March 6, 2010 at 7:07 pm #33429March 7, 2010 at 4:55 pm #33430
Be careful in believing some of the math on that website!!!
Some of it is WRONG, and so this puts his trustworthiness into question.
His information about the “TRUE VALUE OF PI” etc. is utter nonsense.
Just in case anyone was/is lured in,
and may actually believe his claim,
let me now provide a “MATHEMATICAL SMACK-DOWN” and debunk it!!!
Note: This argument can be followed by anyone with a basic
background in calculus that we teach to college freshman!
Even if you don’t have such a background, you should still be
able to get the gist by skimming . . .
For a circle of FIXED RADIUS, you can inscribed a regular n-gon
inside it and circumscribe another regular n-gon around it.
(I.E. Imagine a stop sign neatly fitting inside a circle and
another surrounding the circle)
This gives the relationships:
#1: Perim of inner n-gon <= Circum of circle <= Perim of outer n-gon
#2: Area of inner n-gon <= Area of circle <= Area of outer n-gon
Using ONLY the relationships of triangles, you can use simple geometry
and classical triangle trigonometry to turn #1 and #2 into formulas:
2 * r * n * sin( 180deg / n )
<= circumference of the circle
<= 2 * r * n * tan( 180deg / n )
n * r^2 * sin( 180deg / n ) * cos ( 180deg / n )
<= Area of the circle
<= n * r^2 * tan ( 180deg / n )
By using the Pythagorean Theorem, you can show that cos ( 180deg / n )
goes to the number 1 in the limit as n goes to infinity.
From this fact, and the fact that tan x = (sin x) / (cos x)
you can show that both n * sin ( 180deg / n) and n * tan ( 180deg / n)
have the same limit number L as n goes to infinity.
By the Sandwich Theorem of Limits from Calculus (again doesn't use
pi at all), taking limits in #1 and #2 gives:
Circumference of the Circle = 2 * r * L
Area of the Circle = r^2 * L
where L again is this common limit number.
Note: It is this step where
Jain's comments about "infinitesimal area under
the curve that does not disappear" is complete BS and
false mumbo-jumbo. The argument above using the Sandwich
Theorem proves that.
At any rate, to continue:
By using similar triangles, a simple geometry proof demonstrates that the
limit L is independent of the size of the circle.
This fixed constant which is the ratio of the circumference of the
circle to 2 * r (the diameter) can be used as the DEFINITION OF PI.
Circumference of ANY Circle = 2 * pi * r
Area of ANY Circle = pi * r^2
where the specific VALUE of pi is yet to be determined.
Using the basic formulas for Circumference and Area, one can
derive Circle Trigonometry using the unspecified value pi as
already defined above.
One can then derive special Inverse Trigonometric Identities
These functions can be evaluated numerically to as high as
degree of accuracy as desired using infinite series representations
of the Inverse Tangent Function (Calculus II material)
Such computations have been done by computers to an accuracy of
a trillion digits.
THERE IS NOTHING INACCURATE WHATSOEVER WITH THE ESTABLISHED VALUE OF PI.
OF COURSE, this is just ONE example of a false mathematical claim
on that website. So I don't know that I'd trust much of what he says.
Bottom line: Be VERY skeptical of people who claim to be
math "experts" that don't even have bachelor's degrees in mathematics.
StevenMarch 7, 2010 at 10:02 pm #33432
ok, warning heeded.
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