This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set.
Abu Kamil was the first mathematician to introduce irrational numbers as valid solutions to quadratic equations.
For example, the inverses of elements of Q(√c) are of the same form as the above algebraic numbers: Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all real quadratic irrationals, and only real quadratic irrationals, have periodic continued fraction forms.
For example The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers.
It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function.
Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map
, since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.
Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail.
Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail.
There are finitely many equivalence classes of quadratic irrationalities in
is a bijection that respects the matrix action on each set.
The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.
in a continued fraction corresponds to reducing the quadratic form.
The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.
The square root of 2 was the first such number to be proved irrational.
Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17.
Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes.
The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma.
Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae.
This asserts that every integer has a unique factorization into primes.
For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator.
When the numerator is squared that prime will still not divide into it because of the unique factorization.
His proof is in Euclid's Elements Book X Proposition 9.
[4] The fundamental theorem of arithmetic is not actually required to prove the result, however.
The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of 2 found by Theodor Estermann in 1975.