To be compliant a UTF-8 decoder MUST detect and reject any token which is a non-minimal representation of that UTF-16 character. (Non-compliant decoders have been -- and still can be? -- hijacked, e.g. by creating impossible file names.)