Re: PHYS-L digest 914

[Charlie Crummer <ccrummer@cats.ucsc.edu>]

> Dear listmembers,
>
> Reading Bob Sciamanda's quote, I recall one question I had during
> schooldays that I was afraid to ask the teacher about:
>
> We were told that irrational numbers are numbers that cannot be expressed
> as a fraction (or quotient of two numbers).
>
> But looking at the circumference formula, pi is the quotient of
> circumference and diameter...
>
> any thoughts on this?

The evolution of the concept of number seems to be this:  The most
primitive concept comes from counting things and results in the concept of
positive, non-zero integers.  The next step was to include zero as a
number.

		X + 0 = X  (0 is the identity of addition.)

(The Greeks had serious philosophical problems with using a symbol to
represent nothing at all!)  Negative numbers had to be introduced to
accomodate solutions to equations such as

		X + 3 = 0.

The concept of multiplication also must have primitive origins.  The
identity of multiplication is 1.

		X * 1 = X.

The concept of multiplication makes the counting of groups of things
easier. Next comes division.

		X * (X inverse) = 1 where (X inverse) is defined to be 1/X,
i.e. 1 divided by X.  This gave rise to the rational numbers, ratios of
integers.

It was discovered that there are numbers which cannot be expressed as
ratios of integers.  These are the irrational numbers; square root of 2,
pi, e, ....


Charles A. Crummer, PhD
Lower Division Laboratory Manager
Physics Department, Kerr Hall
University of California, Santa Cruz 95064
(831)459-4154 (office), (831)459-3043 (FAX)