Using the common form of rational numbers, prove that the sum, difference, product and quotient of any two rational numbers is again a rational number.

Sum of two rational numbers is a rational number:

As we know that, any rational number exists in the form of where p is the numerator and q is the denominator (q≠0), p and q are both integers.

Let us take two rational numbers as ‘a/b’ and ‘c/d’ where (b,d ≠0).

‘a’ and ‘c’ are the numerators while ‘b’ and ‘d’ are the

denominators. a,b,c and d are integers.

Sum of the above rational numbers =

= …………eq(1)

As we know that sum, product and division of two integers are

always integers.

So, (ad), (bc),(bd) and (ad + bc) are integer values.

Therefore, is fraction with integers in the numerator

and denominator.

As we know that, by definition, a rational number can be expressed as a fraction with integer values in the numerator and denominator (denominator not zero).

So, is a rational number (bd ≠0).

Therefore, Sum of two rational numbers is a rational number.

Difference of two rational numbers is a rational number.

Let us take two rational numbers as ‘ ’ and ‘ ’ where (b, d ≠0).

‘a’ and ‘c’ are the numerators while ‘b’ and ‘d’ are the

denominators. a,b,c and d are integers

Difference of the above rational numbers

= …………eq(1)

As we know that sum, product and division of two integers are

always integers.

So, (ad), (bc), (bd) and (ad-bc) are integer values.

Therefore, is fraction with integer values in the numerator

and denominator.

By definition, a rational number can be expressed as a fraction with integer values in the numerator and denominator (denominator not zero).

So, is a rational number (bd ≠0).

Therefore, difference of two rational numbers is a rational number.

Product of two rational numbers is a rational number.

Let us take two rational numbers as ‘a/b’ and ‘c/d’ where(b,d≠0).

‘a’ and ‘c’ are the numerators while ‘b’ and ‘d’ are the

denominators. a,b,c and d are integers.

Product of the numbers

From the above statements, we can say that (ac) and (bd) are also integers with (bd ≠0).

So, ac/bd is a fraction with integer values in the numerator and denominator (denominator not zero) making it a rational number.

Quotient of any two rational numbers is again a rational number:

Let us take two rational numbers as ‘ and ‘’ where(b, d≠0).

‘a’ and ‘c’ are the numerators while ‘b’ and ‘d’ are the

denominators. a,b,c and d are integers.

By, dividing the rational numbers we have =

=

=

From the above statements, we can say that (ad) and (bc) are also integers with (b,c ≠0).

So, ad/bc is a fraction with integer values.

Let .

Thus, X /Y can be expressed as a quotient of two integers and by definition, a rational number .