Answer:
<em>Why might he think that and what would you tell him?</em>
He thought perhaps that numbers were formed by rational numbers of the form  , where
, where  . I would tell him that there are rational numbers
. I would tell him that there are rational numbers  , such that
, such that  , where
, where  ,
,  ,
,  .
.
<em>Is there a number that falls between these?</em>
 is a rational number between
 is a rational number between  and
 and  .
.
Step-by-step explanation:
<em>Why might he think that and what would you tell him?</em>
He thought perhaps that numbers were formed by rational numbers of the form  , where
, where  . I would tell him that there are rational numbers
. I would tell him that there are rational numbers  , such that
, such that  , where
, where  ,
,  ,
,  .
.
<em>Is there a number that falls between these?</em>
Indeed, the average number of  and
 and  , for instance. That is:
, for instance. That is:



 is a rational number between
 is a rational number between  and
 and  .
.