# Prove directly that between any two rational numbers there exists a rational number.

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Let a and b are two rational numbers such that a<b .

`a,b in Q` , Q is set of rational numbers.

Let

`a=p/q and b=r/s`

by def of rational numbers .

consider

`(a+b)/2=(1/2)(p/q+r/s)`

`=(1/2)((ps+rq)/(qs)) in Q`

`a<(a+b)/2<b` , this proves there exist at least one rational number between a and b.