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Mugwump101
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 Early Counting Systems   « on: Dec 1st, 2006, 2:20am » Quote Modify

I have two questions

1. How did Cro-Magnon man count?
2. How are numbers counted using string?
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towr
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 Re: Early Counting Systems   « Reply #1 on: Dec 1st, 2006, 3:32am » Quote Modify

Cro magnon man was much like us, so he could probably count in his head or at least on his fingers.

And with strings, you could use knots to count.
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rmsgrey
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 Re: Early Counting Systems   « Reply #2 on: Dec 1st, 2006, 7:10am » Quote Modify

Evidence has been found of what's thought to be an early counting system used by shepherds - the way it's thought to have worked is:

A supply of pebbles is kept near the gate to the sheep pen. As the sheep are let out, the shepherd takes one pebble per sheep and carries them with him in a pouch all day. When he returns with the sheep at night, he puts the pebbles back into storage, one as each sheep passes.

This is one of the "big idea" moments in mathematics - the idea that you can match pebbles to sheep and have the same number of each. Without that key jump - the idea that two collections of very different objects have a common property, the idea of number just doesn't happen. Counting seems natural to us, but imagine if we hadn't grown up with the idea. Where would we get the idea that "an apple and a banana" has anything in common with "a lake and a star"?
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Icarus
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 Re: Early Counting Systems   « Reply #3 on: Dec 1st, 2006, 4:08pm » Quote Modify

That is not just an early counting system. It is still used by primative tribes today. Even when the herdsmen know how to count, they often rely on this method to keep track of their herds. This allows them to not have to remember how many animals they have right now - a number that regularly changes, making it easy to get confused about the current value.

rmsgrey about this idea being the basis of mathematics. And this is probably what whoever gave these questions to you wants for question 1.

And towr is right about how to count with string. More herdsmen may use knots in string than use pebbles for keeping track of their herds. After all, it is easy to drop a pebble, and not even realize it.
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ecoist
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 Re: Early Counting Systems   « Reply #4 on: Dec 11th, 2007, 8:20pm » Quote Modify

I have to disagree with rsmgrey.  I read that shephards knew their flocks like they knew their children.  No need to count to determine if all are there or some are missing.  However, I agree that matching had great value.  Indeed, for example, how can a lender loan money, brass rings, at 25 percent interest without counting?  And Icarus is right about matching being used today!  Croupiers at gambling casinos use matching to pay winnings to blackjack players, including when the winnings are not equal to the bet!

To answer question 1, I read that there was a tribe that initially could count up to three only: one, two, three, many.  Then it devised a number system that began I, V, N.  Since a symbol for zero was contradictory for the tribe (can't represent nothing by something), what could that number system have been?
 « Last Edit: Dec 11th, 2007, 8:29pm by ecoist » IP Logged
Grimbal
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 Re: Early Counting Systems   « Reply #5 on: Dec 12th, 2007, 2:13am » Quote Modify

You can use that system to check your drinking when partying.

Put a handful of beans in your left pocket.  Every time you order a drink, take a bean from the left pocket and put it in the right pocket.  At the end of the evening (or more likely the next morning) count the beans in the right pocket.  Then decide if you have a problem.
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towr
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 Re: Early Counting Systems   « Reply #6 on: Dec 12th, 2007, 2:51am » Quote Modify

If you drink too much you might no longer be able to distinguish left from right somewhere along the way.
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JiNbOtAk
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 Re: Early Counting Systems   « Reply #7 on: Dec 12th, 2007, 3:59am » Quote Modify

on Dec 12th, 2007, 2:13am, Grimbal wrote:
 At the end of the evening (or more likely the next morning) count the beans in the right pocket.  Then decide if you have a problem.

on Dec 12th, 2007, 2:51am, towr wrote:
 If you drink too much you might no longer be able to distinguish left from right somewhere along the way.

I'd say you'd probably have less than the total you started out with.

One do wonder, what happens to the Cro-Magnon guy if one day, the number sheeps exceeds the number of pebbles ? What then ?
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Ghost Sniper
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 Re: Early Counting Systems   « Reply #8 on: Dec 12th, 2007, 5:41am » Quote Modify

Get more pebbles   .
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 Re: Early Counting Systems   « Reply #9 on: Dec 12th, 2007, 12:22pm » Quote Modify

on Dec 11th, 2007, 8:20pm, ecoist wrote:
 I have to disagree with rsmgrey.  I read that shephards knew their flocks like they knew their children.  No need to count to determine if all are there or some are missing.

On school trips, where the teacher knows all their students well, it's still routine to count heads rather than run through a roll-call when you want to check everyone's still together. It's significantly faster and easier to look at a group of kids and count how many of them are "yours" than it is to run through a list of about 20-30 names in order to check that each individual is present. Of course, if you discover that you have too few, you then go through the matching process to identify the missing individuals, but, provided you rarely have people missing, it's a significant time-saver to just count.
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ecoist
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 Re: Early Counting Systems   « Reply #10 on: Dec 12th, 2007, 2:09pm » Quote Modify

rsmgrey, we are not on the same page!  I questioned your reference to history; you seem to be comparing the effectiveness of different methods of keeping track of the members of a set.

And no one seems to be taking my bait: how can you do mathematics with one hand tied behind your back?  Specifically, show how to lend money with interest without counting?  Or, name the natural numbers without a symbol for zero?
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 Re: Early Counting Systems   « Reply #11 on: Dec 12th, 2007, 2:55pm » Quote Modify

on Dec 12th, 2007, 2:09pm, ecoist wrote:
 rsmgrey, we are not on the same page! I questioned your reference to history; you seem to be comparing the effectiveness of different methods of keeping track of the members of a set.
But only the setting is different. Even if the sheep from the herd were like children to the shepherd, checking each sheep in the flock corresponds to a pebble would still be a much better way to make sure you haven't lost any. Recognizing them may be easy enough, but spotting one is missing is not.

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 And no one seems to be taking my bait: how can you do mathematics with one hand tied behind your back?
A metaphorical hand? (Because I can do some basic arithmetic in my mind)
(Maybe I should reread your earlier post to see what you're getting at..)

Quote:
 Specifically, show how to lend money with interest without counting?
Grouping?
Well, ok, you might have to count a little, but not more than up to ten or so. If I want 10% interest, then for every group of 10 I gave you I need a group of 10 plus a group of 1 back.
The actual number isn't important to know as long as you keep track of a proper correspondence. (i.e. "for every pebble/chip/token I get a group of 11 *whatever* back")

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 Or, name the natural numbers without a symbol for zero?
I II III IV V VI ... ?
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pex
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 Re: Early Counting Systems   « Reply #12 on: Dec 12th, 2007, 3:02pm » Quote Modify

on Dec 12th, 2007, 2:55pm, towr wrote:
 I II III IV V VI ... ?

Ah! It's been way to long since I've seen this debate here - isn't zero a natural number?

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ecoist
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 Re: Early Counting Systems   « Reply #13 on: Dec 12th, 2007, 3:15pm » Quote Modify

You're getting there, towr.  You can avoid counting altogether.  Use your imagination, or watch croupiers do their magic at casinos.

As for your naming the natural numbers (except zero, pex), your names do not uniquely identify a number; e.g., III=N.  (I'm also hoping your number naming scheme allows efficient means for addition and multiplication)
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JiNbOtAk
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 Re: Early Counting Systems   « Reply #14 on: Dec 12th, 2007, 5:50pm » Quote Modify

on Dec 12th, 2007, 2:55pm, towr wrote:
 A metaphorical hand? (Because I can do some basic arithmetic in my mind)

Reminded me of this joke :

Lil Billy had just learned to count on his fingers, and he's showing off to his uncle. "Look uncle, I've learn to count. One, two, three, four, five, six, seven, eight, nine, ten." "That's good Billy," says his uncle. "Why don't you try counting without using your hands. Put them in your pockets." So Billy put both his hands in his pockets, and started again, "One, two, three, four, five, six, seven, eight, nine, ten, eleven.."

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mikedagr8
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 Re: Early Counting Systems   « Reply #15 on: Dec 14th, 2007, 12:53am » Quote Modify

on Dec 12th, 2007, 3:02pm, pex wrote:
 Ah! It's been way to long since I've seen this debate here - isn't zero a natural number?

Hmm, I DON'T want to be confused here. I thought it wasn't. Just like 1 is a prime, is it not.
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 Re: Early Counting Systems   « Reply #16 on: Dec 14th, 2007, 1:05am » Quote Modify

on Dec 14th, 2007, 12:53am, mikedagr8 wrote:
 Hmm, I DON'T want to be confused here. I thought it wasn't. Just like 1 is a prime, is it not.
1 is not a prime.
And whether 0 is a natural number or not has been a long contested fact in mathematics. It depends on the circumstances. sometimes things work better if you include it, sometimes when you don't.
If you construct natural numbers from sets, then 0 should be natural, as you start with the empty set. {}=0, {{}}=1, {{}, {{}}}=2, {{},{{}},{{},{{}}}}=3 etc (if memory serves me right).
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mikedagr8
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 Re: Early Counting Systems   « Reply #17 on: Dec 14th, 2007, 1:10am » Quote Modify

on Dec 14th, 2007, 1:05am, towr wrote:
 1 is not a prime. And whether 0 is a natural number or not has been a long contested fact in mathematics. It depends on the circumstances. sometimes things work better if you include it, sometimes when you don't. If you construct natural numbers from sets, then 0 should be natural, as you start with the empty set. {}=0, {{}}=1, {{}, {{}}}=2, {{},{{}},{{},{{}}}}=3 etc (if memory serves me right).

Ok thanks for clarifying. I was just recently retold it was not, but it seemed as though it should. Why is 1 not a prime? It is divisible by itself and 1.
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 Re: Early Counting Systems   « Reply #18 on: Dec 14th, 2007, 1:15am » Quote Modify

It is actually a matter of how you define it.

The notion of prime numbers is more useful if you exclude 1, because then the expression of a number as a product of primes is unique.  If you accepted 1 as a prime, you would have to make endless provisions for that.

So while as a lawyer you could argue that 1 is a prime, as a mathematician, you would not.
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mikedagr8
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 Re: Early Counting Systems   « Reply #19 on: Dec 14th, 2007, 1:32am » Quote Modify

Thanks again.
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 Re: Early Counting Systems   « Reply #20 on: Dec 14th, 2007, 5:36am » Quote Modify

Alternatively, the definition I've often heard is that a prime number is a positive integer which has "exactly two unique (positive integer) divisors: one and itself."  Under that definition one is not a prime as it has only a single unique divisor.

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 Re: Early Counting Systems   « Reply #21 on: Dec 14th, 2007, 6:09am » Quote Modify

on Dec 12th, 2007, 12:22pm, rmsgrey wrote:
 On school trips, where the teacher knows all their students well, it's still routine to count heads rather than run through a roll-call when you want to check everyone's still together. It's significantly faster and easier to look at a group of kids and count how many of them are "yours" than it is to run through a list of about 20-30 names in order to check that each individual is present. Of course, if you discover that you have too few, you then go through the matching process to identify the missing individuals, but, provided you rarely have people missing, it's a significant time-saver to just count.

You've never watched Home Alone, have you?
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 Re: Early Counting Systems   « Reply #22 on: Dec 14th, 2007, 3:01pm » Quote Modify

on Dec 14th, 2007, 6:09am, Ghost Sniper wrote:
 You've never watched Home Alone, have you?

And how likely is a shepherd to have a strange sheep passing through?
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Ghost Sniper
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 Re: Early Counting Systems   « Reply #23 on: Dec 14th, 2007, 3:16pm » Quote Modify

OK. In the first part of the first Home Alone, the parents did a head count. However, a neighbor's kid was mixed in with the 11 other kids, and since he was wearing a hat, he looked like Kevin. So, the parents counted 12, and with Kevin sleeping in, he got left behind.

So, if the strange sheep looks in any way similar to one of the shephard's sheep, and the shephard is in a hurry, he might mix up the strange sheep with his own.
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rmsgrey
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 Re: Early Counting Systems   « Reply #24 on: Dec 14th, 2007, 3:19pm » Quote Modify

on Dec 14th, 2007, 3:16pm, Ghost Sniper wrote:
 OK. In the first part of the first Home Alone, the parents did a head count. However, a neighbor's kid was mixed in with the 11 other kids, and since he was wearing a hat, he looked like Kevin. So, the parents counted 12, and with Kevin sleeping in, he got left behind.   So, if the strange sheep looks in any way similar to one of the shephard's sheep, and the shephard is in a hurry, he might mix up the strange sheep with his own.

Yeah, I've seen it, but getting strange sheep mixed up in the flock is gonna be even less common - people tend to mix fairly indiscriminately out in public; sheep very rarely get the opportunity to mingle.
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