Little known fact: The square root of two is
also known as Pythagoras' constant.
Now that we know there are irrationals in the set of
real numbers, one might ponder
how many other
irrational numbers exist. One existance of an
irrational constant implies that there are at least infinitely many other irrationals. This follows from the fact that given an
irrational k and a
rational r, kr is irrational. (
Proof: Suppose there exists
integers a != 0, b, c, d such that k * c / d = a / b. Then k = ad/bc.
Contradiction.)
Also, the entire set of irrational numbers is
uncountably many.
Proof: Suppose there exists an indexing on the
set of all irrationals
K, such that K = {k
n}. Since
the set of rational numbers is countably infinite, there exists an indexing Q = {q
n}. The set of
real numbers
R then has an indexing:
r
n = k
n/2 if n even,
r
n = q
(n-1)/2 if n odd. This contradicts the fact that
the real numbers are uncountable.
QED.