Phil:
> If I choose a small enough unit of measurement I will get a whole
numbers for both the diameter and circumference.
That assumption is wrong. You won't. Ever.
Suppose you choose a measure such that the diameter is 1000 of your
units. The circumference will be 3141.6+ of those units, not an
integral number.
Suppose you choose a measure such that the diameter is 1 million of your
units. The circumference will be 3141592.6+. Again not an integral number.
Suppose you choose a measure such that the diameter is 1000 million of
your units. The circumference will be 3141592653.5+ units. Still not
integral.
and so on. Forever.
It doesn't help to choose some other division such that the diameter
isn't a nice even power of ten. You can choose any unit you like such
that the diameter is an exact multiple and the circumference will not be
an exact multiple.
Same argument applies to the measurement of the diagonal of a square.
Philip S Hall wrote:
Jon
In one of your documents you say: "Since by definition the value for
Pi is the ratio between the circumference of the circle to its
diameter, it must be representable in the form a/b ..."*
*
This is your fundamental mistake. The *vast* majority of numbers are
simply not expressible as integer ratios. Pi is just one of those.
There is no pair of integers (p, q) such that p/q = pi. None.
As a matter of interest Jon, how would you answer this challenge if
someone put it to you:
Suppose I have a disc and am able to measure it's diameter and its
circumference with complete accuracy using a unit of measurement as
small as I like. If I choose a small enough unit of measurement I will
get a whole numbers for both the diameter and circumference.
If I divide the measured circumference by the diameter I will get a
rational number. So how come pi is irrational?
Phil Hall
