Proportionality And Its Consequences

[MrAl]

Hello there,


I'd like very much to hear other readers ideas about what proportionality
really is, and what exactly is their idea about what a
"constant of proportionality"
really is.

For something to think about, proportionality can be depicted even without
a constant such as like this:
y : x
which would mean that x is proportional to y.

Note there is another symbol used for this too, which looks like this:
y ∞ x
except that symbol in the center there looks like that only with an
open end (couldnt find the actual symbol on the list for this board).


Another way to represent this is to use an equation like:
y=K*x
where K is a constant. Because K is a constant it makes y proportional
to x.

Your ideas/comments?

 

[BrownOut]

y=K*x;

Do you think proportionality only refers to linear relationships?

How about y=e^x? Is that a proportional relationship?

Just asking.

 

[dknguyen]

**broken link removed**)

Seems the wiki says proportionality is linear. Personally, I always thought of "directly proportional (or just proportional)" as no more than "If one gets increases the other also gets increases ." And "inversely proportional" would be "as one increases, the other decreases". No more, no less. NOthing implied about linearity or anything like that.

To talk about the relationship more specifically than just "the trend of increasing and decerasing", I always used more explicit terms. Like linearly proportional or [linearly] proportional to the squared (I would leave out the linearily because I would be that statement doesn't make much sense if it's not linear.

I guess the wiki's definition is more consistent than my definition. It's a not very consistent if I say "proportional" doesn't necessarily imply linearity, but then say that "proportional to the squared of something" does imply linearity.

Now I have to find a new word to describe the increasing/decreasing trend between two things. Mainly because sometimes I am talking about a relationship and I know the increasing/decreasing trend, but I don't actually know whether it's linear or not and I don't want to say more than I know.

 

[Tesla23]

Hello there,


I'd like very much to hear other readers ideas about what proportionality
really is, and what exactly is their idea about what a
"constant of proportionality"
really is.

For something to think about, proportionality can be depicted even without
a constant such as like this:
y : x
which would mean that x is proportional to y.

Note there is another symbol used for this too, which looks like this:
y ∞ x
except that symbol in the center there looks like that only with an
open end (couldnt find the actual symbol on the list for this board).


Another way to represent this is to use an equation like:
y=K*x
where K is a constant. Because K is a constant it makes y proportional
to x.

Your ideas/comments?

Another semantic question like ohms law!

I was always taught that proportionality means y=K*x, and my reference books support this. The most authoritative thing I can find online without wasting too much time is Proportionality

 

[MrAl]

Hi,

There are sometimes other types of proportionality but the one i am talking
about is usually referred to as "Directly Proportional" and is sometimes written
as the attached picture shows. This is the linear relationship y=K*x yes,
but one thing i would like to point out here is that when it is written in this
way (see the picture) there is no constant shown (such as the K), but rather
just the two 'variables' x and y (or whatever they are).

What this is all about is an attempt to establish just what proportionality
means (to be more exact, direct proportionality or when something is said to
be "in direct proportion to" something else).

As noted above, the picture shows how the relationship y=K*x is sometimes
written, but very important here is to note that when written this way we dont
even have to write the K.

The reason i write this is because this relationship has far reaching consequences
in electronics and other branches of science.
This thing to think about is, when we look at it this way what does this relationship
imply (without resorting to writing the 'K')?

 

[dknguyen]

It's just one of the many shorthand symbols involved in math proofs. Why write "K" when you don't have to? especially when you can't define it's actualy value because you don't know what it is? I think you're just overthinking it.

The reason i write this is because this relationship has far reaching consequences
in electronics and other branches of science.
This thing to think about is, when we look at it this way what does this relationship
imply (without resorting to writing the 'K')?

Why bother defining it without "restorting" to writing the K? Like I said overthinking. It just means that any ratiometric change to one will incur the same ratiometric change to the other. Using K instead of english leaves no room for misinterpretation and is much shorter to write.

 

[MrAl]

It's just one of the many shorthand symbols involved in math proofs. Why write "K" when you don't have to? especially when you can't define it's actualy value because you don't know what it is? I think you're just overthinking it.


Why bother defining it without "restorting" to writing the K? Like I said overthinking. It just means that any ratiometric change to one will incur the same ratiometric change to the other. Using K instead of english leaves no room for misinterpretation and is much shorter to write.

Hi,


Thanks for the reply. I was hoping for some discussion like this.


I wanted to bring to the attention of many Electro Tech users the
significance of not resorting to using the "K" when defining or just
talking about what proportionality really means. It's not overthinking
as much as boiling off all the water in order to get to the heart of the
matter.
More to the point, if we allow an equation like y=K*x we might be
tempted to allow K to change, but when we think of proportionality
without the K as in y:x we have no K to change so it must be the
same no matter what x and y are.
Another way to say this which also helps to clarify a bit is this:
2 is to 1 as 4 is to 2. This means the first number is proportional
to the second number, same as saying y:x and again K is implied
rather than explicitly stated and so can not be changed at will.

I wanted to draw attention to this to help others understand what
it means to have proportionality. When we say y:x we are mainly
talking about two variables that have a unique relationship, and
if they were not proportional we could not say y:x or y=K*x.
In other words, if we let K vary then we do not have proportionality
anymore. It's just a little more evident when we say y:x than
when we say y=K*x because with y:x there is no K to deal with
so we must understand it from looking only at what y and x are.
For example, if i say 2:1 then we know right away that y is 2 and
x is 1 so we know that y is proportional to x, and it doesnt matter
what x is set to as long as y remains proportional:
2:1
4:2
6:3
15:7.5
All of the above have the same basic relationship, in that they are
all in proportion of 2:1.
Note that even though we allow x to change y changes in a way that
holds this proportionality.
In other words, proportionality involves not only the two numbers
(such as the 2 and the 1 in 2:1) but ALL numbers in existence
(other than zero) from -infinity to infinity. When we talk about
a proportional relationship we are talking about a relationship that
exists between countless sets of numbers, not just the two used
to show the relationship (2:1). When we say 2:1 we also mean
4:2, 6:3, etc, or more usual, when we say something like 6:3 we
also mean 2:1, or ANY two numbers that hold this unique relationship.
So this means that the concept of proportionality involves more than
just two numbers even though only two numbers are used to show
the relationship.


Once everyone understands this it will make another concept in
electronics that depends on proportionality (which i intend to reveal soon)
more clear.

If anyone else too would like to comment i would like to hear.

 

[dknguyen]

This is beginning to sound like the resistance thread.

V=I*R or V=I*R(V,I,T,t,f,...)?

But yes, if we let K vary then the term "proportional" no longer applies. We tend to write K(...) if that is the case though rather than just K. But K = X/Y which says the same thing as X:Y. Depending on interpretation and context some are less or more foolproof. Different ways of writing different things. That's all. I don't exactly see the problem with all this.

 

[BrownOut]

Why should k be constant for there to be proportionality? If my mortage payment is based on the formula:

I = R*D; where I in the interest, R is the Rate of interest and D is the debt, then the relationship is exactly the definition of proportionality, where I represents Y, D represents X and R represents K. But R isn't necessarily constant. From the start of my loan repayment, R has varied anywhere from 5.5% all the way to 8%.

 

[dknguyen]

Hmm. I suddenly have an urge to watch the movie after reading your signature.

 

[BrownOut]

Heh! Was watching it yesterday. Love a good swashbuckler!

 

[shimniok]

Why should k be constant for there to be proportionality? If my mortage payment is based on the formula:

... umm... well, only because that's how proportionality is defined mathematically?!

I = R*D; where I in the interest, R is the Rate of interest and D is the debt, then the relationship is exactly the definition of proportionality, where I represents Y, D represents X and R represents K. But R isn't necessarily constant. From the start of my loan repayment, R has varied anywhere from 5.5% all the way to 8%.

I am going to go out on a limb and say that, no, I = R*D is not "exactly the definition of proportionality." And neither is y=K*x ...

Not without specifying that K is a constant!!

That y is always a constant multiple of x is exactly the definition of proportionality.

That I is always some varying multiple of D is exactly not the definition of proportionality. It just shows that there's a relationship between I, R, and D.

When K is a constant and y=K*x, this absolutely means that the relationship between y and x is linear.

Meanwhile, y=e^x is not an example of proportionality because the ratio of y to x is not constant. Which is because the equation is not linear. You cannot say that y is proportional to x. You can say that y is related to x.

What proportionality means, simply, is that the value of one variable (x) is dependent on the other (y) and that the ratio of one variable to another is always a fixed number. That is, one variable increases (or decreases) in proportion to the increase (or decrease) in the other variable.

This isn't something mystical. It's really very, very simple.

Michael

 

[MrAl]

Hello again,


Thanks Michael. That explains it very well. That's exactly the point i want to
get across too, that when proportionality is defined as "directly proportional"
there is no flexibility allowed to change the "constant of proportionality". It
MUST remain fixed or else we loose the quality of direct proportionality.

dknguyen:
This is going to seem rudimentary to you because you already understand what
proportionality is and so y:x seems exactly the same as y=K*x to you. To
some people however, y=K*x means y=K(...)*x where K is a function of other
variables or even of x or y. So in another sense i guess i am trying to draw a
very strong distinction between the difference between y=K()*x and simply
y=K*x. I am doing this in order to establish first what "direct proportionality"
really means and how strict this kind of definition is. I believe that once this
is fully understood (by people who now believe that K() is the same as K for
whatever reason) there is another concept in electronics that can be immediately
understood. I believe you have also guessed what this is already

Before continuing, it might help to say that there are 'other' types of proportionality
too besides the kind where we define a constant of proportionality and write either
y:x or y=K*x, but the type that this discussion is limited to will be the y:x or y=K*x
kind of proportionality which is also called "direct proportionality", where two numbers
are said to be "in direct proportion to" each other.

The best statement so far has been Michaels:
"That is, one variable increases (or decreases) in proportion to the increase (or decrease) in the other variable."

and what that says is that when we deal with proportionality, we are not dealing with only two numbers set
apart from all other numbers (such as 4 and 2), but really ANY two numbers that satisfy that SAME relationship
such as 8:4, 50:25, etc. It's not about just two numbers alone. In fact, the relationship holds for all numbers
that x can take on, in that y must remain "in direct proportion to" x or else, lets face it simply, there is no
proportionality in the first place!
This is very simple.
Also, just in case we have two numbers like 4 and 2 again, then later two other numbers say 9 and 3, we can
also note that although 9 and 3 are two numbers that can also be said to have a proportional relationship they
do not have the SAME proportional relationship that the numbers 4 and 2 had. 9 and 3 are in proportion 3:1,
while 4 and 2 are in proportion 2:1. 9 and 3 are definitely NOT in proportion 2:1.

Another key point might be this:
When we find two numbers that are believed to be in direct proportion, we usually want to divide through to
find out what the proportionality constant is. For example, for the numbers 12 and 3, we want to divide them
both by 3 to get 4:1. By dividing through, we make one number equal to 1 and so the proportionality relationship
between the two numbers becomes clear. This makes it easier to compare to other sets of numbers like
120 and 30, where if we divide both of these by 30 we again get 4:1, but for the two numbers 150 and 30
we get 5:1, which is a different ratio.
This is very very important in many application areas not only electronics. Calculating gear ratios is another
very simple yet important application area. We count the teeth of the two gears and figure out what the
gear ratio is, after which we can compare to other sets of gears to see if we have the same ratio. If we do,
then the speed of the output shaft is the same for both gear sets. If the ratio is different, then the speed
is different.

Another view on direct proportionality is this:
We can define a certain proportionality by using two numbers like 4 and 2 as in 4:2, but we CAN NOT
define the CONCEPT of proportionality ITSELF with only two numbers. In order to understand what
proportionality really is in the first place, we have to look at more than two numbers. I believe the
minimum count of numbers that have to be considered would be four, but really all the numbers
on the number line should be considered including the negative ones.

Still yet another view is this:
Proportionality is not a quantity of any kind, as in 3, 6, 15, 3234, etc. It does take on the properties
of a 'direction' however, in that a given proportionality relationship between two numbers can also
be expressed as a direction angle equal to the inverse tangent of two numbers. Since this view
makes the relationship an angle, there are many many sets of numbers that satisfy this relationship.
For example, the 4:2 set again, the angle is invtan(4/2)=63.435 degrees (approximately) and the
angle for 2:1 is invtan(2/1)=63.435 degrees. Note that both sets produce the same direction angle.

 

[MrAl]

Hi again,


Ok i can drop the bomp shell now...

The above discussion was meant to talk about and understand exactly what
proportionality means, and the consequences of that kind of relationship.

Assuming we are beyond that now, i can restate Ohm's Law...

The law stating that the direct current flowing in a conductor is
directly proportional to the potential difference between its ends.

Thus, the law is not talking about random resistance, but rather a resistance
that obeys the law of proportionality.

 

[BrownOut]

You're mistating Ohm's Law.

the current I in a circuit is directly proportional to the potential difference V, and inversely proportional to the resistance R, or I = V/R.

Resistance, given by the symbol R, is not the constant of proportionality.

 

[MrAl]

You're mistating Ohm's Law.



Resistance, given by the symbol R, is not the constant of proportionality.

i=v*(1/R)

1/R can still be said to be the constant of proportionality. In other words,
i is proportional to v.

I do prefer to call R the constant of proportionality too however as in:

v=i*R

where we would then have to say that i is inversely proportional to R.

 

[BrownOut]

You can call it the Sultan of Arabia; that won't make it so. V and R are both varaibles.

 

[shimniok]

The law stating that the direct current flowing in a conductor is directly proportional to the potential difference between its ends.

For a specific conductor? Or any conductor? I guess you could say that for a given conductor, where R is fixed (constant), the current is directly proportional to the voltage for that specific situation.

But you can't (correctly) say the same when you generalize to any conductor or resistor(s) in the circuit. Do you see the distinction?

Ohm's law doesn't describe a directly proportional relationship anymore than the ideal gas law (PV=nRT) describes a directly proportional relationship between pressure and temperature. With Ohm's law, current depends on voltage and resistance. All three are variables. With the I.G.L., pressure depends on volume, temperature, and the number of moles of the gas.

When x varies proportionally to y, by definition, there can't be any other variables in the equation.

There are in Ohm's Law and the ideal gas law.

Therefore they are not equations describing direct proportionality.

Michael

 

[MrAl]

You can call it the Sultan of Arabia; that won't make it so. V and R are both varaibles.

Hello again,


It's not 'me' that is calling it anything...it's the way that law is written.
If you want to mess with Ohm's Law then you have to first mess with
proportionality and what it means when two variables are proportional.


shimniok:
Then i guess you should argue with the writers of those definitions because they all
say proportionality exists in the Ohm's Law. R is a CONSTANT. You have to know
when 'variables' are really constants and when they are not. Just because R is a
letter like v and i that doesnt automatically make it a variable in an equation.

I suppose that you would also want to say then that in the equation:
y=A*x^3+B*x^2+C*x+D
that A,B,C, and D are variables?

 

[BrownOut]

You have to know
when 'variables' are really constants and when they are not. Just because R is a
letter like v and i that doesnt automatically make it a variable in an equation.

Both R and V are variables. There isn't anything special about V that gives it some special status over R. In any real cirucit, you can vary V AND R to affect the current, which is also a variable. Constants are things like the Universal Gravitation Constant, Avogrado's Number and Boltzmann Constant, as in the previously mentioned, PV=nRT. In the case of Ohm's law, the constant of proportionality is unity.

If R was a constant, they it wouldn't be included on the little triangle that noobs use to calculate these quantities. Do you ever see anyone calculating Boltzmann's constant from pressure and temperature?