Why octal is better than decimal

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Kurushimi
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Re: Why octal is better than decimal

Postby Kurushimi » Thu Dec 23, 2010 1:49 pm UTC

Zamfir wrote:
Disanidi wrote:That idea that counting started with fingers and toes is a fallacy to begin with. Society has certainly veered in that direction, but counting in general used to be 1, 2, 3, and a lot. A lot being broke into a little lot or a big lot.

Do we have any evidence for this? Low counting numbers in languages seem to have extremely old origins, going back beyond any written source material. And they are nearly always base-10.


I'm pretty sure Disanidi is right. I'm reading my copy of Mathematics: From the Birth of Numbers, and here it says "Hunting and gather peoples - such as the aborigines of Australia, Tasmania, and Papua New Guinea - generally have had, or still have, specific names only for the numbers one and two and, sometimes, three, but they can count to as many as six by combining numbers [like two-two for 4 or three-three for 6]... In these languages, speakers refer to everything beyond six as many, much, or plenty." (pg 3)

But where he says that the idea that counting started with fingers and toes is a fallacy, he is certainly wrong. Why do you think Roman Numerals have a special letter for 5 and 10? The Greenlandic counting system uses a base 20 system because it counts on its fingers and toes. Sure, not ALL counting systems began by counting on your fingers and toes but certainly a great deal of them did. Saying that there that that idea is a "fallacy" is just plain wrong.

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Re: Why octal is better than decimal

Postby Disanidi » Thu Dec 23, 2010 2:13 pm UTC

Kurushimi wrote:But where he says that the idea that counting started with fingers and toes is a fallacy, he is certainly wrong. Why do you think Roman Numerals have a special letter for 5 and 10? The Greenlandic counting system uses a base 20 system because it counts on its fingers and toes. Sure, not ALL counting systems began by counting on your fingers and toes but certainly a great deal of them did. Saying that there that that idea is a "fallacy" is just plain wrong.


"But where he says that the idea that counting started..."

gmalivuk wrote:Well with factors, all I can say is, duh. When 1, 2, 3, and 4 are all factors of a number, then of course factors of that number are going to show up an awful lot. But where in nature do you see 12s, exactly?


gmalivuk wrote:Rather, those all just fall out of 12 being an easy number to *calculate* with, on account of being divisible by 2, 3, and 4. But even though 12 crops up all over the place, pretty much all the counting numbers still seem to be done in 10 (or occasionally 8 or 20).


You answer your own question. ;)

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Re: Why octal is better than decimal

Postby Zamfir » Thu Dec 23, 2010 2:36 pm UTC

Kurushimi wrote:
Zamfir wrote:
Disanidi wrote:That idea that counting started with fingers and toes is a fallacy to begin with. Society has certainly veered in that direction, but counting in general used to be 1, 2, 3, and a lot. A lot being broke into a little lot or a big lot.

Do we have any evidence for this? Low counting numbers in languages seem to have extremely old origins, going back beyond any written source material. And they are nearly always base-10.


I'm pretty sure Disanidi is right. I'm reading my copy of Mathematics: From the Birth of Numbers, and here it says "Hunting and gather peoples - such as the aborigines of Australia, Tasmania, and Papua New Guinea - generally have had, or still have, specific names only for the numbers one and two and, sometimes, three, but they can count to as many as six by combining numbers [like two-two for 4 or three-three for 6]... In these languages, speakers refer to everything beyond six as many, much, or plenty." (pg 3)

But I am skeptical of the claim that because some modern-day hunter-gatherers do something, it must therefore be the oldest way of doing things. There is a lot of potential for observer bias here: if we find an isolated tribe that counts to relatively high numbers, we call their counting system "advanced", and if it counts to low numbers we call it "primitive". But then the assumption of progression from one system to the other is already baked in.

For all we know, ancestors of people who have counting names up to 4 used to have counting names up to a higher number, and some of them fell in disuse. Over thousands of years, languages of illiterate or non-agrarian people might well vary in the complexity of their counting system. Sometimes bare-bones simple, sometimes more intricate than useful in normal life, then going back again. Perhaps sometimes taking over counting systems from other languages, either simpler or more complex.

As possible example: most languages have counting words for very low numbers that are completely unrelated to the standard counting system. In English, words like single, pair, couple, few, many, half. Suppose groups of English-speaking people were left in near isolation for thousands of years, in very simple circumstances. Some groups might drop the formal one-two-three-four system but keep the other terms (or their descendants). Future anthropologists could then wrongly assume that only-couple-few-lots was what other counting systems look like in their infancy.

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Re: Why octal is better than decimal

Postby Kurushimi » Thu Dec 23, 2010 3:16 pm UTC

Disanidi wrote:
Kurushimi wrote:But where he says that the idea that counting started with fingers and toes is a fallacy, he is certainly wrong. Why do you think Roman Numerals have a special letter for 5 and 10? The Greenlandic counting system uses a base 20 system because it counts on its fingers and toes. Sure, not ALL counting systems began by counting on your fingers and toes but certainly a great deal of them did. Saying that there that that idea is a "fallacy" is just plain wrong.


"But where he says that the idea that counting started..."


Alright, counting did not start on your fingers and toes. Only when people decided to count higher than three, did they turn to their fingers and toes. Almost every system that counts to high numbers uses a base that is a multiple of five.

Zamfir wrote:
Kurushimi wrote:
Zamfir wrote:
Disanidi wrote:That idea that counting started with fingers and toes is a fallacy to begin with. Society has certainly veered in that direction, but counting in general used to be 1, 2, 3, and a lot. A lot being broke into a little lot or a big lot.

Do we have any evidence for this? Low counting numbers in languages seem to have extremely old origins, going back beyond any written source material. And they are nearly always base-10.


I'm pretty sure Disanidi is right. I'm reading my copy of Mathematics: From the Birth of Numbers, and here it says "Hunting and gather peoples - such as the aborigines of Australia, Tasmania, and Papua New Guinea - generally have had, or still have, specific names only for the numbers one and two and, sometimes, three, but they can count to as many as six by combining numbers [like two-two for 4 or three-three for 6]... In these languages, speakers refer to everything beyond six as many, much, or plenty." (pg 3)

But I am skeptical of the claim that because some modern-day hunter-gatherers do something, it must therefore be the oldest way of doing things. There is a lot of potential for observer bias here: if we find an isolated tribe that counts to relatively high numbers, we call their counting system "advanced", and if it counts to low numbers we call it "primitive". But then the assumption of progression from one system to the other is already baked in.

For all we know, ancestors of people who have counting names up to 4 used to have counting names up to a higher number, and some of them fell in disuse. Over thousands of years, languages of illiterate or non-agrarian people might well vary in the complexity of their counting system. Sometimes bare-bones simple, sometimes more intricate than useful in normal life, then going back again. Perhaps sometimes taking over counting systems from other languages, either simpler or more complex.

As possible example: most languages have counting words for very low numbers that are completely unrelated to the standard counting system. In English, words like single, pair, couple, few, many, half. Suppose groups of English-speaking people were left in near isolation for thousands of years, in very simple circumstances. Some groups might drop the formal one-two-three-four system but keep the other terms (or their descendants). Future anthropologists could then wrongly assume that only-couple-few-lots was what other counting systems look like in their infancy.


You have here a lot of speculation. "What-ifs" and what not. I'd prefer to go on the actual evidence that shows many different tribal people counting this way.

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Re: Why octal is better than decimal

Postby Jyrki » Thu Dec 23, 2010 3:26 pm UTC

Well. A 'reliable relic' of old names of numerals is staring at us in our languages. The English language is a mixture of bases 10 and 12 in the small sense that 'eleven' and 'twelve' have an execptional form. The 'teens' run from 13 to 19 only. The same holds true for German and Swedish (all three of these languages have a common root, so no surprise). Yes, I'm aware of the fact 'thirteen' stands for ten+three et cetera, so the teens use 'ten' a basic point of reference. Therefore this is not necessarily evidence for 12 having ever been a base for a number system, but rather reflecting the fact that the numbers 11 and 12 occurred frequently enough to warrant having special words.

This is not universal, though. In my native Finnish the 'teens' range from 11 to 19, and the corresponding words all use to ten as the basic point of reference. Other exceptions that I am aware of are that French and Danish use a somewhat mixed system in that in the range 60-99 something resembling a base 20 system is used.

All the languages I ever bothered to check out have a word for 'dozen'. I guess the old-timers were selling eggs (or whatever) in 3x4 crates?

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Re: Why octal is better than decimal

Postby Zamfir » Thu Dec 23, 2010 3:48 pm UTC

Kurushimi wrote:You have here a lot of speculation. "What-ifs" and what not. I'd prefer to go on the actual evidence that shows many different tribal people counting this way.

That's the point. Nearly everything about the linguistic history of illiterate people is speculation. Assuming all languages started with one-two-three-many and then progressed to a more complicated system is just as much speculation. I am just pointing out that many different theories could explain the current existence of tribes that have few counting words, and that their existence is not a reason to assume they are typical for "early" societies.

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Re: Why octal is better than decimal

Postby Kurushimi » Thu Dec 23, 2010 5:02 pm UTC

It is not just as much speculation. It makes more sense. One-two-three is obviously more simple so, it makes sense for them to start with that. We even have examples of tribal people using that. Many of these didn't affect each other and were not affect by the advanced world. They developed and used these are on their own. They are obviously good examples of what older tribal people did. Sure, probably not EVERYONE started counting the same way, but it is obvious that the one, two, three, many system was common.

Jyrki wrote:Well. A 'reliable relic' of old names of numerals is staring at us in our languages. The English language is a mixture of bases 10 and 12 in the small sense that 'eleven' and 'twelve' have an execptional form.


"Eleven is a derivative of Old English endleofan, where the terminal -fan is believed to carry the meaning of "to leave" or "left", that is, one left after counting ten." - Mathematics: From the Birth of Numbers

Twelve developed similarly, but obviously means "two left over". This is indicative of a base 10, not base 12.

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Re: Why octal is better than decimal

Postby gmalivuk » Thu Dec 23, 2010 5:43 pm UTC

Yakk wrote:I there was also 60 via Babylonians, which is where we get hours:minutes and minutes:seconds and degrees in a circle.
Yes, I am aware. Which is why I've mentioned them a couple of times already as an exception to the general trend of using the number of fingers. And even the Babylonians used base-10 for each "digit"! So it's an exception that proves the rule, rather than any kind of true anomaly.

Kurushimi wrote:We even have examples of tribal people using that. Many of these didn't affect each other and were not affect by the advanced world. They developed and used these are on their own. They are obviously good examples of what older tribal people did. Sure, probably not EVERYONE started counting the same way, but it is obvious that the one, two, three, many system was common.
Plus, the singular-dual-plural forms in many languages (or at least relics of them, as we have in English) also suggest that 1-2-many was a precursor to more complex mathematical systems.

Kurushimi wrote:
Jyrki wrote:Well. A 'reliable relic' of old names of numerals is staring at us in our languages. The English language is a mixture of bases 10 and 12 in the small sense that 'eleven' and 'twelve' have an execptional form.


"Eleven is a derivative of Old English endleofan, where the terminal -fan is believed to carry the meaning of "to leave" or "left", that is, one left after counting ten." - Mathematics: From the Birth of Numbers

Twelve developed similarly, but obviously means "two left over". This is indicative of a base 10, not base 12.
Right, so it really only means that those numbers were used frequently enough to maintain their older forms, while other numbers gained more standardized forms. Which is a very common trend in all language change.
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Re: Why octal is better than decimal

Postby Xanthir » Wed Dec 29, 2010 5:23 pm UTC

gmalivuk wrote:
Yakk wrote:I there was also 60 via Babylonians, which is where we get hours:minutes and minutes:seconds and degrees in a circle.
Yes, I am aware. Which is why I've mentioned them a couple of times already as an exception to the general trend of using the number of fingers. And even the Babylonians used base-10 for each "digit"!

To clarify, iirc the babylonians wrote their numbers using a mixed radix system, where their digits alternated between base-6 and base-10. This can also be seen as writing numbers in base 60 where each "digit" is a base-10 digraph.

So it's an exception that proves the rule, rather than any kind of true anomaly.

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Re: Why octal is better than decimal

Postby gmalivuk » Wed Dec 29, 2010 6:15 pm UTC

Nah, it's something like the scientific sense here, in that what seems at first glance like an exception to the rule actually isn't one at all.
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Re: Why octal is better than decimal

Postby Xanthir » Wed Dec 29, 2010 6:57 pm UTC

You are forgiven.
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Re: Why octal is better than decimal

Postby Arariel » Wed Dec 29, 2010 10:51 pm UTC

Obviously base φ is the most superior.

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Re: Why octal is better than decimal

Postby Kurushimi » Thu Dec 30, 2010 2:22 am UTC

What about base 16? In base 10, you can easily determine if a number is divisible by 2,3,5,9,10 by looking it at the last digit or adding up the digits. In base 16, you can see if a number is divisible by 2,3,4,5,8,15,16 by looking at the last digit or adding the digits. Plus, it helps to understand base 16 in understanding computers. So, yeah. That's pretty cool.

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Re: Why octal is better than decimal

Postby elitekross » Sat Jan 08, 2011 2:25 am UTC

the Egyptians had base 60, and as far as learning other bases, i created a base 5 system with new math symbols and number symbol . it was quite easy to learn int to a point i can use it without thinking. during the period that i created this i was very intersted in the propriteds of the bases themselves and have a notebook about it somewhere. a base twelve would be easy to transition to

as far as the origin of the bases, i remember reading somewher that in all natural symbolic representations, the first three numbers are representations of 1 2 and 3
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or _
1 /_ 2
_
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_) 3

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Re: Why octal is better than decimal

Postby kernelpanic » Sat Jan 08, 2011 3:01 am UTC

You do notice that in the octal example you divide 6410? It's as arbitrary as 10010, and it also has the nice property of giving 32, 16, 8, 4, 2, 1, 1/2, 1/4, 1/8...
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Re: Why octal is better than decimal

Postby silverhammermba » Sat Jan 08, 2011 3:41 am UTC

Throwing in my twelve cents:
Disanidi wrote:
gmalivuk wrote:I have a hard time believing that humans would innately be at a disadvantage for a base system because we have 10 fingers. I *believe* we are creatures of habit, not patterns, despite our pattern abilities. If, from a young age, we learned to count to 12 instead of 10, we would not have any more problems than with counting to 10.

There's a tribe in the Amazon (appearing on one of David Attenborough's "Life" documentaries) that counts using two hands - but in a very different way from most cultures.

One to five are indicated as usual: by pointing to the corresponding digit with the other hand. Six is indicated by pointing at the wrist, seven by pointing at the crook of the elbow, eight the shoulder, nine the neck. However I don't know how high their counting system goes nor if its base is something other than 10. Still, I think this lends weight to the argument that there may be other bases that are as easy to work with as 10.

And to lend weight to the argument that 2, 3, and 4 are more "important" divisors than 5, consider time. Hours are divided into 60 minutes, 60 has 2 though 5 as divisors, yet no one ever talks about a fifth of an hour. There are a few possible reasons for this: it could be that dividing into fifths is too granular to be convenient, or that fifths of an hour are multiples of 12 and not too many people have their 12 times table memorized.

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Re: Why octal is better than decimal

Postby gmalivuk » Sat Jan 08, 2011 5:27 am UTC

silverhammermba wrote:And to lend weight to the argument that 2, 3, and 4 are more "important" divisors than 5, consider time. Hours are divided into 60 minutes, 60 has 2 though 5 as divisors, yet no one ever talks about a fifth of an hour. There are a few possible reasons for this: it could be that dividing into fifths is too granular to be convenient, or that fifths of an hour are multiples of 12 and not too many people have their 12 times table memorized.
Halves, thirds, and quarters are so common as divisors because it's easy to eyeball them. And yet, time, yards/feet/inches, pounds/ounces, gallons/quarts/pints/cups/ounces, and so on were all developed by people who were at the same time using the decimal system for doing all their actual counting.
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Re: Why octal is better than decimal

Postby letterX » Sat Jan 08, 2011 7:15 am UTC

Everyone wrote:Mathematics: From the birth of numbers

Seriously: how many of us own this book?

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Re: Why octal is better than decimal

Postby silverhammermba » Sun Jan 09, 2011 2:31 am UTC

gmalivuk wrote:Halves, thirds, and quarters are so common as divisors because it's easy to eyeball them. And yet, time, yards/feet/inches, pounds/ounces, gallons/quarts/pints/cups/ounces, and so on were all developed by people who were at the same time using the decimal system for doing all their actual counting.

I don't know about that. Halves and quarters are definitely easy, but eyeballing accurate thirds takes practice. In fact I'd argue that eyeballing fifths isn't significantly harder than thirds. Sevenths (and higher) are arguably difficult but personally I think I can divide into 3 and 5 just about equally well.

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Re: Why octal is better than decimal

Postby Kurushimi » Mon Jan 10, 2011 12:14 pm UTC

I think people would say thirds if it's less than a half, but more than a fourth. You don't have the ability to do that with a fifth.

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Re: Why octal is better than decimal

Postby Velifer » Mon Jan 10, 2011 7:25 pm UTC

We have a great deal of practice estimating to the nearest tenth. I think we could be just as good if we had the same practice measuring to the nearest eighth. Probably better at estimating fifths.
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Re: Why octal is better than decimal

Postby MHD » Mon Jan 10, 2011 10:27 pm UTC

I think we should all use base -10, that would eliminate the need for pesky negatives.

POP QUIZ:
What base? 1+1=10.01
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Re: Why octal is better than decimal

Postby kernelpanic » Mon Jan 10, 2011 11:33 pm UTC

MHD wrote:POP QUIZ:
What base? 1+1=10.01

Base (sqrt(5)+1)/2, aka golden ratio base.
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Re: Why octal is better than decimal

Postby Yakk » Mon Jan 10, 2011 11:44 pm UTC

Unary!
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Re: Why octal is better than decimal

Postby Arariel » Tue Jan 11, 2011 1:41 am UTC

kernelpanic wrote:
MHD wrote:POP QUIZ:
What base? 1+1=10.01

Base (sqrt(5)+1)/2, aka golden ratio base.

AKA phinary.


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