Irreducible fraction of a given rationalGiven a rational r ∈ Q , how to find...

spockmonkey40

spockmonkey40

Answered

2022-07-13

Irreducible fraction of a given rational
Given 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?

Answer & Explanation

Johnathan Morse

Johnathan Morse

Expert

2022-07-14Added 18 answers

You have two cases.
If the number of decimals is finite, let'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 a
and hence
x = a a 1 . . . a p a 10 p 1
which you can also simplify.

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