Wikipedia:Reference desk/Archives/Mathematics/2013 June 12

= June 12 =

What is the difference (if any) between a matrix and a table?
Practically speaking I guess refering to "tables" in a paper about linear algebra would be confusing. Also refering to matricies in a school text book would also be confusing. But if you were to get very semantic about the whole thing, what would be the difference? 80.254.147.164 (talk) 09:42, 12 June 2013 (UTC)


 * Semantically (in the proper sense of the word, meaning "of or relating to meaning"), a matrix (in the linear algebra sense) is a coordinate representation of a linear transformation. A "table" might have such a meaning, but in general does not. --Trovatore (talk) 09:51, 12 June 2013 (UTC)


 * A matrix is merely a two-dimensional grid of numbers. Matrices can be used to describe coordinate representations of a linear transformations. Jason Quinn (talk) 04:31, 16 June 2013 (UTC)


 * A table of numbers can be used as a matrix and sometimes is used that way. A matrix is usually thought of as a complete unit which can be involved in operations, whereas a table is something where one looks up values and may add a row or column to when you find out something new. Dmcq (talk) 09:54, 12 June 2013 (UTC)


 * Table (information) is a general way to arrange and display data in rows and columns. There are no rules about the structure or content of a table. For example, cells may span multiple rows or columns and have content of different type or be empty. Nothing is implied about operations you can perform on something just because it's written in table form. And a table is often viewed as a graphical object including use of images, colors, different fonts and text alignment, and so on. Matrix (mathematics) is an abstract mathematical object with fixed rules. A matrix is usually written in a form resembling a table but that's just a representation of the matrix. Mixing the words table and matrix would definitely be bad in most contexts. PrimeHunter (talk) 10:58, 12 June 2013 (UTC)


 * From the perspective of object oriented programming, a table and matrix are different objects. There are different operations that can be performed on them.  Matrices of the appropriate dimensions, for instance, can be multiplied whereas tables cannot.  The implementations also differ.  Tables are often represented as associative arrays, matrices can be represented differently depending on whether they are generic, sparse, symmetric, etc.  The choice of representation is closely tied to efficiency for the kinds of operations being performed.   Sławomir Biały  (talk) 11:56, 12 June 2013 (UTC)

No Two Real Numbers Adjacent
This is something I've thought of before and have finally decided to ask for expert advice on the matter. Given any two real numbers a and b, with a<b, assume that they are 'adjacent', in that there is no real number c such that a<c<b. However, if you take the arithmetic mean of a and b, you find such a c. So this means that there are no two real numbers adjacent to each other. Is this true? What are the implications of this? Is it meaningless to talk about adjacency of the reals? Thanks. meromorphic  [talk to me]  11:19, 12 June 2013 (UTC)
 * You are right: It is meaningless to talk about adjacency of the reals. For exactly the reason you said. It is also meaningless to talk about adjacency of the rationals, and of the decimal fractions. Bo Jacoby (talk) 11:52, 12 June 2013 (UTC).
 * This property is important enough that there is actually a name for it: the standard ordering on the real numbers is a dense order. « Aaron Rotenberg « Talk « 13:23, 12 June 2013 (UTC)
 * Yeah this is the wrong place to talk about two numbers ever getting adjacent if you keep halving the distance between them but find "for all practical purposes" at ;-). Dmcq (talk) 13:26, 12 June 2013 (UTC)
 * The name of this meaninglessness is that the reals are a dense set; it gets a bit technical when saying related things like "the rationals are dense in the reals" (vs. the reals dense in themselves). --Tardis (talk) 13:52, 12 June 2013 (UTC)
 * I have to disagree with one aspect of what the other commentators have said. No, it's not meaningless at all, to talk about two reals being adjacent.  No two reals are adjacent; that's a meaningful statement, and to make it meaningful, then the concept of two reals being adjacent has to be meaningful.  It just never actually happens. --Trovatore (talk) 17:11, 12 June 2013 (UTC)
 * What is the meaning of talking about events that never happen? Bo Jacoby (talk) 07:25, 13 June 2013 (UTC).
 * What's the meaning of saying that X never happens, if you can't assign a meaning to X in the first place? --Trovatore (talk) 10:12, 13 June 2013 (UTC)
 * From meaningsful expressions like adjacent integers and reals you may construct meaningless expressions like adjacent reals. Bo Jacoby (talk) 11:05, 13 June 2013 (UTC).


 * Integers do not enter the picture in any way. The notion of adjacency is perfectly meaningful for elements of any partially ordered set, so it is in particular meaningful for reals. If you go to a grocery shop and ask for tomatoes, which as it happens they don’t have in stock, the shop assistant will tell you “sorry, we haven’t got any”, not “I don’t understand what you are asking for”.—Emil J. 14:18, 13 June 2013 (UTC)
 * If a proof in a math book starts like this: “let A and B be two adjacent real numbers”, I will think: “I don’t understand what you are asking for” rather than: “sorry, I haven’t got any”. Bo Jacoby (talk) 20:19, 13 June 2013 (UTC).
 * Here’s one:
 * Lemma: No two real numbers are adjacent.
 * Proof: Let A and B be two adjacent real numbers. Put C = (A + B)/2, and assume without loss of generality that A < B. Then A < C < B, hence A and B are not adjacent after all, a contradiction. QED
 * You claim not to understand this proof, but it is in fact meaningful (and valid).—Emil J. 13:21, 14 June 2013 (UTC)


 * You just proved that the idea of adjacent real numbers leads to a contradiction, which means that it is meaningless to talk about adjacency of the reals. QED. The OP's question is answered in the affirmative, and the nit-picking is not helpful at all. Bo Jacoby (talk) 16:43, 14 June 2013 (UTC).
 * No, proving that the existence of something with property P leads to a contradiction does not mean that property P is meaningless. This is not nitpicking.  This is essential to clear thought.  If P were meaningless, then it would be meaningless (rather than true) to say that there is nothing with property P.  --Trovatore (talk) 20:33, 14 June 2013 (UTC)
 * The question was not if the property is meaningless. The questions was if it is meaningless to talk about it. Which it is. Bo Jacoby (talk) 21:04, 14 June 2013 (UTC).
 * No, if a property is meaningful, then it is ipso facto meaningful to talk about it. --Trovatore (talk) 21:06, 14 June 2013 (UTC)


 * I agree with Trovatore here. The very fact that we're trying to test the OP's proposition - and he himself easily disproved it - means that it is testable, which means it is conceptually possible, which means it is meaningful.  I'd say the same thing about the propositions that 1 + 1 = 2, or 1 + 1 = 9, or 1 + 1 = 378.  Only one of them is mathematically correct, but all are testable and all have meaning.  --   Jack of Oz   [Talk]  21:13, 14 June 2013 (UTC)
 * If it is meaningful for you guys to talk about adjacency of reals, then go ahead and talk about it. It is meaningless to me. Bo Jacoby (talk) 07:48, 15 June 2013 (UTC).


 * You could call "a pair of adjacent real numbers" an empty name - a description which is well-formed but refers to something that doesn't exist. « Aaron Rotenberg « Talk « 17:25, 14 June 2013 (UTC)


 * I thought this was the property that the real numbers are dense meaning that if A<>B then there is a number (thus an infinite number of numbers) between A and B. So if A<>B then (A+B)/2 is between A and B and it is easy to show that if A<B then A < (A+B)/2 < B.  Note that the real number and the rational numbers are dense but the integers are not dense.  RJFJR (talk) 19:14, 12 June 2013 (UTC)
 * Right, dense, but unfortunately the word dense means a number of different things in mathematics, some closely related, some not so closely. The most directly relevant link in this case is to dense order.  The dense set link given by Tardis is not quite as on-point, but not entirely unrelated either (for example, if you delete any point from the reals, what remains is dense in the reals as a whole, and the fact that the reals form a dense order can be used to prove that).
 * Exactly what "dense in itself" means I'm not quite sure &mdash; literally any topological space is dense in itself. However the phrase is certainly known in the literature (Cantor used it, if I'm not mistaken).  The only purely topological meaning I can think of is the one I alluded to above (having no isolated point), but that applies to, say, the union of the closed intervals [0,1] and [2,3], which are not a dense order. --Trovatore (talk) 20:49, 12 June 2013 (UTC)
 * We have the article dense-in-itself. « Aaron Rotenberg « Talk « 02:16, 13 June 2013 (UTC)