Talk:Subtraction

From Wikipedia, the free encyclopedia

	This article is within the scope of WikiProject Mathematics.
	Mathematics grading:	Start Class	Top Importance
big diference between this and Addition A vital article.

[edit] Problems?

This article needs to be fixed. The way it's written a and c are points while b is a distance. This is not a proper geometric analogy to the way subtraction is carried out, and the original writer is subtracting two different classes of objects.

Perhaps you've heard it mentioned that you can't add or subtract apples and oranges.

1) Start with 3 apples

2) Take away 2 oranges from those 3 apples

3) One shouldn't expect to have 1 apple left, as subtraction in this way just doesn't make sense.

Points are quite different than distances, and just as in the case with apples and oranges, one should not expect to be able to subtract distances from points. Instead a, b, and c should all be distances from the origin, a point who's numerical value is zero. In this way the distance to c minus the distance to a is the distance b.

Could somebody who posesses both the tools to generate decent images as well as a strong math background fix this article? -User:AlfredR

I'm responsible for the ambiguity in the article. I was adding and subtracting vectors but I did not make it clear. When I wrote the article, people were discussing the need for a simple explanation of subtraction for those with little or no math background. My best attempt was to use two position vectors (a,c) and a displacement vector (b) to illustrate addition and subtraction on a number line. I hid the details since those who wanted an explanation of subtraction would not appreciate them.

For example, if vector a = (1, 1), b = (2, 0), c = (3, 1) then

a + b = c

c − b = a

We need to determine a simple (but obviously rigorous) subtraction explanation or decide that a simple explanation is not needed. The Addition page does not offer a simple explanation so we could ignore simple explanations on the Subtraction page as well.

I removed the "Basic subtraction" section that I wrote since it is ambiguous.

jaredwf 02:59, 27 October 2005 (UTC)

AlfredR is misinformed, and I'm going to restore jaredwf's Basic subtraction section. I'm also going to give a lengthy argument here on the talk page, because the good name of mathematics itself is at stake, and becuase I have nothing better to do.

Geometric vectors (in our case, signed distances) are equivalence classes of "differences of points"; they are exactly the correct objects to add to, or subtract from, points. We need not summon the chimeras that are "position vectors". In some sense, it is more fundamental to subtract a vector from a point than from another vector.

On a more general note, the apples-and-oranges analogy is worse than wrong. Teenagers don't even blink if you subtract the number 2 from the letter x. No one complains if I multiply a matrix with a column vector or an algebraic number with a spinor field. Half the fun of geometric algebra is adding quantities with different dimensions, and hang the taboos. Some of the most interesting mathematics arises when fundamentally different objects interact and when familiar concepts are revealed to tolerate a whole lot more ambiguity than they let you know about in school. Moral: mathematicians are clever. If you think something is meaningless, they will give it meaning.

(By the way, if we make the usual assumption that apples and oranges are unrelated, then 3 apples minus 2 oranges is an element in the free abelian group over apples and oranges. If, for some application, we later find it useful to equate apples and oranges, we can pass to the quotient group by this equivalence through the canonical projection and, yes Virginia, we will have one apple left.)

Now that I'm done ranting, I should comment that the section in question is a little unencyclopedic in tone. It should be condensed or rewritten, but not deleted. Melchoir 10:34, 27 November 2005 (UTC)

Your ranting aside, I stand by my previous statement. Your notion of "differences of points" requires that "subtraction of points" is already defined in a manner consistent with the formation of a vector space in which each point labels a vector. With this in mind, it is an obvious tautology to say that these are "exactly" the correct objects to add to or subtract from "points" because you've already assumed that "points' behave like vectors under subtraction.

Now since we're doing vector like addition and subtraction the obvious geometric interpretation is to treat each point like a geometric vector. A mixed interpretation is unclear without further information on how to pass between or identify one space with the other. for instance, a point + a vector might be intuitive, but what's intuition tell you about a vector plus a point? (assuming you know nothing about commutativity - just looking at a picture).

There's no reason to, geometrically, expect it to be anything in particular. for instance you may think it places a point at the end of a vector, but where should that vector start? why should one starting point be better than any other?

Thus my original point, that the geometric interpretation of subtraction employed in this article is not proper, stands. I did not claim, however, that it was incorrect. Once we build the appropriate, trivial, isomorphism we can identify the spaces and go about abusing notation however you see fit.

AlfredR 21:37, 3 October 2006 (UTC)