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Re: [Phys-L] irrationals



There are lots of examples, Alex. Pi, e, and the golden ratio are among
the most famous.

Your argument is about the constructability of a representation or a
physical object (like a circle) showing the ratio precisely. We could, for
example, calculate the billionth digit of Pi. But any physical experiment
to demonstrate the ratio to that precision would be impossible.

Bob, you might try searching for readings on countable and uncountable
infinities.
Here's a simple start at counting the rational numbers:
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
Here's something a little more accademic:
https://natureofmathematics.wordpress.com/lecture-notes/cantor/
The latter includes a nice proof that the real numbers are a superset of
the rational numbers.

Just to mess with your brain some more:
Finite polynomials have roots.
Infinite polynomials (series) do not.

Happy mathing,
Paul


On Mon, Sep 11, 2017 at 1:38 PM, Alex. F. Burr via Phys-l <
phys-l@mail.phys-l.org> wrote:


IRRATIONAL NUMBERS
Interesting. But why do you want a rational explanation of an irrational
concept?
-
I do not think I can really solve your (real) problem, but I can say that
one will have a difficulty finding an actual real example of an irrational
number. There is, of course, no real pair of lines with an irrational
ratio. Every real line will have an integer for its length. The length
unit
cannot be made as small as you wish because quantum mechanics will rear
its
ugly head when the unit gets small enough.
What about the square root of 2? It has no “reality”. It is a concept. A
useful concept, true; but you cannot write it down in any number system.
(You can define a symbol to represent the concept.) You CAN write down a
number which when multiplied by itself comes as close to 2 as you wish,
but it
will never be exactly 2.


In a message dated 9/11/2017 11:46:19 A.M. Mountain Daylight Time,
phys-l@mail.phys-l.org writes:

I have recently been pondering the concept of the irrational number. This
is a number which cannot be expressed as the ratio of two integers. The
Pythagoreans rejected this concept and, according to legend, executed a
member who asserted (and proved) its existence. An example is the square
root
of 2 - the diagonal / side ratio of any square. A little thought leads to
the following logical implication of the irrational number concept
(quoting a Wikipedia author):

" When the ratio of lengths of two line segments is an irrational number,
the line segments are also described as being incommensurable, meaning
that
they share no "measure" in common, that is, there is no length ("the
measure"), no matter how short, that could be used to express the lengths
of
both of the two given segments as integer multiples of itself."

To me, this is a weirdity that rivals (perhaps exceeds) the weirdities of
Quantum Mechanics. Quantum weirdities are concerned with the weird
behavior of material objects. The irrational number is a weirdity of
CONCEPT -
apart from any problems of measurement, construction or material
existence.
I simply cannot form a valid and self consistent CONCEPT of such a
quantity.

Somebody please straighten me out!

Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
treborsciamanda@gmail.com
www.sciamanda.com
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