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

Amira Serrano 2022-10-13 Answered
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 < 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?
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Answers (1)

Kaylee Evans
Answered 2022-10-14 Author has 20 answers
2 x + 3 y + 2 x 3 y 2 x + 3 y 2 x 3 y = p
Apply 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't factorise equation using factorisation method.
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