inverse proportionality.

[dstebbins]

I'm currently taking pre-calculus and I'm learning that, when there is a constant c, then the equation of c = xy is the equation for an inversely proportional relationship of x and y. This makes sense, because if x decreases, y has to increase to keep c at its current constant rate.

 

However, is this really the only inverse proportionality equation? It seems to me that c = x + y would also be inversly proportional, since 7 = 2 + 5 as well as 7 = 6 + 1. Notice how when one goes up, the other goes down, which is really the only requirement for an inverse proportionality relationship.

 

Am I right, or am I stoned?

[timo]

x ~ y means x = c*y. So proportionality means that one value is a multiple of another. The inverse of multiplication is division. That means that inverse proportionality should be x = c/y.

 

Applying that to your question we see that you´re stoned:

From c = x*y => x = c/y.

From c = x+y => x = c - y.

 

Both connections of x and y have the property that for an increasing y, x gets smaller. But proportionality and antiproportionality are more than just this rather weak statement. In fact (anti-)proportionality is already a pretty strong statement about the connection between two values x and y. See it this way: You have reduced the huge amount of possible combinations (x,y) to one degree of freedom: c.

[dstebbins]

x ~ y means x = c*y. So proportionality means that one value is a multiple of another. The inverse of multiplication is division. That means that inverse proportionality should be x = c/y.

 

Applying that to your question we see that you´re stoned:

From c = x*y => x = c/y.

From c = x+y => x = c - y.

 

Both connections of x and y have the property that for an increasing y, x gets smaller. But proportionality and antiproportionality are more than just this rather weak statement. In fact (anti-)proportionality is already a pretty strong statement about the connection between two values x and y. See it this way: You have reduced the huge amount of possible combinations (x,y) to one degree of freedom: c.

I'm confused. If proportionality doesn't mean as x gets larger, y gets smaller, then what is the definition of proportionality?

[timo]

x being proportional to y means that x = c*y with some constant c (which should not depend on y). In words you could call it "x is a multiple of y".

 

 

EDIT: The statement below might not be correct; I am not sure how common it is to speak of proportionality if c<=0.

 

In fact, this does not even necessarily mean that x gets bigger as y does. If c is negative then in fact the opposite is true (although the magnitude of x will still increase if the magnitude of y does).

[dstebbins]

I understand this. I'm just saying there's a whole crapload of ways to write a proportionality equation. For example, a directly proportional relationship can be written as y/x = c, cx = y, cy = x, or x/y = c. An inverse relationship can be written as c = xy, c/y = x, or c/x = y. I'm just asking if we could use addition in the form of c = x + y or any other manipulation of that equation.

[timo]

No, not for proportionality. y/x=c, cx=y and x/y = 1/c is the same equation (in the sense that it describes the same relationship between x and y) - it´s just written down in different ways. c = x + y is a different equation (you cannot manipulate it into one of the equations above) and none that describes a proportionality between any of the letters involved.

[Bignose]

dstebbins, you have to remember that mathematics is defined very, very exactly. Unlike the english language where words have different shades of meaning, one word can mean different things, and different words can mean the same thing. Mathematics is defined so that when one says "proportional" that has exactly the one meaning and only the one meaning it is supposed to have.

 

So, when one says "proportional" this is defined to mean one and only one thing. As you know, proportional means x = cy. If you mean c = x + y, you have to say something else (I don't know if the additive relationship has a special name... "linear?"). That is so that confusion is kept to a minimum when using mathematics. As a farcical example, we wouldn't want + and - change meanings... they need to mean one and only one thing.

[EvoN1020v]

A good example of inverse proportionality is [math]F_{net}=ma[/math].

 

Expansion of the formula: [math]F \propto \frac{1}{m}[/math].