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Hello:

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.

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