Question

the mean of two rational number is rational and lies between them​

Answer 1

Answer:

Let x and y be two rational numbers then x and y can be express in the form of p/q where p and q belongs to integers and q ≠ 0.

Let

x = s/t   where s and t ∈ Z and t ≠ 0

y = u/v  where u and v ∈ Z and v ≠ 0  

Then

Mean of x and y = ( x + y ) / 2

Putting values we get

Mean of x and y = ( s/t + u/v ) / 2 = (sv + ut) / 2uv

Since sv + ut is again belongs to Z

So

Let

sv + ut = r

And similarly

Since 2uv again belong to Z

So

Let 2uv = w

Thus

Mean of x and y = r / w  

where r and w belong to Z and w ≠ 0 because 2uv ≠ 0

Hence mean to two rational numbers is again a rational number.

Now To show Means lies between them

Let

y > x

Then

⇒ y/2 > x/2

⇒ y/2 > x - x/2             Since x - x/2 = x/2

⇒ y/2 + x/2 > x

And

y/2 + x/2 < y                 Since y > y /2 and y > x / 2.

Thus

    x < y/2 + x/2 < y    

⇒ x < ( y + x ) / 2 < y

⇒ x < Mean of x and y < y

Hence prove mean of two numbers lies between them.