Choices of Mathematical Base

[zelos]

I have seen various people deal with this and all, having acquired some knowledge about math as I study it and its history I thought I would give a dollar or two on the subject of Choice of mathematical base.


Numbers are about as ancient as we humans have been sapient in any form similar to ours, it is doubtful that neanderthals didnt have some concept of it considering they used clothing and more hence had atleast some simple understanding of numbers. It is then equally doubtful that our conpeople wont have some understanding of numbers aswell.

The simplest form is Unary system, 1 line or something means 1 and 2 lines means 2 and so on. That system is very simple to understand and is used in modern terms through the way of I+1=II, IIII+1=IIII. It is conceptually easy to understand and develop but as anyone can see it is a hige hassle once you start dealing with large numbers, how do you really see the differens ebtween 142 lines and 143 lines without starting to count the individual lines themselves? One way to solve this is by the system of every fifth line croses over the previous four creating a bundle of 5 thats easy to recognise. But even that goes only so far because if you got 245 and 250 you'll still need to count 50 of those and you cant see that easily (assuming it isnt perfectly organised in forexample rows of 10 in each but even there it'll come to a limit where you start counting the individual rows then). That is why all societies really abonden this with some exceptions for various reasons.

So people start choosing bases for their math and ways to represent it in a more coherent fashion that doesnt require counting beyond 5-7 in the individual symbols themselves. But the question becomes then "Which base do I choose?"

Well I would say there is one of two reasons any given base is really picked by people, both really does go back to "Convienience" in some sense.

Biological Reason, This mostly refers to that the base is choosen purely from what is aviable by the people, 10 fingers for example. But traces of things have given us reasons to thinkt aht europe used to have base 8 prior to base 10s entrance, mostly because 9 has often a name that coincide with "new" which might not be a coincidence on its own. But why 8? Same reason though there are 8 SPACES between fingers on human hands. Base 20 is the same, 10 fingers and 10 toes. This is an easy way, natural and understandable for any creature to pick.

Mathematical Convinience, this one is also popular in ancient civilizations, they werent nearly as dumb as we wanted to make them. 12 and 60 are common in this class but exacly waht do I mean with it? Mathematical Convinience is mostly that from a mathematical point of view it is considerbly easier dealing with in fractions decimal system and all. For example in a civilization where decimal system has yet to be developed a base with alot of divisions in it is alot easier dealing with than one with few. Why? if I got base 60 and divide a cake by 3 I know it should be (0 . 20 , 0) parts in each piece. That is of course in a decimal system, but even babylonians used it well for division, 1/13=7/91~7/90=7*40/3600=7*40/60², Sure its an approximation buit its 1% off and most things cant be solved exacly. Having a more rich divisionable base makes approximations easier because its easier to reach a good appriximation with a finite decimal string or division.

12, 24, 48 and 60 all have one thing in common, they are highly composite numbers. Meaning they got more ways of being divided with no rest than any other number prior to them. This is the reason why 60 for example was picked by some, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Thats a rather immense amount of possible divisions, 12 even if its small got a good amount of divisors, 1, 2, 3, 4, 6 and 12.

If we include all possible finite decimal expansion up to the base number itself the amount grows even more. This is why many prime numbers in the base number itself is so valueble.

This is the reason these numbers have been loved by ancient people and why certain numbers come time and time again in various forms of bases or subbases while others are not that prone (but not impossible).

An example that is often a nail in my eye is many seem to take base 16. Sure it works well in our digital age of computers but for people when the choice is acctually made and not just used base 16 is horrible, 1 it doesnt have any natural representation on our own bodies (though of course I dont know how your species is, erhaps 8 fingers per hand? or 4 fingers and toes?) and mathematicly it is a horrific number to pick, why? It only got 1 prime number namely 2 in it, 2^4=16 meaning it can only properly deal with divisions of power two as finite fractions or eliminations in the number itself. While it is true all decimal expansions that are finite are considerbly less than those with inifnite expansion why would any people really want to limit themselves more?

Base 16 therefor in most cases will be one that is reserved to binary computer ages or some species where it occures naturally on their body as it possesses no value mathematicly.

I could go on forever with various bases but I shant I merely took 16 as an example.

I'll post about how numbers are represented later on if people are interested

[MrKrov]

Whether you wanted a simple discussion of mathematical bases or which ones people here like to pick: viewtopic.php?f=6&t=348&hilit

[zelos]

Whether you wanted a simple discussion of mathematical bases or which ones people here like to pick: viewtopic.php?f=6&t=348&hilit

I didnt, I just wanted to give poeple something to think on :3

[Ear of the Sphinx]

I'm thinking about making base-5 and base-10 symbols for my conlang. That's, it has already base-10 system, but this case seems unlikely.

[Systemzwang]

anyone except me considered bijective numeration?
http://en.wikipedia.org/wiki/Bijective_numeration

(bryatesle writes its numbers with the opposite endianness to ours, and bijectively numerated, and octally, so 213 would come out as:

523

64 would come out as
87

72 would come out as
88

73 would come out as
111

[Ear of the Sphinx]

anyone except me considered bijective numeration?
http://en.wikipedia.org/wiki/Bijective_numeration

Yhm, I'm using 10-adic in Nikau, but I'm not sure of it. I'm rather going to change it to normal base-10.

[zelos]

anyone except me considered bijective numeration?
http://en.wikipedia.org/wiki/Bijective_numeration

(bryatesle writes its numbers with the opposite endianness to ours, and bijectively numerated, and octally, so 213 would come out as:

523

64 would come out as
87

72 would come out as
88

73 would come out as
111

Question you should ask then is show and why did such an odd system arise?

[Systemzwang]

anyone except me considered bijective numeration?
http://en.wikipedia.org/wiki/Bijective_numeration

(bryatesle writes its numbers with the opposite endianness to ours, and bijectively numerated, and octally, so 213 would come out as:

523

64 would come out as
87

72 would come out as
88

73 would come out as
111

Question you should ask then is show and why did such an odd system arise?

trivial, really.
the opposite-endianness: our number system was actually written the same way in Arabic, which is normally read the other way, so Arabian Arabic numerals are opposite-endian to ours. Endianness can go either way, anyhow, there's no big reason why we have the biggest number the farthest to the left - in fact, this is a bit counterintuitive, as you can't know how big it is without first counting the number of digits to the right of it; this way, you get that automatically.

bijective numeration: well, it's essentially just a base-whatever system where there's no 0, and this flaw is fixed by having a placeholder for a full tally in a position; so if we had a decimal bijective numeration, it'd go:
1
2
3
4
5
6
7
8
9
T <- full set, but we can't write it as 10 since we don't have any zero to put in the first column
11 <- one full set + 1
...
1T <- one full set + one full set = 20
9T = 9 full sets + one full set = 100
TT = T full sets + one full set = 110
111 = 1 full set of full sets, plus one full set, plus one = 111
11T = 120
...
1T1 = 201
1TT = 210
211 = 211
TTT = T*T*T + T*T + T = 1000 + 100 + 10 = 1110
1111 = 1111

counter to normal base-X systems, this kind of system has a symbol for X.

[Nortaneous]

I'm probably going to end up giving Kannow base 8. Should make things interesting, since Kett and Arve, both of which eventually become significant substrates, use base 10. (I'm thinking Tharu will mostly preserve base 8, Enzielu will retain some aspects of it, and Renzell will lose it almost completely.)

[Tyhoeciyh]

I thought that base four would be nice for the reason that it is the normal number that a person can immideatly recognize without doing a simple mental calculation. The problem with it is that with four being so small, the numbers would become enormously long.

[òmhò]

My number systems usually end up being base 9. Something to do with the sacred removal of one finger at birth probably.

[Micamo]

I thought that base four would be nice for the reason that it is the normal number that a person can immideatly recognize without doing a simple mental calculation. The problem with it is that with four being so small, the numbers would become enormously long.

IIRC, that number is actually 5.

[zelos]

I thought that base four would be nice for the reason that it is the normal number that a person can immideatly recognize without doing a simple mental calculation. The problem with it is that with four being so small, the numbers would become enormously long.

IIRC, that number is actually 5.

with base 6 you can count to 35 on your fingers

[eldin raigmore]

I thought that base four would be nice for the reason that it is the normal number that a person can immediately recognize without doing a simple mental calculation.

IIRC, that number is actually 5.

I thought it was six, because that's what I can do.

[Micamo]

I thought it was six, because that's what I can do.

In any pattern, with the different units spread out any distance relative to each other (within your visual range)? And are you sure your brain isn't just parsing it as 2 groups of 3 first, then adding these to make 6?

[teh_Foxx0rz]

are you sure your brain isn't just parsing it as 2 groups of 3 first, then adding these to make 6?

That's what my brain does at least.

[zelos]

are you sure your brain isn't just parsing it as 2 groups of 3 first, then adding these to make 6?

That's what my brain does at least.

some people are able to go as high as 6 or 7 or even 9 before they start grouping

[thetha]

I like base 6. I've used it for all my projects that make it to numbers d:

[eldin raigmore]

If the main base is twenty or higher, there's always a "sub-base".
Even if the main base is less than twenty, there sometimes is a "sub-base".

Also, there's often a "super-base". A language might have a base and a sub-base and a superbase.

For instance, its main base might be six; its subbase might be three; and its superbase might be twenty-four.

Or, its main base might be twelve; its subbase might be four; and its superbase may be one-hundred-twenty.

Or, its main base might be twenty; its subbase might be five; and its superbase may be three-hundred-sixty.

Or, its main base might be thirty; its subbase might be six; and its superbase may be eight-hundred-forty.

Or, its main base might be forty-two; its subbase might be seven; and its superbase may be 1680.

Or, its main base might be fifty-six; its subbase might be eight; and its superbase may be 3024.

Or, its main base might be sventy-two; its subbase might be nine; and its superbase may be five-thousand-forty.

_______________________________________________________________________________


For such a language, if it has only multiplicative bases, and doesn't have an exponential sequence of bases, the largest number that has a unique phrase for it would be something like

((s*A + t)*B + u*A + v)*C + (w*A + x)*B + y*A + z

where B is the base, A is the subbase, and C is the superbase,
and each of s, t, u, v, w, x, y, and z is less than A,
each of sA+t, uA+v, wA+x, and yA+z is less than B,
and each of (s*A + t)*B + u*A + v and (w*A + x)*B + y*A + z is less than C.

_______________________________________________________________________________


For an exponential base, I'd prefer some multiple of 6 or some multiple of 10. For instance, thirty or sixty, or forty-two, or sixty-six, or seventy, or seventy-eight.

If not one of those, then my favorites, from smallest to largest, would probably be:

two, three;
four;
six, eight, ten;
twelve, eighteen, twenty;
twenty-four, thirty;
thirty-six;
forty-eight;
sixty, seventy-two.

[zelos]

in my owlrd its base 12 and superbase 144

as each incrament after 144 is a power ofi t. to say 1 728 one say 12 144s
35831808 (12^7) is 12 144 20736 and so on