Question

Fractions between 1 and 2


Is it finite or infinite

​

Answer 1

infinite.................

Answer 2

Consider numbers of the form a + 1 / b, where a is a specified integer and b is any positive integer > 1. You can also see that it is an improper fraction of the form (ab + 1) / b.

It’s clear that for every positive integer b there is a corresponding fraction of this form, meaning there is an infinite number of fractions of this form. It is also clear that fractions of this form must be between a and a + 1, because 1/b is necessarily smaller than 1 and larger than 0. There is also, by definition, no whole number that could exist between a and a + 1.

Therefore, for any arbitrary interval between two integers, let’s say x and y, where y > x. One can prove that any number between the interval x and x + 1 must necessarily also be between x and y, because if y > x and both are integers, y >= x + 1. Therefore, there is an infinite number of fractions between x and y, because we’ve proven that there is an infinite number of fractions between x and x + 1.

For the example interval [1, 2], you can simply set a to 1, and obtain 3/2, 4/3, 5/4 …. every one of those fractions will be different and between 1 and 2, and there is an infinite number of them.