Difference Btwn Base, Radix, Numeral/number System

PinbackerCasual Member
 Joined: Feb 7 2014, 03:32 PM
Hi,
What are the differences between these four terms?
 base
 radix
 numeral system
 number system
I always use them interchangeably since I don't know the differences between them...
Thx in advance.
What are the differences between these four terms?
 base
 radix
 numeral system
 number system
I always use them interchangeably since I don't know the differences between them...
Thx in advance.

DanDozens Disciple
 Joined: Aug 8 2005, 02:45 PM
"Base" and "radix" are interchangeable AFAIK.
A "numeral system" or "number system" is a way of writing down numbers. It doesn't have to be place value. Numerals systems include tally marks, Egyptian numerals (ð“Ž‰ð“»), Roman numerals (XLII), Hebrew gematria (×ž×´×‘), Greek ionic numerals (ÎœÎ’Ê¹), and the familiar HinduArabic numerals (42).
A "numeral system" or "number system" is a way of writing down numbers. It doesn't have to be place value. Numerals systems include tally marks, Egyptian numerals (ð“Ž‰ð“»), Roman numerals (XLII), Hebrew gematria (×ž×´×‘), Greek ionic numerals (ÎœÎ’Ê¹), and the familiar HinduArabic numerals (42).

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
That's correct. One thing, Pinbacker: If Wendy tries to tell you that "radix" means the period (or comma) that separates the integer digits from the fractional digits, don't listen. Tell her that's just the radix point. The radix is the base.jrus @ Mar 20 2016, 11:42 PM wrote:â€œRadixâ€ is just the Latin word for â€œrootâ€. In this context, as Dan says, itâ€™s synonymous with â€œbaseâ€.
But, of course, Wendy is the most intelligent and wellinformed person here, right? Just ask her.
As of 1202/03/01[z]=2018/03/01[d] I use:
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KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Ugh, don't ye know? Ãžose be not Engelsc! Ãžose be all fremd Ladin words! Hwaet Ãže fecc be ye Ãžinking? Just say "rune" for all of Ã¾em!Pinbacker @ Mar 30 2016, 02:48 PM wrote: Thank you.
Also, what's the difference (if there is any) between a "numeral", a "digit" and a "symbol"? Can I use these three words interchangeably?
As of 1202/03/01[z]=2018/03/01[d] I use:
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icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
A symbol has a wide meaning that should be selfexplanatory; it is some picture or word that conveys a concept, and it could be applied to numbers. I've used "numeral" to signify how we write or say a number and "digit" to mean place value. I had been using "digit" in the place I now use "numeral". When I used the term "digit" to refer to the numbers 0 <= m < n of base n, mathematicians at the OEIS would take it as the wrong use, that "digit" at least there is used to mean place value.
Decimal 387 has three digits. But decimal as a base has 10 numerals.
There is no written standard though and so long as you explain yourself or that it can be gathered from your writing, you'll be understood.
No one I know except Wendy uses the word rune for any of these; "rune" is even more strictly defined than "hieroglyph" (which seems to apply both to Egyptian pictograms and Mayan glyphs. For this, when I hear the word "glyph" when talking typefaces, the image of a painstakingly handcarved stone picture of a severed head on three dots meaning "3", the carver then immediately sacrificed on the stone temple to the sungod, comes to mind.) Runes are northern European futhark letters. When she writes rune I gather what she means but I think an even heavier image comes across to most people. This would be that of a Viking fresh from the sea, with a dagger comin' acarvin' an inscription over a pillaged English town, the priests and men gored and the womenfolk ravaged, the village afire.
Decimal 387 has three digits. But decimal as a base has 10 numerals.
There is no written standard though and so long as you explain yourself or that it can be gathered from your writing, you'll be understood.
No one I know except Wendy uses the word rune for any of these; "rune" is even more strictly defined than "hieroglyph" (which seems to apply both to Egyptian pictograms and Mayan glyphs. For this, when I hear the word "glyph" when talking typefaces, the image of a painstakingly handcarved stone picture of a severed head on three dots meaning "3", the carver then immediately sacrificed on the stone temple to the sungod, comes to mind.) Runes are northern European futhark letters. When she writes rune I gather what she means but I think an even heavier image comes across to most people. This would be that of a Viking fresh from the sea, with a dagger comin' acarvin' an inscription over a pillaged English town, the priests and men gored and the womenfolk ravaged, the village afire.

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Well, I thought from the context of the question that Pinbacker was asking it rhetorically and tongueincheek, and already knew the answer. But in case he wasn't, I'm glad Icarus chimed in. My take is that "numeral" can mean any word, phrase, symbol, or string of symbols that means a number, of any magnitude, not just the digits of a base. But I can see that some people might use it to mean one of the digit characters used in a given base, or one of the characters used in a nonplacevalue system, like the "Roman Numeral" I or V or X, although I would say a Roman numeral was any string of such characters that the Romans would have written. And yes, there seems some differences in usage of "digit" to mean a positional place or one of the symbols occupying a place (the latter is my preference).
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
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PinbackerCasual Member
 Joined: Feb 7 2014, 03:32 PM
I wasn't asking this question rhetorically. I genuinely didn't know the answer.
And now Icarus introduced a fourth term that means approximately the same thing: "glyph".
This confuses me even more.
I'm not interested in how one specific person personally uses these terms.
I'm only interested in how the majority of people use these terms.
Okay, in order to make things clearer, I'll give a few small sentences.
Please try to replace the "XXX" with one or more of the four terms {digit, glyph, numeral, symbol}.
If the "XXX" can be replaced by more than one term, please put all the possible terms under brakets.
And obviously if none of the terms fits correctly, don't replace the "XXX" by anything.
1) "The dozenal number 4A73 has four XXX."
2) "The dozenal system is made up of twelve XXX."
3) "In the dozenal system, to represent a number between twelve and one hundred and fourty three, we need to use two XXX."
4) "In the dozenal alphabet, there are twelve letters, and these letters are more precisely called XXX."
5) "The spoken word 'seventy three' is a XXX."
6) "The twelve basic units that the dozenal system is built upon are called XXX."
7) "In the dozenal system, when you write a string of XXX, it represents a number." ... "And each XXX (character) in this string can take one of twelve possible values."
8) "Written in the dozenal system, 4A73 is a XXX."
9) "The integers n such that 0 <= n <= base1 are called XXX."
And now Icarus introduced a fourth term that means approximately the same thing: "glyph".
This confuses me even more.
I'm not interested in how one specific person personally uses these terms.
I'm only interested in how the majority of people use these terms.
Okay, in order to make things clearer, I'll give a few small sentences.
Please try to replace the "XXX" with one or more of the four terms {digit, glyph, numeral, symbol}.
If the "XXX" can be replaced by more than one term, please put all the possible terms under brakets.
And obviously if none of the terms fits correctly, don't replace the "XXX" by anything.
1) "The dozenal number 4A73 has four XXX."
2) "The dozenal system is made up of twelve XXX."
3) "In the dozenal system, to represent a number between twelve and one hundred and fourty three, we need to use two XXX."
4) "In the dozenal alphabet, there are twelve letters, and these letters are more precisely called XXX."
5) "The spoken word 'seventy three' is a XXX."
6) "The twelve basic units that the dozenal system is built upon are called XXX."
7) "In the dozenal system, when you write a string of XXX, it represents a number." ... "And each XXX (character) in this string can take one of twelve possible values."
8) "Written in the dozenal system, 4A73 is a XXX."
9) "The integers n such that 0 <= n <= base1 are called XXX."

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Sorry, Pinbacker, let me see if I can help:Pinbacker @ Mar 31 2016, 02:20 AM wrote:I wasn't asking this question rhetorically. I genuinely didn't know the answer.
From dictionary.com:
So "symbol" is a very general term for anything we might write or draw that represents something else or has a meaning. An individual character is an example of a "symbol", but combinations of characters can be a "symbol" too, depending on what subject we are talking about. Or something like an emoticon, like or is a "symbol'. Or for instance, these are also symbols, but I can't type them in from a keyboard, I can only show you pictures of them:symbol
...
2. a letter, figure, or other character or mark or a combination of letters or the like used to designate something:
the algebraic symbol x; the chemical symbol Au.
or or or
So "character" is a more specific term meaning an individual letter or other mark we use in writing. In the computer industry this usually means an individual character you can type on the keyboard, that gets shown on the screen as an individual element of a piece of text. Unicode classifies different kinds of characters, distinguishing "letters" versus "digits" versus "punctuation" etc.character
...
12. a symbol as used in a writing system, as a letter of the alphabet.
13. the symbols of a writing system collectively.
14. a significant visual mark or symbol.
...
The dictionary meaning for "glyph" is very specific, usually referring an ancient symbol carved on a stone momument, usually an Egyptian hieroglyph. But the computer industry has started to use this as a technical term, making a distinction between a "character" and a "glyph". A "character" is the abstract code that specifies a particular text element, but a "glyph" is a bitpattern for depicting it on the screen using a particular font. Different fonts provide different "glyphs" to represent the same character. So for instance we can think of the letter "A" abstractly as the same character no matter what it looks like, while A, <span style='fontfamily:times'>A, A, A</span> are different "glyphs" for it from different fonts.glyph
1. a pictograph or hieroglyph.
2. a sculptured figure or relief carving.
...
So this is specifically anything that represents a number, but it's still a pretty broad term. It can be an individual character like "1" or "2", or a word like "one" or "two", or any longer string representing a bigger number, like "2016" or "two thousand sixteen". It can even be something like "MMXVI" which is a Roman numeral. It could be in any human language for that matter, such as "dos mil diecisÃ©is" or "deux mille seize" or "Î´ÏÎ¿ Ï‡Î¹Î»Î¹Î¬Î´ÎµÏ‚ Î´ÎµÎºÎ±ÎÎ¾Î¹". These are all numerals.numeral
1. a word, letter, symbol, or figure, etc., expressing a number; number:
the Roman numerals.
Here we are very specifically talking about a single character used in a positionalplacevalue representation of numerals, such as HinduArabic decimal. Sometimes we mean "digit" abstractly, when we name the set of characters we use for a particular base, but sometimes we are referring to specific instances in use, and the places where they appear, such as "1234" is a "fourdigit" numeral, and the "2" is the "hundreds digit". (That's assuming this is decimal. If it's dozenal we would say it's the "grosses digit".)digit ...
3. any of the Arabic figures of 1 through 9 and 0.
4. any of the symbols of other number systems, as 0 or 1 in the binary.
...
Here are my answers. Perhaps others have a little different opinion:Okay, in order to make things clearer, I'll give a few small sentences.
...
1) "The dozenal number 4A73 has four digits." or: "... has four positional places."
2) "The dozenal system is made up of twelve digits." or: "... twelve digit characters."
3) "In the dozenal system, to represent a number between twelve and one hundred and fourty three, we need to use two digits." or: "... two positional places."
4) "In the dozenal alphabet, there are twelve characters, and these characters are more precisely called digits."
5) "The spoken word 'seventy three' is a numeral." But the written text of it here is also a numeral. So are "73" and "LXXIII".
6) "The twelve basic units that the dozenal system is built upon are called dozenal digits." "The two basic units that the binary system is built upon are binary digits, 0 and 1."
7) "In the dozenal system, when you write a string of dozenal digits, it represents a number." ... "And each positional place in this string can take one of twelve possible values."
8) "Written in the dozenal system, 4A73 is a dozenal positionalplacevalue numeral."
9) "The integers n such that 0 <= n <= base1 are called digits."
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
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ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
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click on my user name and go to my "Website" link)

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
I completely agree with Kode, except the symbols {0, ... b  1} could also be called "numerals". You see this when you buy numerals for addresses. This range is the reason why, when considered with the az of the alphabet, we get "alphanumeric".

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Well, I suppose, but I think that's in a context where you're already assuming you're talking about characters. Those are "alphanumeric characters" after all, as opposed to, say, spoken or written words. If it's a single character, and it's a numeral, that pretty much guarantees it's a digit. Although not necessarily: I think Unicode has some funky stuff in it.icarus @ Mar 31 2016, 04:23 PM wrote: I completely agree with Kode, except the symbols {0, ... b  1} could also be called "numerals". You see this when you buy numerals for addresses. This range is the reason why, when considered with the az of the alphabet, we get "alphanumeric".
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
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click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

PinbackerCasual Member
 Joined: Feb 7 2014, 03:32 PM
Very helpful, but it's still not completely clear at 100% (maybe I'm just an idiot).
I've got a last test that hopefully should be able to finally make things clear:
10. Each of these eighteen sets in the picture are called XXX.
11. In each set there are ten XXX.
Are all things that are in the same column considered as being the same digit?
Are all things that are in the same column considered as being the same numeral?
I've got a last test that hopefully should be able to finally make things clear:
10. Each of these eighteen sets in the picture are called XXX.
11. In each set there are ten XXX.
Are all things that are in the same column considered as being the same digit?
Are all things that are in the same column considered as being the same numeral?

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Numerically, yes. They all have the same value. Linguistically, probably not. The speakers of these different languages would pronounce them differentlyPinbacker @ Apr 1 2016, 02:34 PM wrote:10. Each of these eighteen sets in the picture are called a set of languagespecific decimal digits.
11. In each set there are ten languagespecific decimal digits.
Are all things that are in the same column considered as being the same digit?
Same answer.Are all things that are in the same column considered as being the same numeral?
If you're still confused by the difference between saying something is a digit and saying it's a numeral, try this: All of us here are members of DozensOnline. That's like saying that these are all digits. But all of us are human beings, too (I hope ) That's like saying all of these are numerals. It's a more general category. All the human beings who post on DozensOnline are forum members. All the numerals you see in these sets are digits. It doesn't mean all human beings in the world are forum members. It doesn't mean all numerals in the world are digits.
.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
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click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

PinbackerCasual Member
 Joined: Feb 7 2014, 03:32 PM
I understand what are "numerals".
What I don't understand is that in every single of my 11 sentences (except nÂ°5) you picked the same word: "digits". And you never picked the word "symbol".
To me it would seem that there should be a difference between a "digit" and a "symbol".
Maybe this is the difference (but I could be wrong):
When we are referring to a base in general we should use "digits".
When we are referring to a specific numeral system we should use "symbols". Maybe not all of the time though, it depends...
"Digits" are the integers n such that 0 <= n <= base1
"Symbols" are the characters/images that we use in a particular numeral system to represent each of those b "digits".
For example:
The Chinese numeral system and the Greek numeral system both have the same ten digits (because they all use base 10). < (I'm not totally sure if this sentence is really correct...)
The Chinese numeral system and the Greek numeral system don't have the same ten symbols.
https://en.wikipedia.org/wiki/Greek_numerals
https://en.wikipedia.org/wiki/Chinese_numerals
I have rewritten the 11 sentences (except nÂ°5).
I have added three extra sentences (BIS) which are exactly like the sentence they are preceding except that there is a small difference at the beginning.
I put in bold words that are different from the word you used.
1) "The dozenal number 4A73 has four digits."
2) "The dozenal system is made up of twelve digits."
2BIS) "This specific dozenal system (pointing to a particular alphabet made by a particular member of this forum for example) is made up of twelve symbols."
3) "In the dozenal system, to represent a number between twelve and one hundred and fourty three, we need to use two digits."
4) "In the dozenal alphabet, there are twelve characters, and these characters are more precisely called digits."
4BIS) "In this particular dozenal alphabet (pointing to a specific alphabet made by a particular member of this forum for example), there are twelve characters, and these characters are more precisely called symbols.
6) "The twelve basic units that the dozenal system is built upon are called dozenal digits." "The two basic units that the binary system is built upon are binary digits, 0 and 1."
7) "In the dozenal system, when you write a string of dozenal digits, it represents a number." ... "And each positional place in this string can take one of twelve possible digits."
7BIS) "In this specific dozenal system (pointing to a particular alphabet made by a particular member of this forum for example), when you write a string of symbols, it represents a number." ... "And each symbol in this string can take one of twelve possible representations."
9) "The integers n such that 0 <= n <= base1 are called digits."
10) Each of these eighteen sets in the picture are called a set of symbols.
11) In each set there are ten symbols.
What I don't understand is that in every single of my 11 sentences (except nÂ°5) you picked the same word: "digits". And you never picked the word "symbol".
To me it would seem that there should be a difference between a "digit" and a "symbol".
Maybe this is the difference (but I could be wrong):
When we are referring to a base in general we should use "digits".
When we are referring to a specific numeral system we should use "symbols". Maybe not all of the time though, it depends...
"Digits" are the integers n such that 0 <= n <= base1
"Symbols" are the characters/images that we use in a particular numeral system to represent each of those b "digits".
For example:
The Chinese numeral system and the Greek numeral system both have the same ten digits (because they all use base 10). < (I'm not totally sure if this sentence is really correct...)
The Chinese numeral system and the Greek numeral system don't have the same ten symbols.
https://en.wikipedia.org/wiki/Greek_numerals
https://en.wikipedia.org/wiki/Chinese_numerals
I have rewritten the 11 sentences (except nÂ°5).
I have added three extra sentences (BIS) which are exactly like the sentence they are preceding except that there is a small difference at the beginning.
I put in bold words that are different from the word you used.
1) "The dozenal number 4A73 has four digits."
2) "The dozenal system is made up of twelve digits."
2BIS) "This specific dozenal system (pointing to a particular alphabet made by a particular member of this forum for example) is made up of twelve symbols."
3) "In the dozenal system, to represent a number between twelve and one hundred and fourty three, we need to use two digits."
4) "In the dozenal alphabet, there are twelve characters, and these characters are more precisely called digits."
4BIS) "In this particular dozenal alphabet (pointing to a specific alphabet made by a particular member of this forum for example), there are twelve characters, and these characters are more precisely called symbols.
6) "The twelve basic units that the dozenal system is built upon are called dozenal digits." "The two basic units that the binary system is built upon are binary digits, 0 and 1."
7) "In the dozenal system, when you write a string of dozenal digits, it represents a number." ... "And each positional place in this string can take one of twelve possible digits."
7BIS) "In this specific dozenal system (pointing to a particular alphabet made by a particular member of this forum for example), when you write a string of symbols, it represents a number." ... "And each symbol in this string can take one of twelve possible representations."
9) "The integers n such that 0 <= n <= base1 are called digits."
10) Each of these eighteen sets in the picture are called a set of symbols.
11) In each set there are ten symbols.

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
But "symbol" is even more general than "character" and in turn more general than "numeral". The usage you're suggesting isn't wrong, and it's pehaps clear in context, but it's not necessarily precise.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
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click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
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click on my user name and go to my "Website" link)

wendy.krieger
One should remember that the terminology is being imposed on a preexisting structure, and different sources will give different words for the same thing. Numbertheory, and the decimrules that dictate modern western thinking, are only late comming labels for preexisting practices.
Germanic twelfty comes from a common source, so they have a common word for twelfty and hundreds etc. But the conversion to latin decimal, done at the time of the church, leads to different words for describing the same thing. Gothic has 'teentywise' for this, whereas the norse has 'tulfcount' (12count) and 'teencount' (10count).
The placenotation (512) apparently descends from the tokennotation (I, X, C), and the use of numerics can be applied to tokennotation (as chinese 5C 1X 2), as well as the raw unmarked places of the sumerians. [The chinese have this too, where they say 5C 1 for 510, and 5C 0 1 for 501.] The systems who find their origins in the heiratic system and alphabetic systems, use a composite symbol like 5C, or simply equate the place/value song with the alphabet song (1, 2, 3, 4, .. , 9, 10, 20, 30, ..) with A, B, C, D, ... , I, J, K, L, ). The greek system was set very early, since they use inherited letters (vow, qoppa, tsampi), that are later fallen out of use.
The columnnotations to the right of the units are ordinals, although the prime might be set to a different word (eg minute). Stevins used prime, second, third, fourth, in Latin and flemish, for his positions, but for sixtymeasure, the order is minute, second, third, of the hour and degree. The french metric of 1792 replicated the sixtydeal of the hour and degree for steps of 100, rather than 10. The CGS prefix system uses exactly the same form for tens, this is preserved in 'tenthmetre' for angstrom.
Modern Western Numbers
A number is comprised of digits, which are letters of the numberword. So 4523 is 'four digits'. Digits are used to refer to used or allowable places, ie 'eight digits' means the calculator can store 12345678. So 'digit' more closely translates the column idea of the abacus, rather than the columncount.
The numeral is a specific letter, like 'A', and represents a columncount. The numerals are the possible occupants of the digits. The canonical form is to have a numeral for each possible remainder against the base, starting with 0, and ending with b1.
While a number is a string of numerals, it is the places that matter, rather than the occupants, and a number is comprised of digits. A telephone pad is for entering the digits of a phone number, rather than the numerals of it, because the order and place of each numeral matters.
While people say decimal has ten 'digits', in fact, it has ten numerals. A decimal number can be set to any length, and a twelvedigit calculator has twelve places, each of which can hold one of ten numerals entered from the keypad. So digit is used as a cover for numerals.
The rule of decim
This is the idea that 'all bases are 10'. In essence we must replicate the ten 'digits' (ie numerals) as far as the base demands, so twelve requires twelve numerals, and base eighteen needs eighteen numerals.
Since the decim idea supposes that each column is the same size, the pattern is to write base 60 as though it is a decimallike base, with sixty numerals, and a numeral in each place where a digit demands one. So you write 1.0 for 60, but and 1.0.3 for 3603, but not required in 1.3;0 for 63.
Since the decim rules colour the way we think of base 60, the tendency is to write it in 'pure 60', or 'largebase' (decimalencoded numerals from 0 to 59), rather than attempting to transliterate.
The rules of centim
The rules that I suppose for twelfty, which I then use for other bases too, is that the numerals do not necessarily occupy a column to itself, and that a column might be two or three numerals. Digit still translates 'numeral in a place', but a column might be several places.
In the sumerian number, the pattern I use is a onetoone transliteration of their numbers, thus AE for 10, 20, .. 50, horizontal notches and 19 for the 19 horizontal notches. A space might be unmarked or marked _, a zero marked 0, replicated at every full stop (the sumerian zero is the same as the sentencestop punctuation).
Radixpoints are not given and not used in the transliteration.
The square root of 2 is given as 1 B 4 E 1 A and on the other part, D 2 B 5 C 5. The significance here is that the individual numerals are s_p_a_c_e_d, but the B (20) is still added to the 4 since the space between B and 4 is the same as between 4 and E.
Writing this as, eg 1 24 51 10, will not reveal the semimedial zero in that the next digit is 7, ie 1B4E1A 7D6 6 4D4 should have two spaces in the first break, and a space in the other two. While 6 can never stand beside 4 (it lies usually between letters), 6 4 indicates 6,04, A7 is a valid combination, but the intent is A..7, ie 10,07 rather than 17.
My form
The general number can be constructed ultimately from a numbercolumn, and if the intent is to describe different formations of this, then it is a matter of runenaming rather than of some 'theoryofbases' naming.
A system is designated baseX if the columncarries are X of one column carries to I of the next right, and I of the column carries to X of the left. Places to the right of the unit column are designated as an radixfraction, for want of better words. So 0.3102 is a 'radix fraction'. It represents an added fraction, in which each subsequent division is into X.
A set of runes which exist that gives the numeric value of any given column, is a numeral, and the columns represent 'digits' of the number. However, one can express numbers without numerals.
The runes representing columns and rows of a counting table, and repeated as many times as the individual counts, are tokens. Roman numbers are written in tokenform, eg CCCII for 302, repeats C three times and I twice.
The runes representing columns, but to which a count (numeric or otherwise) are units, the style of english and chinese numbers are of this form, viz 5C2X1 meaning 5 hundred, 2 tens, 1, or english xvi c lxxxxvi (16 c 96, ie 2016), here the ivxl represents a tokencount, but c is a unit, by which xvi is multiplied.
The runes representing positions in the heiratic notation, are 'letters', since the normal pattern is to map these onto the order of letters in the alphabet (aramaic, greek, gothic).
Mathematical Notation
The supposition im mathematics is that the modern western decimal form applies.
Twelfty
The runes in which twelfty is written are numerals, and occupy digit spots. But a place represents a column or repetition of carry, is two digits of length.
Germanic twelfty comes from a common source, so they have a common word for twelfty and hundreds etc. But the conversion to latin decimal, done at the time of the church, leads to different words for describing the same thing. Gothic has 'teentywise' for this, whereas the norse has 'tulfcount' (12count) and 'teencount' (10count).
The placenotation (512) apparently descends from the tokennotation (I, X, C), and the use of numerics can be applied to tokennotation (as chinese 5C 1X 2), as well as the raw unmarked places of the sumerians. [The chinese have this too, where they say 5C 1 for 510, and 5C 0 1 for 501.] The systems who find their origins in the heiratic system and alphabetic systems, use a composite symbol like 5C, or simply equate the place/value song with the alphabet song (1, 2, 3, 4, .. , 9, 10, 20, 30, ..) with A, B, C, D, ... , I, J, K, L, ). The greek system was set very early, since they use inherited letters (vow, qoppa, tsampi), that are later fallen out of use.
The columnnotations to the right of the units are ordinals, although the prime might be set to a different word (eg minute). Stevins used prime, second, third, fourth, in Latin and flemish, for his positions, but for sixtymeasure, the order is minute, second, third, of the hour and degree. The french metric of 1792 replicated the sixtydeal of the hour and degree for steps of 100, rather than 10. The CGS prefix system uses exactly the same form for tens, this is preserved in 'tenthmetre' for angstrom.
Modern Western Numbers
A number is comprised of digits, which are letters of the numberword. So 4523 is 'four digits'. Digits are used to refer to used or allowable places, ie 'eight digits' means the calculator can store 12345678. So 'digit' more closely translates the column idea of the abacus, rather than the columncount.
The numeral is a specific letter, like 'A', and represents a columncount. The numerals are the possible occupants of the digits. The canonical form is to have a numeral for each possible remainder against the base, starting with 0, and ending with b1.
While a number is a string of numerals, it is the places that matter, rather than the occupants, and a number is comprised of digits. A telephone pad is for entering the digits of a phone number, rather than the numerals of it, because the order and place of each numeral matters.
While people say decimal has ten 'digits', in fact, it has ten numerals. A decimal number can be set to any length, and a twelvedigit calculator has twelve places, each of which can hold one of ten numerals entered from the keypad. So digit is used as a cover for numerals.
The rule of decim
This is the idea that 'all bases are 10'. In essence we must replicate the ten 'digits' (ie numerals) as far as the base demands, so twelve requires twelve numerals, and base eighteen needs eighteen numerals.
Since the decim idea supposes that each column is the same size, the pattern is to write base 60 as though it is a decimallike base, with sixty numerals, and a numeral in each place where a digit demands one. So you write 1.0 for 60, but and 1.0.3 for 3603, but not required in 1.3;0 for 63.
Since the decim rules colour the way we think of base 60, the tendency is to write it in 'pure 60', or 'largebase' (decimalencoded numerals from 0 to 59), rather than attempting to transliterate.
The rules of centim
The rules that I suppose for twelfty, which I then use for other bases too, is that the numerals do not necessarily occupy a column to itself, and that a column might be two or three numerals. Digit still translates 'numeral in a place', but a column might be several places.
In the sumerian number, the pattern I use is a onetoone transliteration of their numbers, thus AE for 10, 20, .. 50, horizontal notches and 19 for the 19 horizontal notches. A space might be unmarked or marked _, a zero marked 0, replicated at every full stop (the sumerian zero is the same as the sentencestop punctuation).
Radixpoints are not given and not used in the transliteration.
The square root of 2 is given as 1 B 4 E 1 A and on the other part, D 2 B 5 C 5. The significance here is that the individual numerals are s_p_a_c_e_d, but the B (20) is still added to the 4 since the space between B and 4 is the same as between 4 and E.
Writing this as, eg 1 24 51 10, will not reveal the semimedial zero in that the next digit is 7, ie 1B4E1A 7D6 6 4D4 should have two spaces in the first break, and a space in the other two. While 6 can never stand beside 4 (it lies usually between letters), 6 4 indicates 6,04, A7 is a valid combination, but the intent is A..7, ie 10,07 rather than 17.
My form
The general number can be constructed ultimately from a numbercolumn, and if the intent is to describe different formations of this, then it is a matter of runenaming rather than of some 'theoryofbases' naming.
A system is designated baseX if the columncarries are X of one column carries to I of the next right, and I of the column carries to X of the left. Places to the right of the unit column are designated as an radixfraction, for want of better words. So 0.3102 is a 'radix fraction'. It represents an added fraction, in which each subsequent division is into X.
A set of runes which exist that gives the numeric value of any given column, is a numeral, and the columns represent 'digits' of the number. However, one can express numbers without numerals.
The runes representing columns and rows of a counting table, and repeated as many times as the individual counts, are tokens. Roman numbers are written in tokenform, eg CCCII for 302, repeats C three times and I twice.
The runes representing columns, but to which a count (numeric or otherwise) are units, the style of english and chinese numbers are of this form, viz 5C2X1 meaning 5 hundred, 2 tens, 1, or english xvi c lxxxxvi (16 c 96, ie 2016), here the ivxl represents a tokencount, but c is a unit, by which xvi is multiplied.
The runes representing positions in the heiratic notation, are 'letters', since the normal pattern is to map these onto the order of letters in the alphabet (aramaic, greek, gothic).
Mathematical Notation
The supposition im mathematics is that the modern western decimal form applies.
Twelfty
The runes in which twelfty is written are numerals, and occupy digit spots. But a place represents a column or repetition of carry, is two digits of length.

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
If you're not a native English speaker, be advised that most of this stuff Wendy is describing is her own peculiar slant and interpretation, colored by her preoccupation with ancient number systems and medieval AngloSaxon usage, or stuff she took from old historical works, or simply made up, either because it suited her or it catered to her strange impediments. It does not necessarily reflect common modernEnglish usage, or dictionary definitions, common mathematical terminology, or what most educated English speakers of today would expect. You need to take everything Wendy says with a major grain of salt. If you can't find corroboration in dictionary.com or wikipedia.com for the things she says, you should discount it.wendy.krieger @ Apr 2 2016, 09:45 AM wrote: One should remember that the terminology is being imposed on a preexisting structure, and different sources will give different words for the same thing.
Although in the latter case, you have to watch out, because she's apparently been busy getting her fingers into certain subjects, such as polytopes and LorentzHeaviside units. You can tell she's been there by the use of the abstract pronoun "one" as in "One should remember...". It's a stilted form of formal speech that is considered extremely stuffy these days and is out of fashion even in serious technical works.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

wendy.krieger
The numerals came to Europe with Leonardo Fibonacci, around 1200. Fibonacci used a series of added fractions, which one does not find in any modern reference. These are fractions of continued numerator. Confusingly, his same notation of fractions serves for continued and egyptian fractions too, without further comment.
The extentions of the digits to the right of the unit point, is what Stevins text is. This was in order by at least 1580. He used a series of ordered places (primes, seconds, thirds, &c), to which Kode and a number of others had made objection to.
The radix point, to separate the units from the fractioncolumns, appears only in 1630. The earliest of number theory appears at this point, when the system is already in place. However, this is not the beginnings of calculation, because we know that the Sumerians were aready calculating three or four millenia before this.
So the terminologies that apply to number systems, and to prenumeral forms of it, are not necessarily the same that the people themselves speak of, and are simply a modern connection of these things.
Heaviside Lorentz
I read Oliver Heaviside. I read H A Lorentz. I am not relying entirely on modern interpretations of either H or L. There is no HeavisideLorentz units in H. What one finds in Heaviside are references to a system whose electrical units conform to the PR scale, a set of decadeunits based on the magnetic side of the HLU. Lorentz actually acknowledges Heaviside, and uses a system GR (modern HLU), but he uses gaussian terminology to it: his electrostatic units are ER and his magnetic units are MR.
The talk page on LorentzHeaviside units suggest that I have an understading that is at least comperable with the editors, and in many ways better. The forms of Maxwell's equations are mine, and you can see them in the talk page first.
The 'Heaviside Lorentz' units in Dresner's dictionary are actually the units that belong to a scheme I call GI. These units never match the true heaviside lorentz units GR.
Polytopes
Kode supposes that since it is a home hobby, it should be a long way behind that poduced at universities etc. University professors are paid largely to instill the subject into postschoolage people, who are there learning things for different reasons. A good deal of what one learns there is things that are never used, but different people use different slices of what they learn.
There are people who make very good models and paint very good pictures as a hobby, and display their models accordingly. But the publishing houses do not really give room for publishing random gems of wisdom outside the schools, because there is more dross then gem.
So when I started communicating the stuff in 2002, I discovered, that far from just being a hobby, or being the very best of modern thinking, it was well in advance of this, because apparently I accessed some forms of radical thinking.
Much of teamikaria uses my notation, this has displaced a very large number of other notations. The shift is every bit as large from moving from Roman numerals describing place to arabic numerals giving value. From singlelevel polytopes to packed arrays describing layers of laceprism.
Unfortunately, the previous major advance in polytope geometry was made over 100 years past, by a woman practicing a hobby, who was cooersed to write a paper in 1910. It was Coxeter's reading of a different paper that referenced it, that it comes to light.
The extentions of the digits to the right of the unit point, is what Stevins text is. This was in order by at least 1580. He used a series of ordered places (primes, seconds, thirds, &c), to which Kode and a number of others had made objection to.
The radix point, to separate the units from the fractioncolumns, appears only in 1630. The earliest of number theory appears at this point, when the system is already in place. However, this is not the beginnings of calculation, because we know that the Sumerians were aready calculating three or four millenia before this.
So the terminologies that apply to number systems, and to prenumeral forms of it, are not necessarily the same that the people themselves speak of, and are simply a modern connection of these things.
Heaviside Lorentz
I read Oliver Heaviside. I read H A Lorentz. I am not relying entirely on modern interpretations of either H or L. There is no HeavisideLorentz units in H. What one finds in Heaviside are references to a system whose electrical units conform to the PR scale, a set of decadeunits based on the magnetic side of the HLU. Lorentz actually acknowledges Heaviside, and uses a system GR (modern HLU), but he uses gaussian terminology to it: his electrostatic units are ER and his magnetic units are MR.
The talk page on LorentzHeaviside units suggest that I have an understading that is at least comperable with the editors, and in many ways better. The forms of Maxwell's equations are mine, and you can see them in the talk page first.
The 'Heaviside Lorentz' units in Dresner's dictionary are actually the units that belong to a scheme I call GI. These units never match the true heaviside lorentz units GR.
Polytopes
Kode supposes that since it is a home hobby, it should be a long way behind that poduced at universities etc. University professors are paid largely to instill the subject into postschoolage people, who are there learning things for different reasons. A good deal of what one learns there is things that are never used, but different people use different slices of what they learn.
There are people who make very good models and paint very good pictures as a hobby, and display their models accordingly. But the publishing houses do not really give room for publishing random gems of wisdom outside the schools, because there is more dross then gem.
So when I started communicating the stuff in 2002, I discovered, that far from just being a hobby, or being the very best of modern thinking, it was well in advance of this, because apparently I accessed some forms of radical thinking.
Much of teamikaria uses my notation, this has displaced a very large number of other notations. The shift is every bit as large from moving from Roman numerals describing place to arabic numerals giving value. From singlelevel polytopes to packed arrays describing layers of laceprism.
Unfortunately, the previous major advance in polytope geometry was made over 100 years past, by a woman practicing a hobby, who was cooersed to write a paper in 1910. It was Coxeter's reading of a different paper that referenced it, that it comes to light.

wendy.krieger
Kode is disengenious to the point of being offensive. He knows that neither wikipedia nor dictionary.com accept original research, and therefore current trends in the subject necessarily are excluded. This means that if wasn't invented by 1970, forget it. Having one's finger in the pie usually is a fairly derogatary statement that suggest that you do not fully understand what you do.
A review of posts here would see that the decim constructs were in use. We see icarrus providing a set of digits suitable for use to bases as far as 120, and the UTL, which continue to chug out runes into the thousands. The idea here is that numbers aswewritethemnow is the best idea to follow, and if your base needs 86 symbols, you MUST PROVIDE THEM.
There are various tables that are placebyplace calculation in summary form, and Kode talks of digit and subdigit in decimform (68 is a digit, 6 is a subdigit in twelfty). The standard mathematical texts I read on the subject suggest exactly that, ae a number is \( d_6 d_5 d_4 d_3 d_2 d_1 d_0 \) where \( b > d_x > 1\) Pictures of the sumerian number system reflect this, where we see 60 > x > 0 giving separate symbols.
Then someone comes along and talks of centim forms where you have many fewer numerals than the decim form says, and say, hay let's look at sumerian numbers in centim form, and the 59 digits become 14. You don't need the full tables of the decim forms, because the carrys are alternated.
But centim forms are not in the literature. You won't find them in Wikipedia or in dictionary.online, because they were not written in 1970. As far as much of the modern world is concerned, it's a cutting edge research into a different paradigm of looking at bases.
I specifically label what I believe to be the modern order, and my interpretation of how they come, and my own forms under quite clear headings. The modern use is supported by examples, and all we see is a spot of mudslinging from Kode, who has nothing to offer here.
A review of posts here would see that the decim constructs were in use. We see icarrus providing a set of digits suitable for use to bases as far as 120, and the UTL, which continue to chug out runes into the thousands. The idea here is that numbers aswewritethemnow is the best idea to follow, and if your base needs 86 symbols, you MUST PROVIDE THEM.
There are various tables that are placebyplace calculation in summary form, and Kode talks of digit and subdigit in decimform (68 is a digit, 6 is a subdigit in twelfty). The standard mathematical texts I read on the subject suggest exactly that, ae a number is \( d_6 d_5 d_4 d_3 d_2 d_1 d_0 \) where \( b > d_x > 1\) Pictures of the sumerian number system reflect this, where we see 60 > x > 0 giving separate symbols.
Then someone comes along and talks of centim forms where you have many fewer numerals than the decim form says, and say, hay let's look at sumerian numbers in centim form, and the 59 digits become 14. You don't need the full tables of the decim forms, because the carrys are alternated.
But centim forms are not in the literature. You won't find them in Wikipedia or in dictionary.online, because they were not written in 1970. As far as much of the modern world is concerned, it's a cutting edge research into a different paradigm of looking at bases.
I specifically label what I believe to be the modern order, and my interpretation of how they come, and my own forms under quite clear headings. The modern use is supported by examples, and all we see is a spot of mudslinging from Kode, who has nothing to offer here.

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Like I said, if you want to know what the common, generallyaccepted definitions and usages of words are, Wendy is an unreliable source. She has invented her own peculiar language for herself, and feels her interpretations are superior to everyone else's, and has an agenda to impose them on everyone else.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

ShaunDozens Disciple
 Joined: Aug 2 2005, 04:09 PM
I am tempted to suggest that you two (Kode & Wendy) set up a subforum for yourselves where you can be as rude as you like to each other and thus refrain from clogging up posts with your continual sniping, 'mudslinging' or whatever you like to call it. It seems to me that your little skirmishes are appearing in every post I look at; and getting nowhere. Imagine you are a visitor to this site ...

KodegaduloObsessive poster
 Joined: Sep 10 2011, 11:27 PM
Shaun, I have tried to do what you suggest in the past but it gets us nowhere, because Wendy insists on jumping into every thread here to push her agenda, even when it is wildly offtopic. Now, in a thread asking for common, generally accepted definitions of terms, she is shouting at us that if we deal in large bases, we must either come up with unique symbols for each digit of the base, or we must swallow her entire concept of "rows and columns" and her techniques for alternating arithmetic, and all the attendant terminology for it that she has concocted. I submit that's a false choice. Mapping large bases recursively down into subbase encodings is nothing particularly "new" or "cutting edge". All Wendy offers is her peculiar language for that. We are not bound to accept her interpretations of things, no matter how many tantrums she throws.
Even Icarus, one of her biggest fans, has asked her to knock it off with the agenda, to no avail.
Even Icarus, one of her biggest fans, has asked her to knock it off with the agenda, to no avail.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Baseneutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)