Editing Quadratic irrational number (section)

==Square root of non-square is irrational==
The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational.  The solutions to the quadratic equation ''ax''<sup>2</sup>&thinsp;+&thinsp;''bx''&thinsp;+&thinsp;''c''&nbsp;=&nbsp;0 are

:<math>\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>

Thus quadratic irrationals are precisely those [[real number]]s in this form that are not rational.  Since ''b'' and 2''a'' are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational.  The answer to this is that the square root of any [[natural number]] that is not a [[square number]] is irrational.

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. Therefore, the square of a rational non-integer is always a non-integer; by [[contrapositive]], the square root of an integer is always either another integer, or irrational.

[[Euclid]] used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in [[Euclid's Elements]] Book X Proposition 9.<ref>{{cite web | url=http://aleph0.clarku.edu/~djoyce/java/elements/bookX/propX9.html |title=Euclid's Elements Book X Proposition 9 |access-date=2008-10-29 |author=Euclid | editor=D. E. Joyce|editor-link=David E. Joyce (mathematician)|publisher=Clark University }}</ref>

The fundamental theorem of arithmetic is not actually required to prove the result, however.  There are self-contained proofs by [[Richard Dedekind]],<ref>{{cite web |author=Bogomolny, Alexander |author-link=Alexander Bogomolny |title=Square root of 2 is irrational | website=Interactive Mathematics Miscellany and Puzzles |url=http://www.cut-the-knot.org/proofs/sq_root.shtml |access-date=May 5, 2016}}</ref> among others.  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.<ref>{{cite journal |first=Colin Richard |last= Hughes |title=Irrational roots |journal=[[Mathematical Gazette]] |volume=83 |number=498 |year=1999 |pages=502–503|doi= 10.2307/3620972 |jstor= 3620972 |s2cid= 149602021 }}</ref><ref>{{cite journal |first=Theodor |last=Estermann |title=The irrationality of √2 | journal=Mathematical Gazette |volume=59 |number=408 |year=1975 |page=110|doi=10.2307/3616647 |jstor=3616647 |s2cid=126072097 }}</ref>

If ''D'' is a non-square natural number, then there is a natural number ''n'' such that:

:''n''<sup>2</sup> < ''D'' < (''n''&thinsp;+&thinsp;1)<sup>2</sup>,

so in particular

:0 < {{radic|''D''}} &minus; ''n'' < 1.

If the square root of ''D'' is rational, then it can be written as the irreducible fraction ''p''/''q'', so that ''q'' is the smallest possible denominator, and hence the smallest number for which ''q''{{radic|''D''}} is also an integer. Then:

:({{radic|''D''}} &minus; ''n'')''q''{{radic|''D''}} = ''qD'' &minus; ''nq''{{radic|''D''}}

which is thus also an integer. But 0&nbsp;<&nbsp;({{radic|''D''}}&nbsp;&minus;&nbsp;''n'')&nbsp;<&nbsp;1 so ({{radic|''D''}}&nbsp;&minus;&nbsp;''n'')''q''&nbsp;<&nbsp;''q''.  Hence ({{radic|''D''}}&nbsp;&minus;&nbsp;''n'')''q'' is an integer smaller than ''q'' which multiplied by {{radic|''D''}} makes an integer. This is a contradiction, because ''q'' was defined to be the smallest such number. Therefore, {{radic|''D''}} cannot be rational.