A student wrote a proof about the product of two rational numbers: 1. Let x=ab and let y=cd, where a and c are defined to be integers, and b and d are nonzero integers. 2. By substitution, xy=acbd. 3. By applying the closure property of integers and nonzero integers on multiplication, ac is an integer and bd is a nonzero integer. What conclusion can the student now make about the product xy? The product xy may be either rational or irrational because the values of a, b, c, and d are unknown. The product xy is rational because it can be written as the quotient of an integer and a nonzero integer. The product xy is a nonzero integer because nonzero integers are closed on division. The product xy cannot be an integer because bd is a nonzero integer.
The Correct Answer and Explanation is:
Correct Answer:
The product xy is rational because it can be written as the quotient of an integer and a nonzero integer.
Explanation
The product of two rational numbers is always rational. Let’s break down the reasoning behind this.
We start with the assumption that the student lets
- x=abx = \frac{a}{b}x=ba,
- y=cdy = \frac{c}{d}y=dc,
where aaa and ccc are integers, and bbb and ddd are nonzero integers. These definitions satisfy the condition of rational numbers, which are defined as numbers that can be written as the quotient of two integers, where the denominator is nonzero.
The student then multiplies the two rational numbers:xy=(ab)(cd)=acbdxy = \left(\frac{a}{b}\right) \left(\frac{c}{d}\right) = \frac{ac}{bd}xy=(ba)(dc)=bdac
Now, consider the numerator and denominator of the resulting expression:
- Since aaa and ccc are integers, and integers are closed under multiplication, acacac is an integer.
- Since bbb and ddd are nonzero integers, and nonzero integers are also closed under multiplication, bdbdbd is a nonzero integer.
Thus, the result acbd\frac{ac}{bd}bdac is a quotient of an integer over a nonzero integer, which fits the definition of a rational number.
Now, let’s examine the incorrect answer choices:
- Saying that the product may be irrational is incorrect because rational × rational = rational.
- Saying that the product is a nonzero integer is too specific and not always true (e.g., 12×23=13\frac{1}{2} × \frac{2}{3} = \frac{1}{3}21×32=31, which is not an integer).
- The statement that “the product cannot be an integer because bd is nonzero” is flawed reasoning. The denominator being nonzero doesn’t prevent the product from being an integer—it depends on the numerator and denominator values.
Therefore, the best conclusion is that the product xy is rational.
