Hi Roger,
I read your online mathematics work at
members.ispwest.com/r-logan/fullbook.html , and I have some questions and
comments relating to it. Any further comments are appreciated.
Excerpts from your page will be bracketed by ***'s.
***
, and sometimes unexpected, as in the determination of the probability of
an occurrence.
***
If you are familiar with how pi arises in some probability distributions
(switching to polar coordinates to evaluate integrals, for example), it
isn't too mysterious.
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When two infinite sequences,
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Where is a plot of the sequences Pn and Gn in the rectangular coordinate
system?
And by a plot do you mean a paired plot, like of the points (Pn,Gn)?
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, a mean proportional relationship, as described above,
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Above this part, an example was given with the proportion being between
the bases and the altitude. Now you have the proportion being between the bases and the hypotenuse, and with infinite sequences! This might be O.K., it just isn't what is "described above".
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, but is also a perfect square, a rational fraction having integers for
both numerator and denominator.
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bm is not an integer. From a little lower, where you give the
relationships among the numbers,
b/m = pi, and b = sqrt((pi^2*phi^2)/(pi^2-phi^2)), so you can solve for m,
and get that bm = roughly 1.1342...
Likewise, 1+m^2 is not an integer. 1+m^2 is roughly 1.36102...
Unfortunately, in your largest "right" triangle (if it is indeed supposed
to be a right triangle), a^2+b^2 does not equal c^2. ie, OC^2+OB^2 does
not equal BC^2, ie. 1.8876...^2 + 2.308...^2 does not equal 3.665...^2. In your smaller triangles the equality holds.
Hi again Roger,
I was incorrect about this part. Your big triangle certainly is right. I goofeed. I wrote 1.8876...^2 + 2.308...^2, when I should have written 1.8876...^2 + pi^2.
OC = pi, but you have OB = b = 3.56... ? Your figure is not to scale then.
You probably made a typo, that is, you meant to type OB = b^2 = 3.56...,
so b = 1.887... as claimed earlier on on your page.
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The definition of a composite number is an integer which is the product of
two other integers greater than 1. This paper clearly indicates that the
fraction bm/(1+m^2) is a perfect square and every perfect square is a
rational fraction with integers for both numerator and denominator and
since bm/(1+m^2) = phi^2/pi, then, by the rules of mathematics, phi^2, pi,
bm, and 1+m^2 are all integers when put in this form.
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bm, 1+m^2, pi, phi^2 are not integers. If you have X/Y = a perfect square,
ie. the square root of X/Y is an whole number, it does not necessarily
follow that X and Y are whole numbers. ie. X/Y = 49, sqrt(X/Y) = 7, but X
could = 2582.3 and Y = 52.7.
Though, if they are postive integers as you claim, are they odd or even?
You know, to be mathematically consistent.
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It should be obvious to all readers that any number raised to the power of
pi*sqrt(-1) will equal -1.
***
That is not obvious, because it is not true. One of many counter examples
is 2^(pi*sqrt(-1)), which is roughly -.5702 + .8214*sqrt(-1).
