Date: Mon, 27 Apr 1998 21:29:08 +0000 From: "Jose Gengler" <jose.gengler@usa.net> To: cp@opus.hpl.hp.com Message-Id: <aabcdefg1420$foo@default> Subject: Mathematical Models about C vs F
Hi!
I am impressed (possitively) in regard to the amount and variety of
response (private and on the list) that C vs F generated. In this
message I want to complement that first message with some general
information.
I am realy certain that the way I have deduced the formula is
somewhat tedious, but it responds to the detailed formal way it is
obtained. But most of all it is related with GENERAL GRAPHICAL
MODELS. You see, many variables are related to each other in many
different ways. When this relation may be represented in a graphical
way that is sufficiently simple to be represented mathematicaly, we
have got the path to obtain a formula.
But my intention was not only to share how the conversion formula is
obtained. My true intention was to share the process by which DIRECT
PROPORTIONALIY between two variables is translated into a general
mathematical formula for those two variables. The conversion of C to
F is a PARTICULAR CASE of that GENERIC DIRECT PROPORTIONALITY
RELATION.
What are the requests for two variables to be in direct proportion?
It is necesary for this relation to be true, that CHANGES IN ONE OF
THE VARIABLES SHOULD PRODUCE PROPORTIONAL CHANGES IN THE OTHER
VARIABLE AND IN THE SAME DIRECTION. If one of the variables grows,
the other variable ALLWAYS grows in the same proportion. If one of
the variables decreases, the other variable ALLWAYS decreases in the
same proportion. An example of direct proportionality is C vs F.
How do I know that two variables are directly proportional?
For direct proportion to be true you need the following condition to
be ALLWAYS true:
(y/x)=K
Where y and x are the variables you want to relate, and K is a
constant. The greater the value of K, the grater the AMOUNT OF CHANGE
IN ONE VARIABLE when a given change occurs in the other variable.
So, if this condition is to be met, and if you see that two variables
change allways in the same direction, but you aren't sure if the
change is proportional, you simply do the following to prove this
condition: you take a few points (pairs of experimental values: for
example: when x1 then y1; when x2 then y2; when x3 then y3; etc),
idealy the most separated on the graphical curve as possibble, so
that with your sapling you cover as much graphical curve as you can
(for example, in nature there are several variables that are
proportional only in a given range, out of which they behave somehow
differently; if your sampling ONLY covers the proportional range, you
may ERRONEOUSLY THIK that they are ALLWAYS proportional). If you now
take each pair of values, and you divide each value of the variable
that goes in the vertical axis (y1,y2,y3, etc.) by their correspondig
respective values of the other variable that goes on the horizontal
axis of the graph (x1,x2,x3,etc), in each division, the result SHOULD
ALLWAYS BE EQUAL TO K. In experimental situations, some slight
variation of K is allowed to tolerate some experimental variation,
but the theoretical result should allways be K.
>From the condition:
(y/x)=K
The following variation may be derived:
y = Kx
In this case the straight line intersects point (0,0), that is, the
point whete the horizontal and vertical axis intersect. Now, if you
want to move the graphical straight line up or down, you add or
substract from this expression a given number that in this case will
be represented by (u):
y = Kx + u
Now this is VERY similar to the GENERIC DIRECT PROPORTIONALITY
FORMULA that I mentioned in the first message on F vs C:
y = mx + u
Here you may easily note that (m = K) and as we said, (m) represents
the SLOPE of the straight graphical line, and it is EQUAL to K, the
constant you got as a result when you tested direct proportionality
with your scattered experimental pairs of values. Now if a line has a
grater slope (m), given variations of one variable will provoke
grater variations in the other variable. As m and K are the same, the
slope is the reason for what I said before, that the grater the value
of K, the grater the proportional change between the two variables.
Direct proportionality relates A LOT of different conditions to each
other in nature. If you detect ANY condition that may be directly
proportional, first THINK if the two variables chang in the same
direction AND ALSO proportionaly. An example of this besides the C vs
F relation is the amount of distance covered by a car that travels
with constant speed, plotted against time. If you THINK direct
proportionality exists, test it using the condition (y/x)=K and
finaly you can follow the steps detailed in my first C vs F message
to generate a generic formula that relates these two variables
mathematicaly!
This may be true at least in a given range.
Another mathematical model is INVERSE PPROPORTIONALITY. These
variables are proportional, but change IN THE OPPOSITE DIRECTION.
When one of them grows, the other decreases, and vice versa. An
example of this would be to plot the amount of gas in the gas tank of
the same car traveling at constant speed, against the distance
covered. That is, the grater the travelled distance the LESS gas in
the gas tank, and I asume that if speed is constant then the change
in the quantity of gas will be proportional (I am not strictly sure
if this is true becouse I am not an expert mechanic; if I am lieing,
please tell me).
How do I know if two variables are inversely proportional?
The generic condition that should be met for all pairs of
experimental values of two variables that are inversely proportional
to each other is:
x*y = K
In this case the result of the MULTIPLICATION (and not the division)
of each pair of values (x1,x2,x3,etc and y1,y2,y3,etc. respectively)
shoud have the same result (K) in each case.
The graphical representation will be a very open "U" shaped curve
that has its "turning curve" close to the intersection between the
vertical and horizontal axis intersect (point 0,0). The two
"branches" of this very open "U" try to approach each axis (that is
one of the "branches" is progressively more and more close to the
vertical axis and the other "branch" is progressively more and more
close to the horizontal axis). All the inverse proportionality curve
is contained in the quadrant of tha graph where both x and y values
are all positive (that is the upper right quadrant). So, if you
drawed the graphical representation of inverse proportionality
correctly, you should notice that this curve NEVER INTERSECTS
NEIGHTHER THE VERTICAL OR THE HORIZONTAL AXIS. If you followed the
instructions correctly, you should have now the graph. If not, simply
plot a lot of values of y against (1/y) and you should get a typical
inverse proportionality curve (looks great using Excel).
How can I get the formula?
If you directly represent in a graph the values of y and x when they
are inversely related and propotional, the graph you get is rather
complex as the above description suggests. But there is a
mathematical trik we can use:
>From this formula:
y*x = K
We can get:
y = K/x
That is:
y = K*(1/x)
If you compare this to the DIRECT PROPORTIONALITY FORMYULA you may
understand that when the relation is INVERSELY PROPORTIONAL, an you
plot y against (1/x) then YOU WILL GET A STRAIGHT LINE!
Now, we can use an AUXILIARY LETTER (w) for a while:
w = 1/x
So we have:
If y = K*(1/x) then:
y = K*w
This is similar to the direct proportionality curve! The only
difference is that as you see, the values of x are plotted 1/x (in
stead of x as was the case before). From here we may do an analogy
between this formula and the direct proportionality formula like
this:
Direct proportionality formula:
y = m*x + u where m = K and (y/x) = K
Inverse proportionality formula:
y = m*w + u where m = K, w = (1/x) and y*x =K
So from this point:
y = m*w + u
we can go ahead and follow the steps for the calculation of the
formula (detailed in the first message of C vs F). When you plot your
experimental points to obtain m, remember to plot y against (1/x) and
NOT against (x).
Finaly, when you calculated m and u, you convert w into its original
representation, namely (1/x). This way, the generic inverse
proportionality formula will be:
y = m*(1/x) + u
Your calculated m and u values will addapt this generic formula to
your particular case.
This extensive discussion on direct and inverse proportionality is
given becouse a lot of variables in nature are related in one of
these two ways, and the math required to obtain extremely exact
formulas is not very complex, but I addmit it is a little tedious. I
think if you master these two mathematical models you will be able to
solve A LOT of different situations.
>From an intuitive point of view, many people think that almost always
two variables change in the same direction the are directly
proportional and when they are changeing inversely that means they
are inversely proportional. Close, but it is not ALLWAYS true. You
see, there are a lot of circumstances where variables are related but
NOT PROPORTIONALY. When proportionality is lost, that usualy means
that mathematics involved will be far more complex. Here I will only
mention a few models not to be confused with proportionality.
For instance the growth of plant cells in ideal tissue culture
conditions is EXPONENTIAL. That means that if we plot the number of
cells against time we will get a curve that initialy has a small
slope (small changes becouse there are only few cells) and as time
passes, the slope of the curve becomes progressievely grater (becouse
there are a lot of cells multiplying).
The generic EXPONENTIAL formula is:
y = e exp(x)
where e is the the natural logarithm base and "exp" means exponent
(becouse here I don't have superscripts or subscripts). Never mind
the formula, only pay attention that since the slope of this curve
progressively grows, proportionality is lost, and if conditions are
mantained and enough time passes, enormous quantities of cells in
very short intervals of time will be produced, becouse if a lot of
time has passed, the slope will be VERY big.
When the food becomes insufficient, the curve will change from an
exponentialy growing population to a progressively stable population,
or even a decreasing population. Part of this process may be
represented by a LOGARITHMIC relation:
y = log (x)
Both exponential and logarithmic relations may be confused with
direct proportionalyty if the approach is exclusively intuitive. But
there are lots of mathematical models, and THE FIRST STEP IN SCIENCE
IS ALLWAYS INTUITION!
There are also other models, for instance quadratic, but it is not my
purpose to cover them here.
I know this is a bit out of subject in relation to CPs, but I think
there are many contexts (including CPs) where this generic knowledge
may be applied. Please be free to use what I wrote as you may. I t
may be incorporated into any website; in fact in that case I would
feel honored.
I sincerely hope this helps. Best regards,
()()()()()
()* || () Jose Gengler
()= []=()
() || () jose.gengler@usa.net
()()()()()
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