Are lowest/irreducible fraction forms unique?

[holezch]

I thought they were unique, but given a fraction a/b, couldn't you always write it as the (decimal representation of a/b without the decimal place)/ 10^(n) ?
thanks

 

[Hurkyl]

Every rational number does indeed have a unique representation of the form a/b, where a is an integer and b is a positive integer.

If what you suggest both makes sense and results in something that is in lowest terms, then you have merely recomputed the form a/b.

 

[holezch]

thanks, but for example, you can write 121/123 = 0.983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398374
...
or you can write (n being the number of digits)
983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398373983739837398374 / 10^(n)

these are both in lowest forms right? but they have different integers a, b for a/b
and I'm curious, I've never seen a proof for that statement

 

[Hurkyl]

Erm... exactly how many digits do you think are in the decimal expansion of 121/123?

 

[holezch]

Well, a lot.. But the problem is that I can't find an example where the number of digits is finite. But if I could, you could apply the same idea?

 

[Hurkyl]

Of course. 1/2 and 127/1000 have only finitely many digits.

(ignoring the infinitely many trailing zeroes, and assuming you choose that decimal representation rather than the one ending in infinitely many trailing nines)

 

[holezch]

D'oh of course, 1/2.. But now in this example, 5/10 is indeed just 1/2.. How can I prove that these irreducible forms are unique? Will it be an easy proof? (I'm guessing that it will be something like supposing you have a' and a then showing that they are equal, a usual strategy)
thanks

 

[Hurkyl]

It follows more or less directly from prime factorization of integers.

 

[HallsofIvy]

I thought they were unique, but given a fraction a/b, couldn't you always write it as the (decimal representation of a/b without the decimal place)/ 10^(n) ?
thanks

Well, first only a small portion of the rational numbers have decimal representations that terminate after a finite number of places (those with only powers of 2 and 5 in their denominators) and those are the only ones for which that is possible. And, of course, that would, in general, NOT be irreducible.

For example, 1/4 = 0.25 so your expression would be 25/100 which is not irreductible because both 25 and 100 are divisible by 25. It reduces to, no surprise, 1/4!