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.