Re: Calculating e
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Re: Calculating e
In article <v01530500b1203426132f@[206.216.138.82]>,
Lisa Ledwith <lledw@GA.K12.PA.US> wrote:
>
> I wanted to illustrate on the calculator how (1+1/A)^A goes towards e as A
> becomes large so I wrote the following program. Everything is fine until I
> get to 10 x 10^13 because the value for B is greater than e. I think this
> is probably a rounding error but I don't know what is really happening. I
> understand what is happening when A is 10x10^14 --- at least I am assuming
> that the overflow in the calculator allows 1/A to be undefined so from that
> point on the calculator thinks it is rasing 1 to a power and thus yeilds 1.
> I added the lines in the program to dispay the value of A and then played
> around a bit. When I increase A's value by multiples of 5 or 2, similar
> things happened. If I change the multiplier to 5 or 3, I eventually get
> numbers on either side of e. Also, if I change my seed, A, to 9, I get
> weird results. Can anyone explain what is happening here? At the very
> least, I have another example of how the calculator can 'lie'.
>
> 10 stor A
> Lbl 1
> (1+1/A)^A sto B
> DISP B
> DISP A
> A*10 sto A
> PAUSE
> GOTO 1
>
> Lisa Ledwith
> Germantown Academy
> Fort Washington, PA
> Mathematics Teacher
>
> lledw@ga.k12.pa.us
>
I am speculating, but I would like to know if you were using a TI-92 in exact
mode, NEITHER approx NOR auto mode. If so, I think the symbolic bug bit you.
Here's what I think might have happened:
The calculation is parsed as follows
1+1e13 is NOT interpreted as 1.0000000000001 in EXACT mode.
Rather, it is interpreted as a NON-reducible:
10000000000001/10000000000000
Now since it is stored as a fraction, when you raise it to an enormous power,
the symbolic algorithm attempts to raise 100000000001 to the 10000000000001-th
power. At which point the calculation terminates with an overflow. If it
were to continue, it would also raise the denominator to a power and then,
a second overflow condition would occur. I have not run the program, but
based on observing the calculation and various intermediate steps, this is
probably what happens. (Please don't tell me that this was NOT a TI-92
problem, because, then I have to make up another theory.)
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