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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Erin,

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.

Dan

Daniel J. Teague

Department of Mathematics

NC School of Science and Mathematics

1219 Broad Street

Durham, NC 27705

teague@ncssm.edu

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