Re: Pi

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Re: Pi

Rhombus wrote:

> One simple explanation is :
>
> PI is the circumference of a circle divided by its diameter.
> But the circumference cannot be measured or calculated exactly.
> Mathematicians get an approximation of this circumference, and hence of PI,
> by calculating the circumference of a regular polygone with, say 1000 sides.
> This can be done exactly with a little trigonometry.

There is another way, using the Talor Polynominal of atan(1). Theoretically,
atan(1)=pi/4, so if you could get a good approximation of atan(1), and then
multiply it by 4, you'd also find pi.
With the Taylor Polynominal of atan(x)=x - x^3/3 + x^5/5 -x^7/7 + x^9/9 + ...
you could calculate pi/4 as accurate as you'd want, simply by taking some more
terms into the sum.

People who own a TI-92, can easily calculate pi theirselves (I don't know
wether
it's possible with other TI's, it depends on wether they can calculate the
Taylor-polynominal. You could ofcourse enter the polynominal by hand...)

First you calculate the Taylor polynominal upto let's say the 21st grade.

    taylor(tan-1(x),x,21)

Then you calculate it for x=1 (using WITH for example). You multiply it by 4,
and there's your own approximation of pi!

   (in this case, it would be 11757173 / 14549535 * 4 = 3.23232)

That's not very close, but the approximation becomes better by taking more and
more terms...

  (up to the 49th term, the result would be 3.18158, that's a little better)

Perhaps this is not the easiest way to calculate pi, but it's certainly a
reliable one!

Within Taylor's theory, it's even possible to calculate how big the deviation
in
your result versus the exact value can be, but that's something for your math's
teacher!
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