Show that Q, the set of all rational numbers, is dense along the number line by showing that given any two rational line by showing that given any two rational numbers r 1 and r 2 with r 1 < r 2 , there exists a rational number x such that r 1 < x < r 2 .

To prove:

Along the number line, the set Q of all rational numbers is dense by showing that, for any two rational numbers r1 and r2 with r1<r2, there exists a rational number between two rational numbers r1 and r2.

Given information:

Q: The set of rational numbers.

Concept used:

The set of rational number is defined as follows.

ℚ={pq|p,q∈ℤ and q≠0}

The set rational number can also be written as ℚ=ℚ−∪{0}∪ℚ+.

Proof:

Let r1 and r2 be any two rational number on the number line such that r1<r2.

Now show that there exists a rational number between r1 and r2.

The midpoint between any two numbers on the number line is its average.

That means the midpoint for the numbers r1 and r2 on the number line will be their average.

That means midpoint=r1+r22 and it lies exactly middle of the numbers r1 and r2 on the number line.

The number r1+r22 is obviously a rational number and it can be shown as follows.

Let r1=p1q1 where p1,q1∈ℤ and q1≠0.

Let r2=p2q2 where p2,q2∈ℤ and q2≠0