Why is rational number not closed under division
Closure property of rational numbers holds good for addition ,subtraction & multiplication but not for division….
As we know, rational numbers are those numbers which can be represented as p/q, where p& q are integers but q not equal to 0.
So, p/q + r/s =( ps +qr) /qs
Here p/q & r/s are rationals, ie q& s can not be zero.
So, when we add 2 rational numbers , the sum is also a rational numbers. As ps & qr both are integers being the product of 2 integers. And their sum will also be an integer. That implies (ps + qr ) is an integer. And qs is also an integer but qs not equal to zero.
This way all the conditions of a rational number are satisfied. So we concluded that Rational numbers are closed for addition.
Similarly we can check for subtraction & multiplication. (p/q) - (r/s)= (ps-qr)/qs
P/q * r/s = pr/qs , closure property holds good.
Now for division: (p/q)÷(r/s) = (p/q) * (s/r)= ps/qr. So here there is a possibility of being r=0 . In that case ps/qr will not be a rational number. So we conclude that rational numbers are not closed for division operation.
Closure property of rational numbers holds good for addition ,subtraction & multiplication but not for division….
As we know, rational numbers are those numbers which can be represented as p/q, where p& q are integers but q not equal to 0.
So, p/q + r/s =( ps +qr) /qs
Here p/q & r/s are rationals, ie q& s can not be zero.
So, when we add 2 rational numbers , the sum is also a rational numbers. As ps & qr both are integers being the product of 2 integers. And their sum will also be an integer. That implies (ps + qr ) is an integer. And qs is also an integer but qs not equal to zero.
This way all the conditions of a rational number are satisfied. So we concluded that Rational numbers are closed for addition.
Similarly we can check for subtraction & multiplication. (p/q) - (r/s)= (ps-qr)/qs
P/q * r/s = pr/qs , closure property holds good.
Now for division: (p/q)÷(r/s) = (p/q) * (s/r)= ps/qr. So here there is a possibility of being r=0 . In that case ps/qr will not be a rational number. So we conclude that rational numbers are not closed for division operation.
