Originally posted by: NichowA
More interesting to me than asking what if pi were a different number is why is pi what it already is? Such an impossible number to represent such a simple relationship.
?
It's not an "impossible" number in any way. It's 'real' number.
It is not a 'rational' number, though. Rational numbers are numbers which can be represented by a relation P/Q, where P and Q are whole signed numbers, integers.
'Rational' numbers, of course, include all finite decimal numbers, since Q can be 10, 100, 1000...
'Rational' numbers, again of course, also include nonfinite decimal sequences, like 1/3 = 0.33333....
'Real' numbers include all 'rational' numbers, but in fact, most 'real' numbers are not 'rational'!
If N is the amount of real numbers and K is the amount of rational numbers, then the relation N/K is infinite!
So you see, -
pi is not a rare case at all. On the contrary, numbers like
pi,
e,
2^½ are the vast majority.
We use the symbols of
pi,
e, etc, in order to be able to represent some important non rational real numbers *exactly*. If we used approximations for these numbers, in mathematic expressions and equations, errors would creep in.