Why octal is better than decimal

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ianfort
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Why octal is better than decimal

Postby ianfort » Sat Dec 18, 2010 7:34 pm UTC

Though I'm happy having 5 digits on each hand instead of 4, the one major downside is that humanity has based its numbering system around this amount. Don't get me wrong. 10 is a perfectly good number, just inconvenient to work with as opposed to 8 (or any power of 2 for that matter). Let me explain.

Lets say you were to divide 100 by 2. You'd of course get 50. Divide that by 2, you're down to 25. But beyond this, you can't divide the same way without having to use fractions. Of course, you could divide it by 5 repeatedly, but that tends to be less useful.

With octal, however, when you divide 100 by 2, you get 40, 40 by 2, 20, 20 by 2, 10, 10 by 2, 4, 4 by 2, 2, and finally you divide that and get 1.

With fractions, of course, you get the pattern 1, 0.4, 0.2, 0.1, 0.04, 0.02. 0.01, 0.004 etc.

Look at decimal by comparison, where it is already very jumbled when you get to a mere 1/64th: 0.015625.

Of course, hexidecimal would work too, but we don't have enough fingers for that, which obviously poses a problem when you don't have a calculator.

I think numbers would be a lot easier to work with if we were to use octal instead of decimal...

By the way, does anyone know where I could find a (long) approximation of pi in octal?

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Re: Why octal is better than decimal

Postby ++$_ » Sat Dec 18, 2010 8:28 pm UTC

It's pretty easy to find the binary digits of pi, and these are easy enough to convert to octal.

My take on this is that we have to use numbers other than powers of 2 sometimes. In those situations, octal is worse than decimal because the numbers have more digits. (Of course, the human ability to recognize, add, and multiply digits in constant time is limited, so you can't just let the base go to infinity, but we can pretty easily handle ten digits.)

In addition, 103 = 210 is a good enough approximation for most purposes, so it is actually not even that hard to divide powers of 10 by 2.

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Re: Why octal is better than decimal

Postby ianfort » Sat Dec 18, 2010 8:56 pm UTC

Well, at the very least, the base should be a perfect power, something which ten is not

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Re: Why octal is better than decimal

Postby firechicago » Sat Dec 18, 2010 9:16 pm UTC

ianfort wrote:Well, at the very least, the base should be a perfect power, something which ten is not


Why? What makes a perfect power easier to use? How often do you have to divide by two that many times? What makes being a perfect power better than avoiding repeating decimals by having lots of different prime factors?

My personal candidate for replacing decimal would be dodecimal, which is still better than base 10 for dividing by 2, but has the advantage of not throwing a tantrum and giving you a repeating expansion whenever you even think about dividing by a power of 3.

Of course the small benefits of switching over could never be worth the massive headache it would be to get everyone to switch.

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Re: Why octal is better than decimal

Postby ianfort » Sat Dec 18, 2010 9:39 pm UTC

firechicago wrote:
ianfort wrote:Well, at the very least, the base should be a perfect power, something which ten is not


Why? What makes a perfect power easier to use? How often do you have to divide by two that many times? What makes being a perfect power better than avoiding repeating decimals by having lots of different prime factors?

My personal candidate for replacing decimal would be dodecimal, which is still better than base 10 for dividing by 2, but has the advantage of not throwing a tantrum and giving you a repeating expansion whenever you even think about dividing by a power of 3.

Of course the small benefits of switching over could never be worth the massive headache it would be to get everyone to switch.


Yeah, I suppose you're right. I didn't really think the whole perfect powers thing through very much.

But the appeal of octal is the fact that 2 is the most common factor of composite numbers apart from 1. Base 12 good as well, though the ability to divide a power of the base by a particular number repeatedly and get 1 is not there, but it still can represent thirds without infinitely repeating, which a strength octal lacks.

Both are better than decimal in my opinion, and yes, I agree that it would be a pain to switch over, but still, its interesting to think about it.

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Re: Why octal is better than decimal

Postby masher » Sat Dec 18, 2010 9:59 pm UTC

ianfort wrote:Lets say you were to divide 100 by 2. You'd of course get 50. Divide that by 2, you're down to 25. But beyond this, you can't divide the same way without having to use fractions. Of course, you could divide it by 5 repeatedly, but that tends to be less useful.


Why do you pick 2 out as being a special divisor? Why not 3? or 7...?

No matter what base you pick, you're always going to get numbers that required a fraction to express; its just a matter of how much you divide...

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Re: Why octal is better than decimal

Postby mdyrud » Sat Dec 18, 2010 10:27 pm UTC

The whole "We have ten fingers, that's why we use base 10" argument has always kind of bugged me. Wouldn't that lead to us using base 11? After all, we have 10 unique digits, plus no digits at all. 11 possibilities.

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Re: Why octal is better than decimal

Postby Mike_Bson » Sat Dec 18, 2010 10:57 pm UTC

I'd rather have base 16.

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Re: Why octal is better than decimal

Postby Dason » Sat Dec 18, 2010 11:03 pm UTC

I thought I read somewhere that e is the optimal base. Which I think meant that 3 was a 'good' base for actually using.

After some investigation I found an article that backs this up. I think it would take quite a lot to move away from base 10 but it's interesting to think about.
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Re: Why octal is better than decimal

Postby ianfort » Sat Dec 18, 2010 11:14 pm UTC

an irrational base? How is that easy to work with? Base three isn't that good either. Odd numbers could end in even numbers and vice-a-versa in every other multiple of the base.

How do non-integer bases work, anyway?

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Re: Why octal is better than decimal

Postby jendral_hxr » Sun Dec 19, 2010 2:38 am UTC

Mike_Bson wrote:I'd rather have base 16.

If I happen to be pre-school or grade-school teacher, I'd love to teach multiplication and division in hexadecimal :twisted:
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Re: Why octal is better than decimal

Postby NathanielJ » Sun Dec 19, 2010 5:05 am UTC

ianfort wrote:Odd numbers could end in even numbers and vice-a-versa in every other multiple of the base.


So? It's still easy to determine whether or not something is even or odd in base 3: just add up all the digits. Then add all the digits of that. Repeat until you get 1 or 2. If you get 1, it's odd. If you get 2, it's even.
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Re: Why octal is better than decimal

Postby ianfort » Sun Dec 19, 2010 5:21 am UTC

NathanielJ wrote:
ianfort wrote:Odd numbers could end in even numbers and vice-a-versa in every other multiple of the base.


So? It's still easy to determine whether or not something is even or odd in base 3: just add up all the digits. Then add all the digits of that. Repeat until you get 1 or 2. If you get 1, it's odd. If you get 2, it's even.


A bit inconvenient and time-consuming, though.

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Re: Why octal is better than decimal

Postby Corun » Sun Dec 19, 2010 11:53 am UTC

As pointed out, there's no reason to single out 2... However, we can apply this principal more generally:

Assume that we want a base between 8 and 16 (lower bases give numbers that are too long, higher bases have too many symbols). Then, if we want to be able to multiply and divide as easily as possible:

Code: Select all

Base, # of Divisors, Divisors / Base
12, 6 (1, 2, 3, 4, 6, 12), 0.5
8, 4 (1, 2, 4, 8), 0.5
9, 4 (1, 2, 3, 9), 0.45
10, 4 (1, 2, 5, 10), 0.4
16, 5 (1, 2, 4, 8, 16), 0.31
14, 4 (1, 2, 7, 14), 0.29
15, 4 (1, 3, 5, 15), 0.27

I list "divisors / base" because it is that which is interesting when it comes to multiplication tables. Any divisors of the base will neatly line up with their base (like 2 and 5 do for decimal). That is, in base 12, the 3s times table goes: 3, 6, 9, 10, 13, 16, 19, 20, 23, ... And further more, a number in base 12 is divisible by 3 if and only if it ends in 0,3,6,9.

I can't really decide which is better out of 8 and 12. 8 is cooler since it's a power of 2, but dammit if 12 doesn't have a lot of divisors.
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Re: Why octal is better than decimal

Postby eSOANEM » Sun Dec 19, 2010 3:52 pm UTC

I'd say 12 was better because it seems to me that only non-trivial divisors should be counted in which case 12 is better than 8 as 12 has 1/3 non-trivial divisors/base whereas 8 only has 1/4.
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Re: Why octal is better than decimal

Postby gmalivuk » Sun Dec 19, 2010 5:17 pm UTC

Yeah, like maybe multiply the number of divisors per base by the number of distinct prime factors a number has, which gives 12 a clear advantage.
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Re: Why octal is better than decimal

Postby Jyrki » Sun Dec 19, 2010 7:17 pm UTC

Yeah, 12 looks good. A possible consideration is also the size of the factors. My gut feeling is that the smaller the divisor, the more probable it is to occur. I am less convinced about the need for the base to be divisible by higher powers of any prime divisor. After all, 1/4, 1/8 etc all have finite length decimal expansions.

Having said that it is highly unlikely that we will move away from base 10 any time soon :)

Then again, Americans might be more willing to consider another base than us Europeans. After all, the metric system is sort of built on base 10.

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Re: Why octal is better than decimal

Postby Thesh » Sun Dec 19, 2010 9:43 pm UTC

mdyrud wrote:The whole "We have ten fingers, that's why we use base 10" argument has always kind of bugged me. Wouldn't that lead to us using base 11? After all, we have 10 unique digits, plus no digits at all. 11 possibilities.


It could have something to do with the fact that the concept of the number zero didn't exist in early civilizations.

http://en.wikipedia.org/wiki/0_(number)#Early_history
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Re: Why octal is better than decimal

Postby mdyrud » Sun Dec 19, 2010 11:25 pm UTC

Hmm, I hadn't thought of that. That is the first time anyone has given me a valid answer to that question. And it makes very good sense. So I think I can put that question to rest.

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Re: Why octal is better than decimal

Postby Luonnos » Mon Dec 20, 2010 2:22 am UTC

You also have to consider the divisors of the base plus one and the base minus one. That way, you can quickly check for divisibility by b+1 and b-1 like you do in base 10 for 9 and 11 (i.e. by looking at the sum and alternating sum of the digits).

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Re: Why octal is better than decimal

Postby pizzazz » Mon Dec 20, 2010 6:29 am UTC

Dason wrote:I thought I read somewhere that e is the optimal base. Which I think meant that 3 was a 'good' base for actually using.

After some investigation I found an article that backs this up. I think it would take quite a lot to move away from base 10 but it's interesting to think about.



The thing is, that article assumes that rw accurately models efficiency, and yet fails to even define efficiency. Humans are clearly capable of working with 10 distinct digits; before anyone asserts that one or another base is more efficient, there should be research to see if humans suffer any detriment from working with more digits.

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Re: Why octal is better than decimal

Postby breadlord » Tue Dec 21, 2010 8:35 pm UTC

I've always thought that 6 would be a good base as it highly divisible for it's size, it's got 2 distinct prime factors, and it's easy to count with using human hands as we have 5 digits on each hand.

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Re: Why octal is better than decimal

Postby Disanidi » Wed Dec 22, 2010 1:07 am UTC

pizzazz wrote:
Dason wrote:... Humans are clearly capable of working with 10 distinct digits; before anyone asserts that one or another base is more efficient, there should be research to see if humans suffer any detriment from working with more digits.


Rather than "research," a large scale social experiment sounds much more exciting.

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Re: Why octal is better than decimal

Postby Dason » Wed Dec 22, 2010 3:29 am UTC

That wasn't me that said that...
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Re: Why octal is better than decimal

Postby SunAvatar » Wed Dec 22, 2010 4:26 am UTC

I've advocated in fun for 'dozenal' before (and probably will again!) and I think it has real advantages, but those advantages are so minor that I don't think the switching costs would ever be made up.

That said, I'd be willing to take part in an internet experiment to see just how long it would take to get used to that system in a 'bilingual' enviornment. How would we set such a thing up?
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Re: Why octal is better than decimal

Postby Yakk » Wed Dec 22, 2010 12:31 pm UTC

pizzazz wrote:Humans are clearly capable of working with 10 distinct digits; before anyone asserts that one or another base is more efficient, there should be research to see if humans suffer any detriment from working with more digits.

If there is no problem with that, we should be using base 9699690.

So lets experiment: Try calculating the inverse of 179699690. How hard was it?
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Re: Why octal is better than decimal

Postby t1mm01994 » Wed Dec 22, 2010 1:44 pm UTC

Yakk wrote:
pizzazz wrote:Humans are clearly capable of working with 10 distinct digits; before anyone asserts that one or another base is more efficient, there should be research to see if humans suffer any detriment from working with more digits.

If there is no problem with that, we should be using base 9699690.

So lets experiment: Try calculating the inverse of 179699690. How hard was it?

now calculate 11-59699690 Not so cool now eh?

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Re: Why octal is better than decimal

Postby Yakk » Wed Dec 22, 2010 2:21 pm UTC

t1m, I suspect you read my comment backwards. Try turning your monitor upside down. :)
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Re: Why octal is better than decimal

Postby Velifer » Wed Dec 22, 2010 2:23 pm UTC

Dozenal.
http://forums.xkcd.com/viewtopic.php?f=40&t=16332&hilit=dozenal

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Re: Why octal is better than decimal

Postby t1mm01994 » Wed Dec 22, 2010 2:47 pm UTC

Yakk wrote:t1m, I suspect you read my comment backwards. Try turning your monitor upside down. :)

Thanks a lot, it helped. But yeah, I believe there is an upper bound, somewhere above 10, but under 9699690...

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Re: Why octal is better than decimal

Postby gmalivuk » Thu Dec 23, 2010 12:12 am UTC

If people are doing math entirely on paper, then I agree that the cusp might be higher than 10. However, counting repeatedly to 10 comes *far* more naturally to our 10-fingered selves than the perhaps theoretically superior 12. And above 12, it takes awhile before you hit another number with as many convenient factors, by which time you've surely crossed the cusp. After all, the size of the addition and multiplication tables that are necessary to memorize for the standard algorithms increases quadratically with the size of the base you're using.
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Re: Why octal is better than decimal

Postby Disanidi » Thu Dec 23, 2010 12:18 am UTC

gmalivuk wrote:If people are doing math entirely on paper, then I agree that the cusp might be higher than 10. However, counting repeatedly to 10 comes *far* more naturally to our 10-fingered selves than the perhaps theoretically superior 12. And above 12, it takes awhile before you hit another number with as many convenient factors, by which time you've surely crossed the cusp. After all, the size of the addition and multiplication tables that are necessary to memorize for the standard algorithms increases quadratically with the size of the base you're using.


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. Our clock system is based around 12, and quite frankly, the foot (12 in) is superior to the meter (100 cm), in my opinion. Creating a "metric" system in base 12 would be easy and more "natural."

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Re: Why octal is better than decimal

Postby Dason » Thu Dec 23, 2010 12:23 am UTC

Disanidi wrote: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.

I don't think anybody would argue against you here.
Our clock system is based around 12, and quite frankly, the foot (12 in) is superior to the meter (100 cm), in my opinion.

Why exactly do you believe the foot to be superior to the meter? I'm an advocate for the US switching the metric system and it's more or less because it's more consistent. If it was base 12 I really wouldn't care. It might take a while to get used to but I really don't think it would be *that* bad... it would just be really hard to convince others to switch.
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Re: Why octal is better than decimal

Postby Disanidi » Thu Dec 23, 2010 12:35 am UTC

Dason wrote:Why exactly do you believe the foot to be superior to the meter? I'm an advocate for the US switching the metric system and it's more or less because it's more consistent. If it was base 12 I really wouldn't care. It might take a while to get used to but I really don't think it would be *that* bad... it would just be really hard to convince others to switch.


Firstly, I wish to say that I advocate switching to the metric system as well. However, a "foot" has always struck me as a more natural unit of measurement. The meter seems unreasonably... large in comparison. I suppose I don't have a better reason. However, I thought I made it rather clear that it was just in my opinion.

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Re: Why octal is better than decimal

Postby gmalivuk » Thu Dec 23, 2010 12:41 am UTC

Disanidi wrote:I *believe* we are creatures of habit, not patterns, despite our pattern abilities.
If you mean to say that we don't have any inborn predispositions to certain behaviors over others, then I'm afraid the sciences of psychology and neuroscience would have to disagree with you there.

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.
Apart from the Babylonians, name another human culture that developed numbers in a base that wasn't a number of fingers and/or toes. If it would be just as easy, how come so few people ever hit upon it independently? (And the fact that various *measurements* are based around 12 isn't an argument in favor of *counting* in base-12. After all, English and Latin alike have decimal counting, despite 12 inches in a foot, 12 Troy ounces in a pound, and so on.)

the foot (12 in) is superior to the meter (100 cm), in my opinion.
But you say this is because of its size itself, which means the fact that it's 12 inches is completely irrelevant. You'd still think the foot was superior even if it was made up of 10 decifeet instead of 12 inches, so I'm not sure how that's even supposed to relate to your argument.
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Re: Why octal is better than decimal

Postby Disanidi » Thu Dec 23, 2010 12:59 am UTC

gmalivuk wrote:If you mean to say that we don't have any inborn predispositions to certain behaviors over others, then I'm afraid the sciences of psychology and neuroscience would have to disagree with you there.


That is not what I mean to say. I said that we are creatures of habit. I was hoping one would infer that I meant that if one is nurtured doing one thing, he will do that one thing.

Apart from the Babylonians, name another human culture that developed numbers in a base that wasn't a number of fingers and/or toes. If it would be just as easy, how come so few people ever hit upon it independently? (And the fact that various *measurements* are based around 12 isn't an argument in favor of *counting* in base-12. After all, English and Latin alike have decimal counting, despite 12 inches in a foot, 12 Troy ounces in a pound, and so on.)


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.

Also, I hope you realize the hypocrisy with saying that natural observations and measurement of those doesn't conclude one thing, while supporting that idea that biological features does. And yes, I realize my own hypocrisy somewhere in my few posts.

To answer the question of, "how come so few people ever hit upon it independently," I say that what matters isn't that amount, but the fact that they did hit upon it independently.

But you say this is because of its size itself, which means the fact that it's 12 inches is completely irrelevant. You'd still think the foot was superior even if it was made up of 10 decifeet instead of 12 inches, so I'm not sure how that's even supposed to relate to your argument.


My point was that we seem to find "12" in nature. I don't mean to become mystical, that would be silly. However, 12 IS a number humans use easily to describe nature and they do so often. (Again, I realize my own hypocrisy.) Lastly, I suspect that you will find another number in which we use to describe nature frequently, I'm just hoping that it will be a factor or multiple of 12.

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Re: Why octal is better than decimal

Postby gmalivuk » Thu Dec 23, 2010 4:42 am UTC

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.
Right, but as soon as a society has need to keep track of bigger numbers, there's a numbering system. And 10 is by far the most common base for those, with the occasional 8 (not counting thumbs?) or 20 (counting toes) thrown in.

Also, I hope you realize the hypocrisy with saying that natural observations and measurement of those doesn't conclude one thing, while supporting that idea that biological features does.
That's not what I said. Or rather, I only said anything about measurement. I doubt 12 actually appears much more than other small integers in nature, and there's *nothing* about nature that led to 12 inches to a foot or 12 ounces to a (Troy) pound or half-12 feet to a fathom or 12 to a dozen or 12*12 to a gross. 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).

I say that what matters isn't that amount, but the fact that they did hit upon it independently.
Can you name a single society that used base-12 without checking Wikipedia first? Because I can think of lots that used 10, one that used 20, and some that may have used 8, but I can't think of a single one that used 12. Of course the amount matters, because you're arguing that it would be just as easy, but that suggests that random flubbing about with numbers should have led an at least vaguely similar number of societies to a duodecimal system as to a decimal one, and that just isn't the case.

Lastly, I suspect that you will find another number in which we use to describe nature frequently, I'm just hoping that it will be a factor or multiple of 12.
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?
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Re: Why octal is better than decimal

Postby Zamfir » Thu Dec 23, 2010 10:09 am UTC

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.

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Yakk
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Re: Why octal is better than decimal

Postby Yakk » Thu Dec 23, 2010 12:58 pm UTC

gmalivuk 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.
Right, but as soon as a society has need to keep track of bigger numbers, there's a numbering system. And 10 is by far the most common base for those, with the occasional 8 (not counting thumbs?) or 20 (counting toes) thrown in.

I there was also 60 via Babylonians, which is where we get hours:minutes and minutes:seconds and degrees in a circle.

(60 = 1*3*4*5, so 1/1, 1/2, 1/3, 1/4, 1/5 and 1/6 are all representable finitely.)
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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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