Show that the set of non zero rational numbers is closed under division
(a/b) / (c/d) = a/b * d/c = (ad)/(bc)
which is a rational number.
To show that the set of non-zero rational numbers is closed under division, we need to demonstrate that dividing any two non-zero rationals gives us another non-zero rational.
Let's suppose we have two non-zero rational numbers, let's call them a and b. We can write these as fractions: a = p/q and b = r/s, where p, q, r, and s are non-zero integers.
Now, we will perform the division a/b, which can be expressed as (p/q) / (r/s). By the rules of division, this can be rewritten as (p/q) * (s/r).
To find the result of this division, we need to multiply the numerators and denominators separately. Multiplying the numerators gives us p * s, and multiplying the denominators gives us q * r.
Since p, q, r, and s are all non-zero integers, the product p * s and the product q * r will also be non-zero integers.
Now, we have the result of the division, (p * s) / (q * r). This fraction is in its simplest form because both numerator and denominator are non-zero integers.
Therefore, we can conclude that the result of dividing any two non-zero rational numbers is another non-zero rational number. Thus, the set of non-zero rational numbers is closed under division.