Irreducible fraction of a given rationalGiven a rational r ∈ Q , how to find the irreducible fraction a b = r? Any direct formula based on the digits of r, instead of successive approximations by increasing numerator and denominator alternatively?

Question

Irreducible fraction of a given rationalGiven a rational   r  ∈      Q  , how to find the irreducible fraction       a    b    =  r? Any direct formula based on the digits of   r, instead of successive approximations by increasing numerator and denominator alternatively?

Johnathan Morse · Accepted Answer

You have two cases.If the number of decimals is finite, let&#039;s say   x  =  a  .      a    1        a    2    .  .  .      a    n  . Then obviously  x  =            a              a        1                    a        2            .      .      .              a        n                    10      n      and then you simplify this fraction by looking for common divisors.If the number of decimals is infinite, then it must have a period   p, that is  x  =  a  .      a    1        a    2    .  .  .      a    p        a    1        a    2    .  .  .      a    p    .  .  .You have      10    p    x  −  x  =  a      a    1    .  .  .      a    p    −  aand hence  x  =            a              a        1            .      .      .              a        p            −      a                      10        p            −      1      which you can also simplify.