OT ... why base 10?

Posted by: Dave Horne

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

I had a discussion with my son-in-law 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?
Posted by: wouter79

Re: OT ... why base 10? - 07/25/13 01:37 AM

>If beings elsewhere in the universe have eight fingers, would they use a base eight system of counting?

No not necessarily. Other civilizations used 12-based (duodecimal), and we also seem to use it for the clock time? In computer science we all use 2-based.

A complete list is here

https://en.wikipedia.org/wiki/List_of_numeral_systems
Posted by: maurus

Re: OT ... why base 10? - 07/25/13 08:02 AM

Interesting that you ask this question on a piano forum. Of course, for us the 10 fingers are crucial... wink

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 'quatre-vingt' or Danish 'firsindstyve').

Otherwise, see the list linked above...
Posted by: Big McLargehuge

Re: OT ... why base 10? - 07/25/13 08:10 AM

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.
Posted by: rnaple

Re: OT ... why base 10? - 07/25/13 08:42 AM

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.
Posted by: dire tonic

Re: OT ... why base 10? - 07/25/13 09:15 AM

Quote:
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 pre-eminence 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.
Posted by: Peter Leyssens

Re: OT ... why base 10? - 07/25/13 10:20 AM

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 three-teen, four-teen, ... In French, it goes up to seize (sixteen) before it continues with dix-sept (ten-seven).
Posted by: anotherscott

Re: OT ... why base 10? - 07/25/13 10:55 AM

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.
Posted by: dewster

Re: OT ... why base 10? - 07/25/13 11:04 AM

Base 2 works best for two state digital logic (which maximizes noise margins and allows for very high speed circuitry). This creates some conflict with our base 10 system because conversion between the two bases is necessary, and there are some exact decimals that form repeating decimals in the other system, thus hurting precision. I believe many older HP calculators performed internal operations in hexadecimal to avoid this.

The world of the digital designer would be a lot easier if we all went to base 4, 8, or 16 (my preference would be 16 because the binary width is 4 - a power of 2 - but that would make learning times tables more difficult for the little ones).
Posted by: de cajon

Re: OT ... why base 10? - 07/25/13 12:19 PM

Originally Posted By: rnaple
... 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 crazy
Posted by: joflah

Re: OT ... why base 10? - 07/25/13 03:11 PM

Originally Posted By: rnaple
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! One-eighth of a piano.
Posted by: Schroeder II

Re: OT ... why base 10? - 07/26/13 01:12 AM

Originally Posted By: maurus
Interesting that you ask this question on a piano forum. Of course, for us the 10 fingers are crucial... wink

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 'quatre-vingt' 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
Posted by: maurus

Re: OT ... why base 10? - 07/26/13 02:52 AM

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.
Posted by: gvfarns

Re: OT ... why base 10? - 07/26/13 09:49 AM

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.
Posted by: joflah

Re: OT ... why base 10? - 07/26/13 10:02 AM

Originally Posted By: dewster
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
Posted by: dje31

Re: OT ... why base 10? - 07/26/13 11:25 AM

Originally Posted By: joflah

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...
Posted by: PianoStudent88

Re: OT ... why base 10? - 07/26/13 11:49 AM

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 0-9 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.
Posted by: joflah

Re: OT ... why base 10? - 07/26/13 02:01 PM

Originally Posted By: PianoStudent88
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 0-9 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.