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

Teacher Community Forum at
and post messages there.

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

To search the list archives for previous posts go to