Question

Alex says that there is no numbers between 1/3 and 1/4 . Why might he think that and what would you tell him? Is there a number that falls between these?

Answer 1

Why might he think that and what would you tell him?

He thought perhaps that numbers were formed by rational numbers of the form
`(1)/(x)`, where
`x\in \mathbb{N}`. I would tell him that there are rational numbers
`(y)/(z)`, such that
`(1)/(x)\le (y)/(z) \le (1)/(x+1)`, where
`x`,
`y`,
`z \in \mathbb{N}`.

Is there a number that falls between these?

`(7)/(24)` is a rational number between
`(1)/(3)` and
`(1)/(4)`.

Explanation:

Why might he think that and what would you tell him?

He thought perhaps that numbers were formed by rational numbers of the form
`(1)/(x)`, where
`x\in \mathbb{N}`. I would tell him that there are rational numbers
`(y)/(z)`, such that
`(1)/(x)\le (y)/(z) \le (1)/(x+1)`, where
`x`,
`y`,
`z \in \mathbb{N}`.

Is there a number that falls between these?

Indeed, the average number of
`(1)/(3)` and
`(1)/(4)`, for instance. That is:

`(y)/(z) = ((1)/(3)+(1)/(4) )/(2)`

`(y)/(z) = ((7)/(12) )/(2)`

`(y)/(z) = (7)/(24)`

`(7)/(24)` is a rational number between
`(1)/(3)` and
`(1)/(4)`.