PROPERTIES OF RATIONAL NUMBER - 3

DENSENESS OF RATIONAL NUMBERS

Theorem: Between any two distinct rational numbers x and y, there exists a rational number.

Proof:  Let x and y be two rational numbers and let x < y.

Adding x to both sides, and adding y to both sides, we get.

                 x < y => x + x < x + y

and                          x +y < y + y.

ð  2x < x and x +y < 2y

ð  x < 1/2 (x + y) and 1/2 (x + y) < y

ð  x < 1/2 ( x + y) < y.

Now, the sum as well as the product of two rational numbers is a rational number.

So, 1/2 (x + y) is rational number.

Hence, between any two distinct rational numbers, there exists a rational number.

Remark 1: If x and y are two distinct rational numbers, there will exist a rational number z between them. Again, between x and z, and between z and y there exist rational numbers. Thus, by the repeated use of the above result, we can say that between two distinct rational numbers, there are infinitely many rational numbers.

Remark 2: Any n rational numbers between two distinct rational numbers x and y are given by

{x + (y – x)/(n + 1)}; {x + 2 (y – x)/(n + 1)};{ x + 3 (y – x)/(n +1)}:…….;{x + n (y – x)/(n + 1)}

Example 1: if x and y are two rational number, show that

(i)                 x + y is a rational number,

(ii)               x – y is a rational number,

(iii)             xy is a rational number,

(iv)             x/y is a rational number, where y ≠ 0.

Solution: Let x = a/b and y = c/d, where a, b, c and d are integers such that b ≠ 0 and d ≠ 0.

(i)                 x + y = a/b + c/d = ad + bc/bd

But, the sum and the product of two integers being integers, follows that (ad + bc) is an integer and bd is an integer. Also, b ≠ 0 implies bd ≠ 0.

Thus, (x + y) is of the form p/q, where p and q are integers and q ≠ 0.

Hence (x + y) is also a rational number.

(ii)               (x – y) = a/b – c/d = ad – bc/bd.

But, the product and the difference of two integers being integers, it follows that (ad – bc) as well as bd is an integer. Also, b ≠ 0 implies bd ≠ 0.

Thus, (x – y) is of the form p/q, where p and q are integers and q ≠ 0/

Hence, (x – y) is also a rational number.

(iii)              xy = a/b.c/d = ac/bd.

But, the product of two integers is always an integer.

So, ac and bd are integers.

Also, b ≠ 0 and d ≠ 0 implies bd ≠ 0.

Thus, xy is of the form p/q, where p and q are integers and q ≠ 0.

Hence, xy is also a rational number.

(iv)             Let y = c/d be a nonzero rational number.

Then, c and d are integers such that c ≠ 0 and d ≠ 0.

So x/y = a/b ÷ c/d = a/b × d/c = ad/bc.

But, the product of two integers being an integer, it follows that ad and bc are integers.

Also, b ≠ 0, c ≠ 0 implies bc ≠ 0.

Thus, x/y is of the form p/q, where p and q are integers, and q ≠ 0.

Hence, x/y is also a rational number.