I have a two-part question:
1. Is 1/2 the largest fraction possible with (1) one as its numerator?
2. What is the closest fraction to the number 1?
I thank you for your reply.
1. If there is a 1 in the numerator, 1/2 is the closest fraction to 1.
2. There is no closest fraction to 1, and here's a proof.
Suppose we had m/n as the closest fraction that was less than 1.
That means we have n in the denominator and m < n in the numerator.
If m < n-1, then (m+1)/n is closer, so what we thought was the closets wasn't.
Now this argument can be applied until m = n-1, so we have the new fraction,
(n-1)/n as the closest fraction to 1.
If we add 1 to the numerator and denominator, we get n/(n+1).
If we take n/(n+1) - (n-1)/n, we need to have a common denominator.
Multiply the 1st by n/n and the 2nd by (n+1)/(n+1).
That makes our fraction nē/[n(n+1)] - n(n-1)/[n(n+1)].
They can now be combined into (nē - n(n-1))/[n(n+1)].
Multiplying out the n(n-1) gives nē-n, and when that is subtracted from nē, the result is n.
We now have n/[n(n+1)].
It is known that n/n is 1, so what we have left is 1/(n+1).
Since n is positive, the fraction is positive, so whatever we had for the largest number,
(n-1)/n, is not the largest, for we can add 1/(n+1) to it and get a fraction even closer to 1.
No matter what n is chosen, a closer fraction can be found,
which says there is no fraction that is closest to 1.
As an example, suppose we had 18/35. Clearly 19/35 is closer, but 20/35 is closer, but 21/35 is closer, ... up to 34/35 is pretty close, for it is 1/35 away from 1.
If we take 35/36, that is only 1/36 away from 1, so it is closer yet. This could be repeated, for 999/1000 is only 1/1000 away from 1, but 9,999/10,000 is only 1/10,000 away from one.
That's pretty close, but we could take 99,999/100,000 and be even closer since that is only 1/100,000 away from 1.
Adding one more digit to the numerator (a nine) and one more digit to the denominator (a 0) puts a even closer, and this can be done indefinitely.