Proportionality

[Agent Smith]

So I was on a science forum and saw an interesting question. Take the equation \(\displaystyle y = -x\).
The slope is \(\displaystyle -1\) and so as \(\displaystyle x\) increases \(\displaystyle y\) decreases and vice versa. This confused the poster, as does it me, because this is how inverse proportionality \(\displaystyle y \propto \frac{1}{x}\) is also described.

How would you to clear this matter up?

 

[Otis]

Sorry, I don't understand the confusion. Different models may share common characteristics; that doesn't mean they're acting like the same model. Do you see some incompatibility?
[imath]\;[/imath]

 

[Agent Smith]

The confusion is, are they the same or are they different?
1. \(\displaystyle y = -x\)
2. \(\displaystyle yx = 1\)

How would a mathematician respond?

 

[lev888]

The confusion is, are they the same or are they different?
1. \(\displaystyle y = -x\)
2. \(\displaystyle yx = 1\)

How would a mathematician respond?

y=-x
y=-2x

Are these the same or different?

 

[Agent Smith]

y=-x
y=-2x

Are these the same or different?

Their slopes are different; I don't see the point.

 

[lev888]

Their slopes are different; I don't see the point.

You posted 2 functions and asked whether they were the same. I posted 2 functions and asked the same question. Why don't you see the point in my question, but presumably see it in yours?

 

[Agent Smith]

You posted 2 functions and asked whether they were the same. I posted 2 functions and asked the same question. Why don't you see the point in my question, but presumably see it in yours?

Well, I assumed you were asking me to compare the 2 sets of equations. I couldn't see the connection.

In the 2 equations I posted, one is linear and the other isn't. They both can be described as \(\displaystyle x \uparrow x \to \downarrow y\); in your equations they both behave the same way as my equations but the slopes are different. My previous reply should've been this; apologies.

 

[lev888]

Well, I assumed you were asking me to compare the 2 sets of equations. I couldn't see the connection.

In the 2 equations I posted, one is linear and the other isn't. They both can be described as \(\displaystyle x \uparrow x \to \downarrow y\); in your equations they both behave the same way as my equations but the slopes are different. My previous reply should've been this; apologies.

So, how can your 2 functions be the same if they are not even of the same type?

 

[Dr.Peterson]

So I was on a science forum and saw an interesting question. Take the equation \(\displaystyle y = -x\).
The slope is \(\displaystyle -1\) and so as \(\displaystyle x\) increases \(\displaystyle y\) decreases and vice versa. This confused the poster, as does it me, because this is how inverse proportionality \(\displaystyle y \propto \frac{1}{x}\) is also described.

How would you to clear this matter up?

The description you refer to is not the definition, just a description that applies when the constant of proportionality is positive (as it usually is), but not in this case.

As I have said elsewhere, "I suspect that any places where you have seen the "one increases when the other does" idea 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." Also, "It is not at all uncommon for a description that applies to the typical case to be mistaken for the definition of a concept; this happens especially at the elementary level, where special cases such as negative numbers are ignored."

 

[Perdurat]

So I was on a science forum and saw an interesting question. Take the equation \(\displaystyle y = -x\).
The slope is \(\displaystyle -1\) and so as \(\displaystyle x\) increases \(\displaystyle y\) decreases and vice versa. This confused the poster, as does it me, because this is how inverse proportionality \(\displaystyle y \propto \frac{1}{x}\) is also described.

How would you to clear this matter up?

the general equation for a line:
y=ax+b
a: the slope or angle(with respect to the x-axis) is given by (tan-1(a))
b: displaces the line up or down
special cases:
a=0->a parallel line with x-axis
b=0->a line through point 0,0
a=1,b=0-> "the mirror line" of the 1/x function

just play around a bit in a graphic calculator (try: y=ax+b, x=ay+b, 1/x, 1/y for different parameters a,b: -1,0,1,n )
and observe what it does to the graph
"inverse proportionality", i would stick to add,substract,divide,multiply

 

[Agent Smith]

So, how can your 2 functions be the same if they are not even of the same type?

Yes, but the same description applies to both.

 

[Agent Smith]

The description you refer to is not the definition, just a description that applies when the constant of proportionality is positive (as it usually is), but not in this case.

As I have said elsewhere, "I suspect that any places where you have seen the "one increases when the other does" idea 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." Also, "It is not at all uncommon for a description that applies to the typical case to be mistaken for the definition of a concept; this happens especially at the elementary level, where special cases such as negative numbers are ignored."

That makes sense. Proportionality is reserved for positive numbers.

What if we do this \(\displaystyle y = -x + 100\). Within the range [0, 100), we have positive values for y

 

[Agent Smith]

the general equation for a line:
y=ax+b
a: the slope or angle(with respect to the x-axis) is given by (tan-1(a))
b: displaces the line up or down
special cases:
a=0->a parallel line with x-axis
b=0->a line through point 0,0
a=1,b=0-> "the mirror line" of the 1/x function

just play around a bit in a graphic calculator (try: y=ax+b, x=ay+b, 1/x, 1/y for different parameters a,b: -1,0,1,n )
and observe what it does to the graph
"inverse proportionality", i would stick to add,substract,divide,multiply

Gracias

 

[Dr.Peterson]

That makes sense. Proportionality is reserved for positive numbers.

No, that's the opposite of what I said.

The constant of proportionality, and/or the input and output, can be negative, though that is rare; when they are, the usual idea that increasing input for a direct proportion causes an increase in the output is false. This is what is "reserved for positive numbers".

It was originally assumed that the numbers were positive, because nothing else was known at the time!

 

[Agent Smith]

No, that's the opposite of what I said.

The constant of proportionality, and/or the input and output, can be negative, though that is rare; when they are, the usual idea that increasing input for a direct proportion causes an increase in the output is false. This is what is "reserved for positive numbers".

It was originally assumed that the numbers were positive, because nothing else was known at the time!

So a more general definition of proportionality is required then, oui?
This example, \(\displaystyle y = -x\) plays havoc with the intuition of proportionality as increase is met with increase
Maybe we should define direct proportionality as ratio (positive/negative) betwen y and x remaining constant.

 

[Dr.Peterson]

So a more general definition of proportionality is required then, oui?
This example, \(\displaystyle y = -x\) plays havoc with the intuition of proportionality as increase is met with increase
Maybe we should define direct proportionality as ratio (positive/negative) between y and x remaining constant.

Yes, that's how it is defined! Did you never look up a definition??

Proportionality (mathematics) - Wikipedia

en.wikipedia.org

In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) ... . Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality.​

Of course, you do have to ignore elementary introductions, which often give the faulty definition you've used, as I indicated:

Directly Proportional and Inversely Proportional

Directly proportional: as one amount increases another amount increases at the same rate.

www.mathsisfun.com

Directly proportional: as one amount increases, another amount increases at the same rate. ...​

Inversely Proportional: when one value decreases at the same rate that the other increases.​

Usually this site is pretty good. This page seems to be using the term "rate" poorly, in addition to assuming positives, as is common.

 

[Agent Smith]

Yes, that's how it is defined! Did you never look up a definition??

Proportionality (mathematics) - Wikipedia

en.wikipedia.org

In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) ... . Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality.​

Of course, you do have to ignore elementary introductions, which often give the faulty definition you've used, as I indicated:

Directly Proportional and Inversely Proportional

Directly proportional: as one amount increases another amount increases at the same rate.

www.mathsisfun.com

Directly proportional: as one amount increases, another amount increases at the same rate. ...​

Inversely Proportional: when one value decreases at the same rate that the other increases.​

Usually this site is pretty good. This page seems to be using the term "rate" poorly, in addition to assuming positives, as is common.

You mean to say then, per our agreed upon definition, that this \(\displaystyle y = -x\) expresses a direct proportionality relationship? Perhaps we should exclude \(\displaystyle y = -ax\) from our set of relationships (this was from a science forum and I haven't encountered negatives, except maybe as the charge on an electron, in physics formulae).

 

[lev888]

Yes, but the same description applies to both.

I can be described as a man. Does it mean I am the same as any other man???

 

[Agent Smith]

I can be described as a man. Does it mean I am the same as any other man???

You mean they belong to the same category but are nonidentical? I get that for your examples \(\displaystyle y = -x\) and \(\displaystyle y = -2x\), but what about \(\displaystyle y = -x\) and \(\displaystyle yx = 1\) in my question?

 

[Cubist]

You mean they belong to the same category but are nonidentical? I get that for your examples \(\displaystyle y = -x\) and \(\displaystyle y = -2x\), but what about \(\displaystyle y = -x\) and \(\displaystyle yx = 1\) in my question?

yx=1 has a discontinuity at x=0...