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*Consider two rational numbers A and B, where B>A. The difference between these number is D = B-A. Hence A+D = B. Note that D is a rational number.*

*Now divide D by a known***irrational**

*number that is >1, such as sqrt(2). Call this E, where: E = D/sqrt(2). E is definitely irrational, and positive, and smaller than D. So A < A+E < A+D. Remember A+D = B. So: A < A+E < B, and A+E is irrational. Hence no matter the values of rational numbers A and B, there is always an irrational number that lies between.*

*hope helped u dear frnd @ :-)*

*thank Q*### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

Let p/q and n/m be two rational numbers. so p, q, n, m are real numbers.

Without loss of generality, let us assume that p/q < n/m ie., p/q - n/m < 0

ie., (pm - nq) / (q m ) < 0. --- (1)

Let x = [ p/q + n/m ] / 2

= [ p m + n q ] / [2 q m ] --- (2)

finding

p/q - x = p/q - (pm+nq) / (2qm)

= [ 2mp - pm - nq ] / (2qm)

= [ mp - nq ] / (2qm)

< 0 as per (1)

n/m - x = n/ m - (pm+nq)/ (2qm)

= (2qn - pm - nq ) / (2qm)

= (qn - pm) / (2qm)

> 0 as per (1)

Hence, p/q < x < n/m

Thus there is always a rational number x between any two rational numbers.

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if there are two rational numbers X, Y (Y > X) then the rational number

( X+Y) / 2 or X + (Y - X) / k, or (m X + n Y) / (m+n)

lies between given X, Y.. here k , m , n are positive integers.

Without loss of generality, let us assume that p/q < n/m ie., p/q - n/m < 0

ie., (pm - nq) / (q m ) < 0. --- (1)

Let x = [ p/q + n/m ] / 2

= [ p m + n q ] / [2 q m ] --- (2)

finding

p/q - x = p/q - (pm+nq) / (2qm)

= [ 2mp - pm - nq ] / (2qm)

= [ mp - nq ] / (2qm)

< 0 as per (1)

n/m - x = n/ m - (pm+nq)/ (2qm)

= (2qn - pm - nq ) / (2qm)

= (qn - pm) / (2qm)

> 0 as per (1)

Hence, p/q < x < n/m

Thus there is always a rational number x between any two rational numbers.

===============

if there are two rational numbers X, Y (Y > X) then the rational number

( X+Y) / 2 or X + (Y - X) / k, or (m X + n Y) / (m+n)

lies between given X, Y.. here k , m , n are positive integers.