logavioc6

2023-03-13

Why is rational number not closed under division

bbiillyyh3l

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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