Can you always find another fraction in between ant two fractions why?

Answer 1

Yes. The simple answer is that rational fractions are infinitely dense. A longer proof follows:

Suppose you have two fractions a/b and c/d where a, b, c and d are integers and b, d are positive integers.

Without loss of generality, assume a/b < c/d.

The inequality implies that ad < bc so that bc-ad>0 . . . . . . . . . . . . . . . . . . . (I)

Consider (ad + bc)/(2bd)

Then (ad+bc)/2bd - a/b = (ad+bc)/2bd - 2ad/2bd = (bc-ad)/2bd

By definition, b and d are positive so bd is positive and by result (I), the numerator is positive.

That is to say, (ad+bc)/2bd - a/b > 0 or (ad+bc)/2bd > a/b.

Similarly, by considering c/d - (ad+bc)/2bd is can be shown that c/d > (ad+bc)/2bd.

Combining these results,

a/b < (ad+bc)/2bd < c/d.