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.