Answer :

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 .


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

<span lang="EN-USKerala Board Mathematics Part-2