Can Constant of Variation Be Negative?
Date: 01/22/2007 at 22:03:17 From: Chris Subject: Definition of directly proportional - can k be negative? In almost all textbooks, "directly proportional" is defined by saying that a is directly proportional to b if and only if a = kb for some constant k. That's perfectly sensible, but taking the definition literally, it would seem to imply that any k will do, even negatives. However, in every example that I have seen to illustrate the concept, the term "directly proportional" is always applied to the relationship between two positive quantities or two negative quantities--never between a positive quantity and a negative quantity. Further compounding the issue is that several sites I have visited try to explain the concept of direct variation by saying that as one quantity increases, so does the other, which implicitly assumes that k > 0. Yet despite all this, I have yet to encounter an explicit statement that the constant of proportionality must be positive. Please clarify the situation for me: in the mathematical community, is it considered accurate to claim that a is directly proportional to b when the constant of proportionality is negative? Or is there even a consensus on that usage?
Date: 01/22/2007 at 22:47:15 From: Doctor Peterson Subject: Re: Definition of directly proportional - can k be negative? Hi, Chris. In my experience, proportionality does not require the constant to be positive. The relation y = -2x is a direct proportion. I suspect that any places where you have seen the "one increases when the other does" either are in contexts where only positive numbers make sense, or accidentally made that supposition because it is so common. Certainly when proportion was first recognized, negative numbers were not even imagined, so much of what is said about it assumes that same context, and most examples likewise would involve positive numbers. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 01/23/2007 at 04:46:57 From: Chris Subject: Thank you (Definition of directly proportional - can k be negative?) Ah, thank you very much. That was extremely fast and helpful.
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