Re: pi as unit real number replacing unity

On Sep 1, 1:57 am, Mark Nudelman
<ma...@xxxxxxxxxxxxxxxxxxxxx> wrote:
On 8/31/2007 12:02 PM, Narasimham wrote:

On Aug 31, 9:40 am, Narasimham
<mathm...@xxxxxxxxxxx> wrote:
If hypothetically the yardstick of unit real
numbers counting is
changed to include (0, pi, 2pi, 3pi,... ) with
its A.P. common
difference of the irrational number pi as basis
instead of the present
(0,1,2,3,... ) with common difference rational
unity 1, what
simplifications or changes could be effected in
all of Real Analysis?

Regards,
Narasimham

Asked this as pi is _ratio_ of the most natural
figure,the circle.
By definition it is non-dimensional and might
have been a natural
number choice for a unit of rotation rather than
translation. In
this system,sin(0) = sin(1) = 0, cos(1)= -1

If I'm understanding your system, cos(1) = -1/pi,
not -1.

--Mark

Yes thanks,my error. Also so many equations in
electrostatic/magnetic
theory and elsewhere with uncomfortable pi related
coefficients could
get tidied up.


Possibly. There is a distinction between abstract
counting, however, which employs dimensionless real
integers--and measure, which represents the
difference in size among counted quantities. The
former lives in ordered relations, the same counting
line in which you substitute pi for the integer 1.
Why pi, though? Why not sqrt(2)? In fact, any positive
non-zero number will work, won't it?

Now, you are taking the constant pi as a geometric ratio
and then saying that "by definition it is non-
dimensional." Do you realize that this is the same
insight by which we get to complex analysis? That is,
the derivation of the Euler Identity e^ix=cosx + isinx,
when x=pi, is in logarithmic terms ln(-1)=ipi.

Engineers find it easier to work in the complex plane
because they are not concerned with counting, but
with measure. The advantage of the two-dimensional
analysis is that more "room" to calculate, in which
pi IS a "unit of rotation rather than translation"
(rotation through the complex plane)allows one to
get real results in desired continuous functions while
ignoring imaginary results. One would
need to know more about how you propose to make real
analysis simpler than complex analysis for measure
problems. What advantage?

By the way, Hans Schwerdtfeger in Geometry of Complex
Numbers (Dover,1979)has a nice section on the analytic
geometry of circles.

Tom
.

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