
#2122598  07/25/13 12:45 AM
OT ... why base 10?

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I had a discussion with my soninlaw about this.
Is our system of counting, base 10, anthropomorphically based (since we have ten fingers)? If beings elsewhere in the universe have eight fingers, would they use a base eight system of counting?
Is there anything special or easier about using a base ten system?
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#2122612  07/25/13 01:37 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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>If beings elsewhere in the universe have eight fingers, would they use a base eight system of counting? No not necessarily. Other civilizations used 12based (duodecimal), and we also seem to use it for the clock time? In computer science we all use 2based. A complete list is here https://en.wikipedia.org/wiki/List_of_numeral_systems

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#2122696  07/25/13 08:02 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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Interesting that you ask this question on a piano forum. Of course, for us the 10 fingers are crucial... And yes, the main reason historically for a base 10 system seems to be the anatomical coincidence and the resulting practice of counting with fingers. There were, and there are, alternative systems that have mathematical advantages. Some of them are also part of the modern baggage here ore there: Systems with base 12 ('a dozen', 12 months, 2x12 hours a day, ...) have their advantages (they allow easy division by 2,3,4 and 6), and the Mesopotamian sexagesimal system with base 5x12=60 (even more easy divisors) was perhaps the best from a mathematical point of view. It is still at work in our measurement of time (an hour has 60 mins, a minute has 60 seconds) and of the circle (6x60 degrees, 1 degree = 60', 1'=60"...). Some languages have remnants of a system with base 20 (see the French 'quatrevingt' or Danish 'firsindstyve'). Otherwise, see the list linked above...

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#2122697  07/25/13 08:10 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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I see other systems in numerous science fiction novels and stories, I always find that kind of thing interesting, authors thinking of all the ways aliens (or changed humans) will be different from us.

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#2122703  07/25/13 08:42 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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I don't know why I mention this. Maybe I'm just stupid. From my very shallow knowledge of string theory. There are 11 strings. Less are incomplete. More are unstable. Here's where I probably fault....just off the top of my head... zero through ten make eleven. There is something special about eleven strings. Just something to think about.
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#2122709  07/25/13 09:15 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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Is there anything special or easier about using a base ten system? No, nothing at all, apart from the ease of multiplying and dividing by 10 which is a doddle. Any integer can be used as a numbering base. It would be hard to ascribe a uniqueness in quality to any integer other than a cultural significance arising through folklore or, as you point out, a match for the number of digits in our 2 hands. Integers do have collective qualites and can become members of sets according to simple arithmetical or algebraic rules and qualities; even, odd, prime, integer squares, almost anything you want to make up really. But none of those qualities give a number preeminence as a base for a numbering system. Years ago when I was messing with simple programming I had to get used to hexadecimal (base 16) which is all well and good until you get to the number 10 (decimal) which requires its own symbol, in this case 'A'. So you have. (dec  hex) 0 = 0 1 = 1 2 = 2 3 = 3 4 = 4 5 = 5 6 = 6 7 = 7 8 = 8 9 = 9 10 = A 11 = B 12 = C 13 = D 14 = E 15 = F 16 (decimal) = 10 (hex) So hex(adecimal) is great for multiplying and dividing by (decimal)16 if that's what you want to do. And there's the rub; for higher magnitude numbering bases, you need even more symbols.

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#2122725  07/25/13 10:20 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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As a matter of fact, you can still find remnants of 12 and 16 base counting in many languages. For example, in English, we count up to twelve before we switch to threeteen, fourteen, ... In French, it goes up to seize (sixteen) before it continues with dixsept (tenseven).

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#2122743  07/25/13 10:55 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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Registered: 02/20/10
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Yup, there's nothing inherently easier about operations in base 10 over 8 or whatever except familiarity. Its dominance probably did come from finger counting.

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#2122777  07/25/13 12:19 PM
Re: OT ... why base 10?
[Re: rnaple]

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Registered: 06/10/13
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... of string theory. There are 11 strings. Less are incomplete. More are unstable. I think there are many more than 11 strings if string theory is correct. However, said strings might vibrate in 11 dimensions
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#2122854  07/25/13 03:11 PM
Re: OT ... why base 10?
[Re: rnaple]

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I don't know why I mention this. Maybe I'm just stupid. From my very shallow knowledge of string theory. There are 11 strings. Less are incomplete. More are unstable. Here's where I probably fault....just off the top of my head... zero through ten make eleven. There is something special about eleven strings. Just something to think about. Oh yeah! Oneeighth of a piano.
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#2123079  07/26/13 01:12 AM
Re: OT ... why base 10?
[Re: maurus]

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Registered: 11/17/12
Posts: 82

Interesting that you ask this question on a piano forum. Of course, for us the 10 fingers are crucial... And yes, the main reason historically for a base 10 system seems to be the anatomical coincidence and the resulting practice of counting with fingers. There were, and there are, alternative systems that have mathematical advantages. Some of them are also part of the modern baggage here ore there: Systems with base 12 ('a dozen', 12 months, 2x12 hours a day, ...) have their advantages (they allow easy division by 2,3,4 and 6), and the Mesopotamian sexagesimal system with base 5x12=60 (even more easy divisors) was perhaps the best from a mathematical point of view. It is still at work in our measurement of time (an hour has 60 mins, a minute has 60 seconds) and of the circle (6x60 degrees, 1 degree = 60', 1'=60"...). Some languages have remnants of a system with base 20 (see the French 'quatrevingt' or Danish 'firsindstyve'). Otherwise, see the list linked above... Circles actually have 360 degrees because until fairly modern times it was beliieved a year was 360 days long Hence a day or 1 degree was 1/360 of that
Edited by Schroeder II (07/26/13 01:12 AM)

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#2123133  07/26/13 02:52 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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Registered: 05/21/11
Posts: 1229

I am sorry, but this is nonsense. On the history of calendars see here. There is a connection between measuring the circle, measuring time, and the calendar of course.

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#2123219  07/26/13 09:49 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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Not so fast. According to the wikipedia entry on degree there are three theories for why there are 360 degrees in a circle (no one knows for sure).  In some early calendars (for example the Persian one) there were 360 days in a year
 Babylonians used base 60 numbers so 6*60 was a natural choice
 360 is nicely divisible by lots of things
So I guess we don't really know but the calendar guess is as good as any.

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#2123226  07/26/13 10:02 AM
Re: OT ... why base 10?
[Re: dewster]

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Base 2 works best for two state digital logic (which maximizes noise margins and allows for very high speed circuitry). In effect, music uses binary fractions for note values, since all the subdivisions are powers of two. whole note : 1 half note : 0.1 quarter : 0.01 eighth : 0.001 dotted eighth : 0.0011
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#2123269  07/26/13 11:25 AM
Re: OT ... why base 10?
[Re: joflah]

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In effect, music uses binary fractions for note values, since all the subdivisions are powers of two. whole note : 1 half note : 0.1 quarter : 0.01 eighth : 0.001 dotted eighth : 0.0011
Wouldn't it be more accurate to say: whole note : 1 half note : 0.5 quarter : 0.25 eighth : 0.125 dotted eighth : 0.1875 Not to be a complete math nerd...but I'm prepared to be...
Edited by dje31 (07/26/13 11:27 AM)
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#2123285  07/26/13 11:49 AM
Re: OT ... why base 10?
[Re: Dave Horne]

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joflah was showing binary fractions, where the digits are 0 and 1 and each decimal place is 1/2 of the preceding place.
You are showing decimal fractions, where the digits are 09 and each decimal place is 1/12 of the preceding place..
joflah's point is that the binary fractions show a very simple pattern for the notes. To be a true math nerd: work out the binary fractions for triplets and for compound meter.

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#2123349  07/26/13 02:01 PM
Re: OT ... why base 10?
[Re: PianoStudent88]

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joflah was showing binary fractions, where the digits are 0 and 1 and each decimal place is 1/2 of the preceding place.
You are showing decimal fractions, where the digits are 09 and each decimal place is 1/12 of the preceding place..
joflah's point is that the binary fractions show a very simple pattern for the notes. To be a true math nerd: work out the binary fractions for triplets and for compound meter. Right. But triplets would be a repeating expression triplet half note (in space of whole note): 0.01010101010... whereas in decimal, it'd be 0.33333... You'd just have to show the division, same as in decimal: 1/11b for each of the three notes. Compound meter? The notes would be the same, I guess.
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