Wikipedia:Reference desk/Archives/Mathematics/2010 April 3

= April 3 =

General variable
Is there any sort of standardized "general variable"? What I mean is a variable that stands for any variable. For example, if I wanted to state that a change in something is the final quantity minus the initial quantity, what variable should I use with that? Of course, i could use x, but I don't wouldn't want to imply that it's just x that works like this. Is there a general variable that you could use that would indicate to the reader that this holds true for any sort variable? Yakeyglee (talk) 05:59, 3 April 2010 (UTC)


 * Don't quite undestand the question but an old joke occurs to me:
 * Teacher Now suppose the number of sheep is x...
 * Student: Yes, I see that, but what happens if the number of sheep is not x?
 * :) Dmcq (talk) 10:31, 3 April 2010 (UTC)


 * The way I interpret the Q is that they want to know what variable would be appropriate to represent an integer, a real number, a complex number, or any other. I'm more familiar with computer programming than mathematics, and in some languages, the convention is that variables starting with I-N are integers, and the rest are reals.  So, what similar conventions are there in mathematics ? StuRat (talk) 18:54, 3 April 2010 (UTC)


 * That's pretty much the convention in maths as well. Also the letters at the beginning of the alphabet tend to be used for constants and the letters at the end for variables. The letter z is often used for a complex number, x can be used for practically any unknown but a real is more common. It is not all that standardized though. The letters p, q and r for instance are often integers but could be rationals or group elements and other times might be reals. Dmcq (talk) 19:17, 3 April 2010 (UTC)
 * Also, in most cases, n and m will be natural numbers, or at least integers. The same applies to k and $$\ell$$ but those are somewhat less used, generally only after you've already used up n and m.  Depending on context, i may either be a natural-number (or at least integer) variable, or a constant (the imaginary unit).  Some barbarians also use j for the imaginary unit; otherwise it's probably a natural-number-or-integer variable.  e may be the base of the natural logarithm, or the identity of a group, or the number of edges in a graph. --Trovatore (talk) 19:42, 3 April 2010 (UTC)
 * (Hmm, it seems that StuRat already said much of that, though in less detail. The perils of reading just the last contribution....) --Trovatore (talk) 19:53, 3 April 2010 (UTC)
 * If x is the quantity, you would often write the change in x as Δx ("delta x"). 66.127.52.47 (talk) 19:56, 3 April 2010 (UTC)
 * I was going more along the lines of a letter that stands for variables in general. Yakeyglee (talk) 05:28, 4 April 2010 (UTC)
 * I don't see why you need such a thing. If you are trying to say that "a change in something is the final quantity minus the initial quantity," just use x or y or whatever, and specify that the variable can represent any quantity. Most mathematical statements and definitions work like this. For example, the formula for the area of a circle is often given as $$A=\pi r^2$$, which is true even for circles in which the radius is called something other than r. —Bkell (talk) 05:38, 4 April 2010 (UTC)

Induction Question
In a question, I was required to prove a given statement for all integers. I did this via induction and it all seemed to come out well but I am now stuck on which integer n I take as my 'first' case. Does the fact that I had to prove this for all integers mean that induction was not appropriate? If not, where should I start? I can give more details of the question if that would shed any light on the matter. Thanks. asyndeton  talk  16:53, 3 April 2010 (UTC)
 * The induction step here consists of showing that from n follows $$n+1$$ and that from n follows $$n-1$$. You can choose any base you want, it is likely that $$n=0$$ is the simplest. -- Meni Rosenfeld (talk) 17:01, 3 April 2010 (UTC)


 * Depending on what the statement is, it may also be possible to first prove the statement for all nonnegative integers (using n = 0 as a base case), and then prove that if it is true for a nonnegative integer n it is also true for −n. —Bkell (talk) 17:12, 3 April 2010 (UTC)

Proving points on same line...
I have points A(1471/264, -1568/33), B(113/132, 3136/33), C(4,0)

These are the circumcenter, orthocenter and centroid of a triangle, respectively. In theory, they should all be on the Euler Line of this triangle. What is the best way to prove that these three points form a line?

Perfect Proposal  18:52, 3 April 2010 (UTC)


 * Calculate the vectors AB and AC and show they are parallel. --Tango (talk) 18:55, 3 April 2010 (UTC)


 * In other words, show that they have the same slope. Also, any pair of vectors would work, it doesn't have to be those two. StuRat (talk) 18:57, 3 April 2010 (UTC)


 * ... and in other words again, this is equivalent to checking that (cy-by)/(cx-bx) = (by-ay)/(bx-ax) ... and yes, A, B & C are collinear. In Wikipedia, "Collinear points" redirects to "Line (geometry)" where the term collinear is not mentioned.  I've added a brief paragraph to the line article but it needs formatting and possibly expanding.   D b f i r s   19:44, 3 April 2010 (UTC)
 * For a problem such as this perhaps the best way is to compute the trilinear coordinates of each point and show they satisfy a linear equation.--RDBury (talk) 09:01, 4 April 2010 (UTC)
 * There is another proof here.--RDBury (talk) 09:09, 4 April 2010 (UTC)