0002 -- lebesgue -- Attachm # 3 -- Response to Post Number ... - Nrich

typoweheeElectronics - Devices

Nov 8, 2013 (3 years and 9 months ago)

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1

of
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Calculations based on


FO = .9999999999 and


r = 1



FO = .9999999999

; i.e. 10 digits of 9
's

to the right of the decimal point.


r = 1


A=

(r**2 + FO**2)**1/2 =
1.4142135623377397097419194062514e
-
5


B =
A

/ FO =
1.4142135624791610659898355128504e
-
5


C

=
ArcTan
{
B
}

=
8.1028468454814783497100949580609e
-
4


D =
C

/ 180

=
4.5015815808230435276167194211449e
-
6


E

=

pi * r**2 *
D
=
1.414213562384880161821729681326e
-
5


Therefore,


E
-

A =
4.7140452079810275074595111990254e
-
16

; i.e. a


positive


quantity.


Admittedly, the
ArcTan

value here is not less than 1/10**6 (10 to the minus 6
-
th power),
however, this is the best I can do using my handheld calculator.

But even here, one can
appreciate the greatening affect the
contributing
factor

pi


has in raising t
he magnitude of E over
that of A.

This should be taken into account when
forming a
rough estimat
e
.


Insofar as your suspicion
go
es
,

lebesque,

you will have to offer a
specific

counter
-
example that
shows S will be
negative

for when ArcTan has a value t
hat is less than 10 to the minus 6
-
th
power.


Perhaps, we can decip
her an answer to your suspicion
, if you
or a friend
have access to a
university
computer
which uses a high precision computational program that accepts inputted
values of, say, r = 1 and F
O = .9999999999999999


(to 16 decimal places or more).

That ought to do it.


I once wrote


fou
r


computational
programs (in PL1 language) that expand

enormously on
the

inherent limitations of the IBM
mainframe

computer to give
exact

results on the operat
ions of
addition, subtraction, multiplication and division.
Needless to say,
IBM quickly
became
enchanted

with

the
se

programs
.


Unfortunately, I presently
do not have

access to one
,

because

I
am
no longer in
volved with

th
at

field
.


Other than trying to
il
legitimize

my equation by nibbling away at it
s

structural edges, as you
have attempted here,
I can't see any

other way of doing it except for delving head
-
on into
the
matter
, whereby either disproving it or vindicating it
through logic
.