Editing Irreducible fraction (section)

==Uniqueness==

Every rational number has a ''unique'' representation as an irreducible fraction with a positive denominator<ref name="unique"/> (however {{sfrac|2|3}} = {{sfrac|−2|−3}} although both are irreducible). Uniqueness is a consequence of the [[Fundamental theorem of arithmetic|unique prime factorization]] of integers, since {{nowrap|1={{sfrac|''a''|''b''}} = {{sfrac|''c''|''d''}}}} implies ''ad''&nbsp;=&nbsp;''bc'', and so both sides of the latter must share the same prime factorization, yet ''a'' and ''b'' share no prime factors so the set of prime factors of ''a'' (with multiplicity) is a subset of those of ''c'' and vice versa, meaning ''a''&nbsp;=&nbsp;''c'' and by the same argument ''b''&nbsp;=&nbsp;''d''.