Why is the decimal system so functional?

[Chillheimer]

Maybe I should better ask how it comes or what leads to the fact that the decimal system is so much easier to work with than any other system.

Or is it?
Is it really so much easier than the mayan system, based on 20, or the babylonian based on 60? Or binary?
Maybe it´s just because we all grew up with it and learned to use it instead of using any other?
Could it actually be totally arbitrary at which point we decide to go to (what we call) the next decimal?
Why after 9?

Or maybe I should just say that the key to all numbering systems is pattern and that I have a strong feel that at the core lies symetry and self similarity due to recursive patterns in the system itself..

So might the decimal system itself be much more self similar than any other system and that this is the reason why it gained most acceptance?

Or are these self similar patterns always relative to the observer/to what we've grown to deal with?
As long as you have invented the zero as mark for the next "deci"-mal, who cares if its deci-, penta-, octa- or whatever-mal?



Or just because we have 10 fingers?


strange stuff keeps me up at some nights..

[Sockratease]

Wouldn't basic mathematical stuffs work the same in any base?  

...So might the decimal system itself be much more self similar than any other system and that this is the reason why it gained most acceptance?

Or are these self similar patterns always relative to the observer/to what we've grown to deal with?
As long as you have invented the zero as mark for the next "deci"-mal, who cares if its deci-, penta-, octa- or whatever-mal?



Or just because we have 10 fingers?


strange stuff keeps me up at some nights..

I remember reading that among the earliest systems, things were base 12.  This was because counting was developed to ensure equal sharing of stuffs and 12 had the most ways to be divided evenly, and the only convenient number divisible by 1, 2, 3, and 4 (not to mention 6 and itself).

I think that has nicer symmetry and more recursive nature things going on than base 10, so it must be the fingers thing.

[asimes]

Personally I find binary and hexadecimal to be more convenient but it is because I work with them on a daily basis in code. In decimal multiplying and dividing by 10 is simple because that is the base, likewise multiplying and dividing by 2 is simple in binary. It probably is just a matter of what you are accustomed to working with

Wouldn't basic mathematical stuffs work the same in any base?

Although your comment is true, it actually can be less efficient to use certain bases for certain problems. As an example, computers use binary behind the scenes and they can multiply or divide numbers by powers of two very efficiently while other numbers are sometimes slower

A computer can do the following typically in a single cycle: x = x*2

On the other hand the following typically requires several cycles: x = x*10. In order to deal with this efficiently it may be converted to: x = x*8 + x*2 or x = x*8 + x + x because then it makes use of power of two

[youhn]

Or just because we have 10 fingers?

Yes. But still a little random. We could have chosen 5 (arms+legs+head / fingers at one hand) or perhaps 8 (only fingers, without thumbs). Or any other number that we see daily.

[Tglad]

Well the Roman system is base 5.

I remember reading that among the earliest systems, things were base 12

Not so long ago... 12 inches in a foot, 12 * 30 degrees in a circle, 12*2 hours in a day, 12*5 minutes in an hour, 12 old pence to a shilling.
Of course 12 is a nice rectangular number, so probably a better choice than 10.

[hobold]

Numbers behave the same way regardless of representation. The specific representation may have a big influence on the efficiency of computation. Not just for computers, as in the power of 2 example above, but for humans as well. Back in the roman empire, reading and writing were part of normal education. Addition and multiplication were not; the roman numerals made these things awkward and difficult.

Sometimes the difference in complexity is so staggering, that algorithms are known for one specific representation, but unknown for other, similar, representations. For example there is https://en.wikipedia.org/wiki/Bailey–Borwein–Plouffe_formula for computation of arbitrary digits of pi in base 16. This idea cannot be adapted to work in base 10; one must use algorithms that compute all preceding digits up to and including the digit at the desired position.

[3dickulus]

arriving at base 10 seems a very logical thing...
when merchants and traders of old had to communicate with a diverse group of cultures and languages, or back far enough - no language, one could always express and understand a numerical value by showing fingers, the decimal place is a convenient representation for div/mul by base, so, after many many generations our brains are wired with favor for base 10... I think?

[Tglad]

I'm not sure decimal or other bases are all that functional, it seems like they're built for positive numbers and the negatives are an afterthought, you need a character that is used nowhere else (-) and it adds conditions to arithmetic rules. Also, even simple operations like increment have unbounded cost.
My stab at it would be to use signed numbers from the start, for each digit, so you get:
[0 - 10] 0 1 2 3 4 5(or 1-5) 1-4 1-3 1-2 1-1 10
[10-20] 10 11 12 13 14 15(or 2-5) 2-4 2-3 2-2 2-1 20
[40-50] 40 41 42 43 44 45(or 5-5) 5-4 5-3 5-2 5-1 50
[50-60] 50(or 1-50) 1-5-1 1-5-2 1-5-3 1-5-4 1-45(or 1-5-5) 1-44 1-43 1-42 1-41 1-40
[90-100]1-10 1-1-1 1-1-2 1-1-3 1-1-4 1-05 1-04 1-03 1-02 1-01 100
And the negatives signs on the digits would be extra small to not be confused with the minus operator between numbers.

As well as the integers, you can represent the unit intervals too, the gaps between the integers, used for representing full periods like years in dates, or your age:

[0 - 10] 0 1 2 3 4 1-4 1-3 1-2 1-1 1-0 10
[10-20] 10 11 12 13 14 2-4 2-3 2-2 2-1 2-0 20
[40-50] 40 41 42 43 44 5-4 5-3 5-2 5-1 5-0 1-5-0
[50-60]1-5-0 1-5-1 1-5-2 1-5-3 1-5-4 1-44 1-43 1-42 1-41 1-40 1-4-0
[90-100]1-1-0 1-1-1 1-1-2 1-1-3 1-1-4 1-04 1-03 1-02 1-01 1-00 100

The two types are distinguished by what they assume the unwritten digits to be. For integers, like with the normal number system, it is assumed that 13 is really ...0000013.0000... For the unit intervals the digits after the decimal point are undefined/unconstrained/all values, I write it with an x, so the third interval 2 is really ...000002.xxxxxx. So to be explicit the integers could be written as 245.0 shorthand and the intervals 245.x shorthand.
A nice extension of this is that the x can be placed anywhere, so effectively considered as an extra digit type. e.g. ...xxx13.x gives the repeating set of intervals at 13,113,213,... or basically an algebra mod 100.

[hobold]

My stab at it would be to use signed numbers from the start, for each digit,

Do you have algorithms to do addition and multiplication in this number system?

[Tglad]

I haven't looked at that, but it should have a better worst case cost. For example, with normal numbers 1000...0000 - 1 is 999...9999 which is as many changes as the length of the number. In the system above 1000...0000 - 1 is 1-000...0001 which only a single sign and digit change. In fact adding or subtracting 1 is always at most a single digit and sign change.

[Tabasco Raremaster]

First answer that came up was 10 fingers.