Question

Construct an argument that shows that the set of rational numbers is closed under division. That is, if x and y are
rational numbers (with y nonzero) and ???? = x / y' prove that ???? must also be a rational number.

Answer 1

Answer:

$\frac{x}{y}$ can also be expressed as a ratio of two integers.

Step-by-step explanation:

Assume x and y are rational number, then x and y can be written as:

$x=\frac{a}{b}$, where a and b are integers and b≠0

$y=\frac{c}{d}$, where c and d are integers and d≠0

Then,

$\frac{x}{y}$ = $\frac{\frac{a}{b} }{\frac{c}{d} }$

= $\frac{a*d}{b*c}$  

Since the set of integers are closed under multiplication, a*d is also an integer. Similarly b*c is also an integer.

$\frac{x}{y}$ can be expressed as a proportion of two integers, provided that y≠0. From this we can conclude that $\frac{x}{y}$  is a rational number, which concludes that the set of rational numbers are closed under division.