Is it possible to find the closest rational number to

Kaleigh Beasley

Kaleigh Beasley

Answered question

2022-06-03

Is it possible to find the closest rational number to an irrational number?

Answer & Explanation

kerutak0emro

kerutak0emro

Beginner2022-06-04Added 3 answers

No. It is a fact that in any open interval ] a , b [ there exists a rational number.
Proof:

Assume, that a > 0. Let n be a positive integer such that 1 b a < n. Now consider the subset of natural numbers { m N | a < m n }. By the well ordering principle, we know that this set has a minimum m 0 . Because of the way m 0 was chosen we know that:
m 0 1 n a < m 0 n
Thus: a < m 0 n a + 1 / n < a + b a = b

Let x be irrational and r be the closest rational number, now get a closer rational from the interval ] r , x [ (or ] x , r [ if x < r.

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