show that the set of nonzero rational numbers is closed under division
Sorry unknown but Im not sure
since a,b,c,d are all nonzero
(a/b) / (c/d) = a/b * d/c = ad/bc
which is a rational number.
There may be other items to consider, depending on what you have already established ...
To show that the set of nonzero rational numbers is closed under division, we need to demonstrate that if we divide any two nonzero rational numbers, the result is still a nonzero rational number.
Let's take two nonzero rational numbers, a/b and c/d, where a,b,c,d are integers and b,d are not equal to zero. The division of a/b by c/d can be written as:
(a/b) ÷ (c/d) = (a/b) × (d/c)
To see if this result is a nonzero rational number, we need to check if the numerator and denominator are integers and the denominator is not zero.
The numerator of (a/b) × (d/c) is ad, which is an integer because the product of two integers is always an integer.
The denominator of (a/b) × (d/c) is bc, which is also an integer because the product of two integers is always an integer.
Now, let's check if the denominator is not zero. Since b and c are nonzero integers, their product bc is also nonzero, which means that the denominator of (a/b) × (d/c) is not zero.
Therefore, we have shown that the division of any two nonzero rational numbers results in a nonzero rational number. Hence, the set of nonzero rational numbers is closed under division.