General Question - Mathematical Bases

[Bodgeups]

Hi everyone,

Quick question - how many mathematical bases are there in existense? Someone posted a question on another board asking:

"If we work on the theory that infinity is a quantifiable thing and can be written down as a number, then *** is the last number before infinity?"

Now he claimed the number in question is 9, but of course that's only going to work under base-10 math. Given base-16, the answer (on the technicality) would be F.

So I wondered, how many bases are there? Total number of bases = total number of possible answers to the question...

[jeroen79]

I have no clue about this last number before infinity thing but wouldn't there be an infinite number of bases?

[Dnereb]

In fact you could in theory use infinety as a base as there are an infinete amount of symbols possible te represent a specific number in your counting system....

[superbovine]

The actually assumation was fasle. In a "proof", i would just prove the hyphothesis wrong through false assumption.

infinity in math means that there is an unbounded limit, if you go about changing the basic rules of math to prove you theory, your theory is probably wrong.

there is an infinity number of bases.

[triggernum5]

The thing is, infinity is by definition defined as any arbitrary uncomprehendably large number.. 1 can be infinately greater than a lesser positive value.. And on the other side of the coin, a number can no longer be defined as infinity once you have a grasp of its magnitude.. Two numbers can be infinitely large, even if its known that one of them is greater than the other by an arbitrary amount.. When I was 5, both my grandparents were infinitely old to me even though I knew gramps had 5 years on grams.
Now bases as well are a matter of definition in a way.. (Or atleast a result of prior definition..) Absolutely ANY number except 1 can be used as a base.. (Actually it wouldn't surprise me if somebody developed some imaginary base-1 math idea just to confuse me and all future highschool students..) Some are common/useful, but the vast majority of base-N possibilities have most likely never been used because there just isn't a purpose (and because its tough to invent 5468578548343743 unique characters to represent unique digits..) I've worked with just about everything up to 16 for certain problems, and a few obscure higher ones..
Actually, a good place to see where obscure typically unuseful bases can be useful is in simple encryption algorithms, and alot of other looping situations actually yield results as though the programmer an obscure arbitrary base.

[Dnereb]

Actually the 1 base numeric system is very old.
Tribal counting systems were based on sticks each stick representing 1 goat.
the arabic invention of zero nor negative numbers was possible ergo a 1 based system.
Sorry future students... don't blame me but blame antopology
A zero based numeric sytem is impossible.... unless somone can explain otherwise.

[triggernum5]

Thats not exactly a base-1 system.. More like binary.. Where the 2 possibilities are goat/no goat.. Base-1 (easier to understand as base-0 maybe) so I suppose there are 2 useless bases the way we perceive math today (0 and 1 which are really the same thing I think because neither allow any logical operations)..

[bushmobile]

I'd have to agree with Dnereb, that is a base-1 system. It's nothing like binary, there is no "no-goat" possibility, i would show you the number of sticks I had - if i had three sticks then I have three goats, the absence of a stick at a particular position does not signify anything.

base-1 is like the tally system (except you don't place a diagonal line for 5) - the only way to compare numbers (beyond about ~7 - which can be judged by eye alone) is to line them up side by side and see who has the longer line.

[eyupjudd]

Wow thats a conversation to get people going isnt it!

[triggernum5]

By 'no goat' I meant 'no stick'.. I get your point.. Multi bit binary notions DO NOT apply.. I still honestly think that existence of any kind is binary (as long as you forget the common operations we typically associate with binary systems in computers).. When dealing with only one bit at a time, Either it exists, or it doesn't..

[superbovine]

The thing is, infinity is by definition defined as any arbitrary uncomprehendably large number..

infinity isn't incomprehensible large number. Its is a:

http://whatis.techtarget.com/definition/0,,sid9_gci809150,00.html
In general, infinity is the quality or state of endlessness or having no limits in terms of time, space, or other quantity.

or
http://en.wikipedia.org/wiki/Infinity
As in real analysis, in complex analysis the symbol , called "infinity", denotes an unbounded limit.

If man wasn't able to understand what infinity was we couldn't do calculus.

[triggernum5]

infinity isn't incomprehensible large number. Its is a:

http://whatis.techtarget.com/definition/0,,sid9_gci809150,00.html


or
http://en.wikipedia.org/wiki/Infinity


If man wasn't able to understand what infinity was we couldn't do calculus.

Sorry I didn't just post a link to a wiki-page that says basically exactly what I said 'in my own words'.. Believe it or not, you aren't the only guy who knows about google..
BTW, you misunderstood what you read.. I never said that infinity was not understandable.. Really, I 'understand' that I can't conceivably spend $1 billion any easier than I could spend $2 billion.. Therefore in the scope of my situation both numbers are in a way infinitely large.. Or perhaps in the future we will measure the density of a black hole.. It would still be valid to refer to its density as infinite in certain contexts where it accounts for an infinitely large portion of total mass in the system..
.. In science we use infinity to define a notion of a number huge enough that it doesn't matter if its any bigger or not..

Edit: I missed it the first time, but BushMobile's application of longer line analysis regarding the sticks representing goats thing shuts me up pretty quick on validity of this base-1 system.. Think you can come up with an application for base-0?:)

[dsheller]

.. In science we use infinity to define a notion of a number huge enough that it doesn't matter if its any bigger or not..


I don't agree with that. 6.02 * 10^23 is fairly large yet we still use it in calculations opposed to stating, "there are infinitely many molecules in <insert favorite compound>."

[triggernum5]

But Avogadro's number isn't huge in the context, or system it's typically applied to.. There are roughly 10^80 atoms in the universe we figure, so ppl thinking along those lines need a really big infinity.. I suppose the absolute smallest number usable in any equation as infinity would be the smallest number that completely overwhelms all other values currently involved to the extent that your sig. digits allow.. Computers are the only brains that really need a value tacked to it.. Infinity is a concept, and a concept can't actually be defined.. Simply understood/pondered.. Like time itself..

[superbovine]

But Avogadro's number isn't huge in the context, or system it's typically applied to.. There are roughly 10^80 atoms in the universe we figure, so ppl thinking along those lines need a really big infinity.. I suppose the absolute smallest number usable in any equation as infinity would be the smallest number that completely overwhelms all other values currently involved to the extent that your sig. digits allow.. Computers are the only brains that really need a value tacked to it.. Infinity is a concept, and a concept can't actually be defined.. Simply understood/pondered.. Like time itself..


OK, I was giving the pure math definintion of infinity, you are mixing the philosphical and math infinity together along with common english usage of the word infinity. They are all mutally exclusive. In a strict math sense infinity is strictly defined as an unbounded limited. Without this strict definition calculus would not work. There is no pondering involved in it, without infinity defined you couldn't do calculus or any math that builds upon it.


In science we use infinity to define a notion of a number huge enough that it doesn't matter if its any bigger or not..


ok bud, i'll give a example here.

what about limits?

graph this function f(n)=n/(n+1) where n is the set of positive numbers. xp powertoys and a having calculator if you want it.

the graph has an asymptote at x=1.
In calculus we would represent this by a limit. lim n->infinity n/(n+1) = 1
or the general form of lim n->infinity f(n) = L.

that means you plug in any real number between (0, infinity) and you answer will be less than 1. That is a huge deal in engineering and physics and not to mention computer science.

if scientist are working with calculation using larger than double precision libraries in c++ the limits of IEEE floating of infinity and negative infinity will be hit without using the special libararies. every wonder what happens if you multiple say an integer with a double floating point percision number with a digital wrong at 1 * 10^-200. The probably gets bigger. Can you imagine if the guys said will the error is close enough out there to infinity and doesn't matter.

[triggernum5]

Totally agree with basically everything you said.. I was making the scientific meaning of infinity because dsheller mentioned moles.. Yet another concept of infinity has nothing to do with numbers.. The infinity symbol represents a mobius strip, which simply has no beginning and no end..

BTW, the special libraries out there aren't even special enough for the huge datatyping needed in some scenarios.. Its almost always most feasible to build specific new ones due to a need for the max efficiency that are often useless for other tasks.. Thats usually supercomputer stuff though..

[Dnereb]

how about this....

I'm not that rich and you can count an infionete number of thigs i do not own
the left front tire of your car, the right front tire of your car,..,...,.., the highest place sand grane in the west sahara, the second highest placed grane in the western sahara,..,..,..,..,..,the first fish in existance, the second fish in existance,..,..,,.. a well if you specify enough it will be an infinate amount.... but now we wil count all things Bill Gates doesn't own.... infinate as well but surely less because he ownes more then I do!

BTW you can count thing in the past and futere that aren't owned by me or Bill so it would be infinate.

[triggernum5]

Dude, you're creeping me out.. I just woke up, so if I ponder infinity before atleast an hour I'll get really bad stomach cramps and drown to death in my thoughts..:)