Date: Sun, 19 Apr 1998 18:36:59 +0000 From: "Jose Gengler" <jose.gengler@usa.net> To: cp@opus.hpl.hp.com Message-Id: <aabcdefg1313$foo@default> Subject: Celcius to Farenheight and vice versa
Hi!
I've got lots of feedback on my question and now I feel it's my turn.
Several people who wrote to me didn't know the formula of conversion
from Farenheight to Celcius and vice versa. It's not a very "nice"
looking formula, that's why it's so difficult to memorize. But the
process to deduce it is more easy to remember if you like a little
math. In this message I include the formulas of conversion but also I
share the mathematical process to deduce them
For those of you who don't like math here are the formulas. Try to
memorize them :-)
F=3D ((9*C)/5) + 32
C=3D(5*(F-32))/9
Or expressed in a more understandable way:
F=3D(9/5)*C + 32
C=3D(5/9)*(F-32)
F=3Dtemperature in Farenheight
C=3Dtemperature in Celsius
The "fancy" way I wrote the formulas is due to my difficulty
representing fractions here. If I would be able to write 9/5 and 5/9
more clearly I would not have to use so many parenthesis.
Now for the mathematical deduction.
To obtain the formula, we should know two things:
-We should know that temperature measured in Celsius (C) and
temperature measured in Farenheight (F) are directly proportional.
This means that if you take a lot of temperatures in Celsius and
find out the respective Farenheight temperature scale values, you can
then make a graphical representation of values in C versus values in
F; if F and C are directly proportional, you will always get a
STRAIGHT LINE (or close to a straight line) as a graphical
representation of theese values. In short, to obtain the formula, we
will calculate the SLOPE of the line in the graph, and also we will
calculate the point where tis line INTERSECTS THE VERTICAL AXIS OF
THE GRAPH. Knowing the slope and this point of intersection we have
got our formula.
- The second thing we have to know, is at least two temperatures
measured both in C and in F. This part of our query is experimental.
Given a temperature, lets say the temperature in your room, we
measure it both with a C thermometer and a F thermometer. Then we
register these values, and then we put both thermometers in the
refrigerator (any other temperature different from the first measured
temperature) and then register the values measured by both
thermometers. The accuracy of our formula, then will basicaly depend
on the accuaracy of our thermometers.
If C and F are directly proportional that means that we can represent
the relation between them using the following generic formula:
y=3Dmx + u
Where:
y is the quantity on the vertical axis of our graph
x is the quantity on the horizontal axis of our graph
m is the slope of the line we get as a graphical representation
u is the intersection of this line with the vertical axis.
If we put the F values on the vertical axis and the C values on the
horizontal axis of the graph, then we get the following particular
formula:
F=3DmC + u
This is very similar to Einsteins' formula but it's not the same :-)
So now we have to calculate m and u to get whole formula. To be able
to do so, now we take our experimental measurements which we got
using our C and F thermometers.
Here is what I got (I addmit I cheated and extracted the values from
a text; I didn't use thermometers this time; so if you are a real
scientist, you wouldn't belive me and you would do the experiment
yourself to find out if what I say is true):
I found out that:
- When the temterature is 20C then the temperature is 68F
- When the temperature is 30C then the temerature is 86F
Now you can get a squared paper and do the graph, putting the Celsius
values on the horizontal axis and the Farenheight values on the
vertical axis. Then you can represent the obtained values, and you
should get two points, which correspondto the two pairs of values
obtained. Since F and C are directly proportional you can draw a
straight line that goes from one of these points to the other. Also
you may prolongue infinitely this line in both directions. You will
note two things:
- That this line has a slope which we now ignore (but not for long)
- That this line intersects the vertical axis (where F values are
represented) in a value that is positive, which we for now also
ignore.
The obtained experimental values may now be gathered in a table that
is more pleasing to the eye, here represented in "glorious
asciivision" (term originaly used by Peter Cole):
C | F
-----------------
20 | 68
30 | 86
Knowing this, we can now calculate the slope of the line, using the
following formula (this formula has to do with a triangle you get in
the graphical representation, where the angle of the slope is alpha,
and then m equals cos(alpha); since I can't represent this graph
here, just trust me and use the formula):
m=3D DC / DF
Where:
DC is the greatest C value (C2) minus the lowest C value (C1)
DF is the greatest F value (F2) minus the lowest F value (F1)
Note: I chose here to write DC and DF for "delta"C and "delta"F which
is a more standard way of representing results obtained from a
substraction. Since I can't write the greek letter "delta", I chose
to replace it for a capital D.
That is:
DC =3D C2 - C1
DF =3D F2 - F1
Then:
DC =3D 30 - 20 =3D 10
CF =3D 86 - 68 =3D 18
Back to our table:
C | F
-----------------
20 | 68
30 | 86
DC=3D10 |DF=3D18
So now we can obtain m:
m =3D DF / DC =3D 18/10 =3D 9/5
Its time to take a glance to our original formula, and see our
progress:
F =3D mC + u
Then:
F=3D (9/5)*C + u
We are close. We only should calculate u and we get our formula. To
this regard, we may use any of the pairs of experimental values we
already know.
For instance let's take "when C is 20 then F is 68". It doesn't
matter if we use the other pair: "when C is 30 then F is 86". In both
cases, the calculated value of u will be the same. But we will use
the first pair of values this time.
In our formula that has the slope already calcualted, we replace the
values of F and C, using the selected pair of values:
68 =3D (9/5)*20 + u 68 =3D 9*4 + u 68 =3D 36 + u u =3D 68 - 36
u =3D 32
=A1Ready!
Now we can write down the complete formula:
F =3D (9/5)*C + u
So the complete formula is:
F =3D (9/5)*C + 32
The subject of this message sais "vice versa"; so we have to keep our
promise. The formula for calculating C from a known F value may be
obtained using simple algebra:
F =3D (9/5)*C + 32 F - 32 =3D (9/5)*C 9*C =3D 5 *(F - 32)
C =3D (5/9)*(F - 32)
Now all our promises are fullfilled! I can sleep well this night! :-)
Final notes:
1.- In experimental science things are a little different. First we
usualy get a lot of experimental values of two things that we thik
are related to each other (in this case F and C). Then we plot the
experimental values on a graph, looking at the shape of all these
obtained points. Finaly we ask ourselves if we know about any shape
(for instance some kind of a line, that not always is traight), that
may be represented by a known generic formula. Once we can fit a
theoric shape to the whole plot of experimental values with the best
aproximation we are able to get, we go ahead and get the formula. In
this case, we knew F and C are best related to eachoter by a formula
that represents direct proportionality.
2.- I wrote this extensive message in as easy terminology as I can.
My purpose is to attract those of you that think that find this too
complicated. I would very much like to read your feedback since as I
like to teach, I would like to know if I succeded or not with this
purpose.
3.- Feel free to distribute this message. I would feel very honored if
it is incorporated into any of the CP FAQ or WEB pages.
I sincerely hope this helps.
Best regards,
()()()()()
()* || () Jose Gengler
()=3D []=3D()
() || () jose.gengler@usa.net
()()()()()
This archive was generated by hypermail 2b30 : Tue Jan 02 2001 - 17:31:31 PST