Why is rational number not closed under division?



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Why is rational number not closed under division

Answer & Explanation



Beginner2023-03-14Added 4 answers

For addition, subtraction, and multiplication but not division, the closure property of rational numbers applies….
Rational numbers, as we are all aware, are those that can be written as p/q, where p and q are both integers but q is 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.
In this manner, every prerequisite for a rational number is met. Therefore, we came to the conclusion that rational numbers cannot be added.
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.

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