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December 14

Numbers

Why are numbers written right to left when everything else is written left to right? --71.153.45.118 (talk) 01:39, 14 December 2009 (UTC)[reply]

One thousand, four hundred and thirty seven = 1437. Seems left to right to me. Do you not read and think about numbers starting with the largest (i.e. "most significant") digit first? Dragons flight (talk) 02:23, 14 December 2009 (UTC)[reply]

(Edit Conflict) I think that most people write a number from left to right (for instance, if you write 568, you will naturally start at the 5 and end at the 8). However, you are right that (some) people write the units digit of a number only after writing each and every other digit. Perhaps that is just the way the convention was adopted. However, most of mathematics is not concerned with how one denotes a number; in fact, the whole of mathematics would run smoothly if we swapped the names of numbers.

Some mathematicians, particularly algebraists, believe that function composition should be written from left to right. Other mathematicians, however, believe that function composition should be written from right to left. Irrespective of this, many mathematicians agree that notational conventions should not lie at the real heart of mathematics; one can worry about them up to a certain point, but then mathematical ideas become of greater value. --PS T 02:26, 14 December 2009 (UTC)[reply]

Hmm... neither Hindu-Arabic numeral system nor Positional notation seem to discuss the historical development of our big-endian convention. What I've learned (but cannot source right now) is that the European convention was taken over from Arabic practice which also had the most significant digit on the left; but as Arabic is written right-to-left, this actually amounted to little-endian notation in their context.

Not to mention a discussion of the confusion brought about by computer representation of Arabic text containing decimal numbers; here (at least in Unicode) numbers are stored in big-endian (Western) order in memory, which must be reversed when rendering the text for display or printing... –Henning Makholm (talk) 02:38, 14 December 2009 (UTC)[reply]

There is some (unsourced) discussion about endianness in Eastern Arabic numerals, which probably ought to be moved to some more general location. 02:42, 14 December 2009 (UTC)

Strangely that article says some European languages traditionally read from the small end like the Indians, however the Egyptians, Babylonians Greeks an Romans all started with the big end just like western languages today as far as I'm aware. Dmcq (talk) 10:41, 14 December 2009 (UTC)[reply]

Well, modern German pronounces two-digit numbers starting with 21 with the units digit first. 23 = drei und zwanzig, etc. Anyone who's ever heard Nena's song 99 Luftballons has heard this (99 = neun und neunzig). Michael Hardy (talk) 22:23, 14 December 2009 (UTC)[reply]

That used to be pretty common in English too as in four and twenty blackbirds, but is there evidence of people writing numbers that way as opposed to writing speech in longhand, for instance in accounts or anything like that? Dmcq (talk) 23:17, 14 December 2009 (UTC)[reply]

Another thing. Were the original numbers from India written left to right? Weren't their languages also written left to right so the Arabs would have reversed the numbers when taking them direct and then they got reverse back to the original in Europe? What order was the original in? Dmcq (talk) 11:00, 14 December 2009 (UTC)[reply]

Roman numerals, and Greek numerals which preceded them[1], were written in a big-endian order, the same as modern western numbers, so the convention pre-dates the arrival of Arabic numerals in the west. --Pleasantman (talk) 16:12, 14 December 2009 (UTC)[reply]

As a left-handed writer I find numbers a welcome opportunity to pull rather than push, so invariably I write them R to L, finding no problem in leaving just the right amount of space no matter how long they are. If I'm transcribing a number from elsewhere I'll read/write it in reverse order a digit at a time, if I'm writing something like 2009 from memory I'll reverse the whole thing mentally then re-reverse while writing. ——86.164.72.171 (talk) 15:31, 15 December 2009 (UTC)[reply]

Cardinal Ordinal

I'm confused about Cardinals and Ordinals in Mathish.

In English , we have "Cardinal : one, two, three; Ordinal : first, second, third."

In Mathish they mean something else Ordinal number and Cardinal number.

I don't understand the Mathish meaning - Could someone explain please. -- SGBailey (talk) 16:31, 14 December 2009 (UTC)[reply]

Cardinal numbers measure the size of a set: how many things there are in it. Ordinal numbers measure the position of a term in sequence. For finite numbers, there is (as long as you adopt the logician's and computer scientist's convention that a sequence starts with the zeroth term, not the first) a direct correspondence between cardinals (zero, one, two, ...) and ordinals (zeroth, first, second, ...), so the distinction is not very important mathematically and is not often made. Specifically, the correspondence is that the nth (ordinal) term in a sequence has n (cardinal) terms coming before it.

In the infinite case, this breaks down and the two notions become very different. Ordinals continue in the obvious way: zeroth, first, second, third, ..., infinityeth (strictly ωth), infinity-plus-oneth, infinity-plus-twoth, ..., but if we work try to use our correspondence, we find that the infinityeth term and the infinity-plus-oneth term both have the same (cardinal) number of terms before them (this cardinal number is called aleph-null). In fact, it turns out that each infinite cardinal number has a great many ordinal numbers corresponding to it (in contrast to the finite case, where there is always exactly one). This leads to the theories of infinite cardinal and ordinal numbers, such as cardinal arithmetic and ordinal arithmetic, being very different from one another. Algebraist 16:51, 14 December 2009 (UTC)[reply]

So, in the finite case, the English and Mathish meanings are the same. Cardinal numbers include zero (for the empty set) while ordinal numbers do not. The 'ordinal' zeroth is confusing and distasteful, both in English and Mathish. Bo Jacoby (talk) 21:02, 14 December 2009 (UTC)[reply]

Starting your indices with zero makes a lot of sense in maths. For example, a general expression for a degree n polynomial will have n+1 constants, it makes far more sense to call them a₀ to a_n than a₁ to a_n+1 since then the index of the constant and the degree of the term can be equal. There are lots of other cases where such notation is also clearer. --Tango (talk) 22:36, 14 December 2009 (UTC)[reply]

The distinction in ordinary language arguably has no referent — it's just a convention that makes things easier to parse, like agreement between verb and pronoun. (This has actually been a bit of a problem, in figuring out how to disambiguate the search term ordinal; the current solution is a bit awkward, but no one has proposed one that isn't.) On the other hand, in mathematics, there's a real distinction, at least in the infinite case. I don't know that there's any real distinction between finite ordinals and finite cardinals. --Trovatore (talk) 21:12, 14 December 2009 (UTC)[reply]

Do the words exist in ordinary language? I know them as technical mathematical terms and technical linguistic terms (and they do agree, really - cardinals are used for counting, ordinals are used for ordering and there is a 1-to-1 correspondence in the finite case [which is the only case linguistics covers]). --Tango (talk) 22:36, 14 December 2009 (UTC)[reply]

The members of a tupple are usually indexed by ordinal numbers like this: (x₁,x₂,x₃). When using other indexes than the ordinal numbers one must distinguish the ordinal number from the index. The constant term a₀ of the polynomial Σa_i xⁱ is indexed by 0, but that does not make it the 'zeroth' term. Bo Jacoby (talk) 09:56, 15 December 2009 (UTC).[reply]

Do you really consider 0 to not be an ordinal? Does anyone agree with you in this perversity? Algebraist 13:07, 15 December 2009 (UTC)[reply]

While I don't have sufficient experience to justify my opinion, I don't consider 0 to be an ordinal. I do this so that the mathematical and English uses of the words "first", "second", etc. will agree in the finite case. If we accept zero as an ordinal, then the first prime number is 3 -- which is inconsistent with the English-language sense of the word, in which the first prime number is 2. Eric. 131.215.159.11 (talk) 17:19, 15 December 2009 (UTC)[reply]

Thanks -- SGBailey (talk) 11:52, 15 December 2009 (UTC)[reply]

Yes, a child counting to three is supposed to say "one, two, three", rather than "zero, one, two". Bo Jacoby (talk) 13:34, 15 December 2009 (UTC).[reply]

Indistinguishable balls in distinguishable boxes

Resolved

What is the quickest route for answering the following sort of question, and what is the proper nomenclature surrounding it: In how many ways can fifteen indistinguishable objects be placed in six labeled boxes (any number of which may remain empty, though the general problems are identical with an offset)? This is something I believe I should have learned to do generally, but my present situation is that I would tackle it on an ad hoc basis. It's a little different from partitions.Julzes (talk) 22:44, 14 December 2009 (UTC)[reply]

I just wrote a 1-line program to just count them in the specific case mentioned, but what I got looks smaller than I would have expected: 15504. Just chime in if it happens to be wrong.Julzes (talk) 22:57, 14 December 2009 (UTC)[reply]

It sounds exactly like partitions to me. The answer should be 20 Choose 5, which is 15,504, so your computer program seems to be correct. The easiest way I know to think about it is that you have 20 objects in a row, 15 of them are balls and 5 of them are dividers and you want to decide where to put the dividers. --Tango (talk) 23:38, 14 December 2009 (UTC)[reply]

Yeah, that's old hat for me. Must be a bad day for this sort of thinking. Thanks.Julzes (talk) 23:59, 14 December 2009 (UTC)[reply]

It's not partitions though; that's actually a much more complicated subject.Julzes (talk) 00:01, 15 December 2009 (UTC)[reply]

Why is it not a partition problem (other than the existence of empty parts, which I think are sometimes excluded from partitions)? There is far more to partitions that a simple problem like this, of course, but that doesn't stop this being a question about partitions. --Tango (talk) 02:45, 15 December 2009 (UTC)[reply]

Conventionally, in number theory, a partition of an integer is an unordered sum - if the boxes were indistinguishable, this would be a partition problem. As the boxes are distinguishable, it is actually a composition problem; to be even more precise, as some of the terms may be zero, the problem involves counting weak compositions. Gandalf61 (talk) 13:01, 15 December 2009 (UTC)[reply]

Well, that's good. I actually learned something even though I've been doing this sort of problem just fine for over thirty years until yesterday.Julzes (talk) 13:21, 15 December 2009 (UTC)[reply]