Date: Oct 24, 2012 2:27 PM
Author: Teague, Dan
Subject: RE: [ap-calculus] e and "lies my calculator tells me."

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To follow up on Doug's nice example and explanation, your students might be interested in the following experiment.

In their favorite language and on their favorite computer, write the following program:

H = 1/2
X=2/3 - H
Y = 3/5 - H
E = (X+X+X) - H
F = (Y+Y+Y+Y+Y) - H
Q = F/E
Print Q

Since the computer is working in base 2, there are five possible values for Q. For me, Q = 2, but other machines using others lengths of representations and using a processor that rounds or truncates can produce values of Q equal to 1, -1 , -2, 2, 4, and 1.5.

Since my machine gives me I know that my machine uses a number of digits that is a multiple of 4 and rounds off at the end.

If you get 4, then your machine is using a number of digits that are 3 more than a multiple of 4 and truncates at the end.

To see this, have them write out the binary representation using 5, 6, 7, or 8 digits and truncating or rounding at the end. Then just do the addition and subtraction.

H = 0.1 X = 0.001010101010... Y = 0.000110011001100...

For example, with 6 digits with rounding, we have:
H = 0.1
X = 0.001011 so X+X+X = 0.100001 and E = 0.000001
Y = 0.000110 so Y+Y+Y+Y+Y = 0.011110 and F = -0.000010
So Q = F/E = -2.


Daniel J. Teague
Department of Mathematics
NC School of Science and Mathematics
1219 Broad Street
Durham, NC 27705

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