Editing Extended Euclidean algorithm (section)

==Simplification of fractions==
A fraction {{math|{{sfrac|''a''|''b''}}}} is in canonical simplified form if {{math|''a''}} and {{math|''b''}} are [[coprime]] and {{math|''b''}} is positive. This canonical simplified form can be obtained by replacing the three output lines of the preceding pseudo code by 
 '''if''' {{math|1=''s'' = 0}} '''then output''' "Division by zero"
 '''if''' {{math|1=''s'' < 0}} '''then''' {{math|1=''s'' := −''s''}};  {{math|1=''t'' := −''t''}}   (''for avoiding negative denominators'')
 '''if''' {{math|1=''s'' = 1}} '''then output''' {{math|−''t''}}        (''for avoiding denominators equal to'' 1)
 '''output''' {{math|{{sfrac|−''t''|''s''}}}}

The proof of this algorithm relies on the fact that {{math|''s''}} and {{math|''t''}} are two coprime integers such that {{math|1=''as'' + ''bt'' = 0}}, and thus <math>\frac{a}{b} = -\frac{t}{s}</math>. To get the canonical simplified form, it suffices to move the minus sign for having a positive denominator.

If {{math|''b''}} divides {{math|''a''}} evenly, the algorithm executes only one iteration, and we have {{math|1=''s'' = 1}} at the end of the algorithm. It is the only case where the output is an integer.