Editing Square root of 2 (section)

===Constructive proof===
While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let  {{math|''a''}} and {{math|''b''}} be positive integers such that {{math|1<{{sfrac|''a''|''b''}}< 3/2}} (as {{math|1<2< 9/4}} satisfies these bounds). Now {{math|2''b''{{sup|2}} }} and {{math|''a''{{sup|2}} }} cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus {{math|{{abs|2''b''{{sup|2}} − ''a''{{sup|2}}}} ≥ 1}}. Multiplying the absolute difference {{math|{{abs|√2 − {{sfrac|''a''|''b''}}}}}} by {{math| ''b''{{sup|2}}(√2 + {{sfrac|''a''|''b''}})}} in the numerator and denominator, we get<ref>See {{citation
 | last1 = Katz | first1 = Karin Usadi
 | last2 = Katz | first2 = Mikhail G.
 | author2-link = Mikhail Katz
 | arxiv = 1110.5456
 | issue = 2
 | journal = [[Intellectica]]
 | pages = 223–302 (see esp. Section 2.3, footnote 15)
 | title = Meaning in Classical Mathematics: Is it at Odds with Intuitionism?
 | volume = 56
 | year = 2011| bibcode = 2011arXiv1110.5456U}}</ref>
:<math>\left|\sqrt2 - \frac{a}{b}\right| = \frac{|2b^2-a^2|}{b^2\!\left(\sqrt{2}+\frac{a}{b}\right)} \ge \frac{1}{b^2\!\left(\sqrt2 + \frac{a}{b}\right)} \ge \frac{1}{3b^2},</math>

the latter [[inequality (mathematics)|inequality]] being true because it is assumed that {{math|1<{{sfrac|''a''|''b''}}< 3/2}}, giving {{math|{{sfrac|''a''|''b''}} + √2 ≤ 3 }} (otherwise the quantitative apartness can be trivially established). This gives a lower bound of {{math|{{sfrac|1|3''b''{{sup|2}}}}}} for the difference {{math|{{abs|√2 − {{sfrac|''a''|''b''}}}}}}, yielding a direct proof of irrationality in its constructively stronger form, not relying on the [[law of excluded middle]].<ref>{{citation |last=Bishop |first=Errett |author-link=Errett Bishop |editor-last=Rosenblatt |editor-first=Murray |editor-link=Murray Rosenblatt |date=1985 |chapter=Schizophrenia in Contemporary Mathematics. |title=Errett Bishop: Reflections on Him and His Research |series=Contemporary Mathematics |volume=39 |location=Providence, RI |publisher=[[American Mathematical Society]] |pages=1–32 |doi=10.1090/conm/039/788163 |isbn=0821850407 |issn=0271-4132}}</ref> This proof constructively exhibits an explicit discrepancy between <math>\sqrt{2}</math> and any rational.