In the Laboratory
Celsius to Fahrenheit and Vice Versa — Quick, Exact, and Neat S. C. Dutta Roy Department of Electrical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi–110016
C. Hester, in a recent issue of this Journal (1), proposes the following approximation for a quick conversion of Celsius (C) to Fahrenheit (F) temperatures: F ≅ 2 C + 30
(1)
As we know, the exact relationship is F = (9/5 ) C + 32
(2)
A quick, exact, and neat method of conversion would be to use the following alternative version of eq 2: F = 2 (C – 0.lC + 16)
(3)
We present here a simple method of reverse conversion—that is, from F to C. The exact formula is, from eq 2, C = (5 /9) (F – 32)
(4)
which can be written as C = (10/ 9) [(F/2) – 16 ] (5) = (1.111 … ) [(F/2) –16] = [(F/2) – 16] + 0.1 [(F/2) – 16] + 0.01 [(F/2) – 16] + … Stated in words, this algorithm amounts to the following steps. 1. Subtract 16 from F/2 and call the result C1 (first approximation).
2. Add C1 /10 to obtain C2 (second approximation). 3. Add C1/100 to C2 to obtain C3 (third approximation).
Continue this until the desired accuracy has been obtained. The reader may try a few numerical examples to get convinced that this procedure is much simpler than it appears to be. It should also be clear that one need not go beyond the second or third step, because there is a distinct pattern of how the successive steps improve the result. We show this by three examples in the table below. F
C1
C2
C3
C4
C5
122
45
49.5
49.95
49.995
49.9995
124
46
50.6
51.06
51.106
51.1106
166
67
73.7
74.37
74.437
74.4437
It is interesting to note that the error in C1 is exactly 10%, that in C2 is exactly 1%, that in C3 is exactly 0.1% and so on. Literature Cited 1. Hester, C. J. Chem. Educ. 1995, 72, 1026.
Vol. 74 No. 10 October 1997 • Journal of Chemical Education
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