Editing Irreducible fraction (section)

==Applications==

The fact that any rational number has a unique representation as an irreducible fraction is utilized in various [[Square root of 2#Proofs of irrationality|proofs of the irrationality of the square root of 2]] and of other irrational numbers. For example, one proof notes that if <math>\sqrt{2}</math> could be represented as a ratio of integers, then it would have in particular the fully reduced representation {{sfrac|''a''|''b''}} where ''a'' and ''b'' are the smallest possible; but given that {{sfrac|''a''|''b''}} equals <math>\sqrt{2}</math> so does {{sfrac|2''b'' − ''a''|''a'' − ''b''}} (since cross-multiplying this with {{sfrac|''a''|''b''}} shows that they are equal). Since ''a''&nbsp;>&nbsp;''b'' (because <math>\sqrt{2}</math>  is greater than 1), the latter is a ratio of two smaller integers. This is a [[proof by contradiction|contradiction]], so the premise that the square root of two has a representation as the ratio of two integers is false.