Deriving the Celsius-to-Fahrenheit Conversion Formula

Date: 12/03/2001 at 23:33:00
From: Calixta
Subject: Temperature conversions

What is 27 degrees above freezing in Fahrenheit?

Thanks.

Date: 12/04/2001 at 01:40:05
From: Doctor Jeremiah
Subject: Re: Temperature conversions

Hi Calixta,

It depends on what temperature scale you are converting from. Since 
you don't say, I am going to assume that you meant to ask what 27 
degrees Celsius is when converted to Farenheit. If that's not what you 
meant to ask, then let me know.

Let's pretend that there is a straight-line (linear) relation between 
the two temperature scales (and it turns out that in fact there is). 
If that is so, then we need two "points" to find the equation for
that line.

Freezing in Celsius is 0 degrees, and in Fahrenheit it is 32.
Boiling in Celsius is 100 degrees, and in Fahrenheit it is 212.

   Farenheit  Celsius
  --------------------
      32         0
     212       100

The equation for a line that converts Celsius to Farenheit is:

  F = mC + b

where C is the Celsius temperature
      m is the "slope" of the line
      F is the Fahrenheit temperature
  and b is the "intercept" of the line

So if we use our two "points" to fill in this equation we get:

   Farenheit  Celsius      equation
  --------------------   ------------
      32         0       32 = 0m + b
     212       100      212 = 100m + b

Now we have two equations and two unknown values (m and b) so we can 
solve for these values and make an equation that converts Celsius to 
Farenheit:

The two equations:

   32 = 0m + b
  212 = 100m + b

The first thing to do is to make both equations equal to the same 
thing. The easy way to do that is to get the b terms alone on one side 
of the equation:

          32 = b
  212 - 100m = b

Notice that we already know the value of b. And since both equations 
are equal to b, they are also equal to each other:

  32 = b = 212 - 100m
  32 = 212 - 100m

And we can solve for m like this:

           32 = 212 - 100m
     32 - 212 = 212 - 100m - 212
         -180 = -100m
  -180 / -100 = -100m / -100
  (-180/-100) = (-100/-100)m
          9/5 = m

If we put the values for m and b into C = mF + b, we can come up with 
an equation that converts Celsius to Fahrenheit:

  F = mC + b   <==   m=9/5 , 32=b
  F = (9/5)C + 32

So this is the conversion function. We want to convert 27 degrees 
Celsius to Fahrenheit, so we plug 27 into this equation for C and
solve for F:

  F = (9/5)C + 32   <==   C=27
  F = (9/5)27 + 32
  F = 80.6

So 27 degrees Celsius is 80.6 degrees Fahrenheit.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/