What is the formula for the decimal representation of a b where b is not coprime to 10?If we want to turn a proper fraction a b into a decimal, then when b is coprime to 10 we just need to rewrite a b in the form m 10 ϕ ( b ) − 1 where ϕ is Euler's totient function. And then the repeatend is m and the period of repetition is ϕ ( b ).But my question is, what do we do when b is not coprime to 10? In that case we would need to rewrite a b in the form k + l m 10 n where k &lt; 10 n , l m is a proper fraction, and m is coprime to 10. Then k would be the non-repeating part, n would be the length of the nonrepeating part, and we can convert l / m into a decimal using the procedure given in the beginning of my post.But is there a formula for the four numbers k, l, m, and n in terms of a and b, short of doing long division to convert the fraction into a decimal?

Question

What is the formula for the decimal representation of       a    b   where   b is not coprime to 10?If we want to turn a proper fraction       a    b   into a decimal, then when   b is coprime to 10 we just need to rewrite       a    b   in the form      m                  10                  ϕ          (          b          )                    −      1      where   ϕ is Euler&#039;s totient function. And then the repeatend is   m and the period of repetition is   ϕ  (  b  ).But my question is, what do we do when   b is not coprime to 10? In that case we would need to rewrite       a    b   in the form            k      +              l        m                    10      n      where   k  &amp;lt;      10    n  ,       l    m   is a proper fraction, and   m is coprime to 10. Then   k would be the non-repeating part,   n would be the length of the nonrepeating part, and we can convert   l      /    m into a decimal using the procedure given in the beginning of my post.But is there a formula for the four numbers   k,   l,   m, and   n in terms of   a and   b, short of doing long division to convert the fraction into a decimal?

Kaylee Evans · Accepted Answer

2        x        +        3        y            +              2        x        −        3        y                            2        x        +        3        y            −              2        x        −        3        y              =  pApply componendo and dividendo,                    2        x        +        3        y            +              2        x        −        3        y            +              2        x        +        3        y            −              2        x        −        3        y                            2        x        +        3        y            +              2        x        −        3        y            −              2        x        +        3        y            +              2        x        −        3        y              =            p      +      1              p      −      1                  2      x      +      3      y              2      x      −      3      y        =            p      +      1              p      −      1      Put x=y            5      y              −      y        =            p      +      1              p      −      1      Squaring both sides,  5.  (  p  −  1      )    2    =  −  1  (  p  +  1      )    2  Sorry I have little mistake here,   5      p    2    −  10  p  +  5  =  −      p    2    −  2  p  −  2  5      p    2    −  10  p  +  5  =  −      p    2    −  2  p  −  1  6      p    2    −  8  p  +  6  =  0  3      p    2    −  4  p  +  3  =  0  D  =      b    2    −  4  a  c=   (  −  4      )    2    −  4.3.3 = 16 - 36 = - 20  p  =            −      b      ±              D                    2      a      =             −      (      −      4      )      ±              −        20              2.3  =             4      ±              20                  i          2                      6  =             4      ±      2      i              5              6  p =             2      +      i              5              3    ,            2      −      i              5              3  Edit-In above formula D is discriminat. It is used when we can&#039;t factorise equation using factorisation method.

What is the formula for the decimal representation of a/b where b is not coprime to 10? (m)/(10^(phi(b))-1)

Answered question

Answer & Explanation

New Questions in Pre-Algebra