Question

based on what we established about the classification of y and using the closure of integers, what does the equation tell you about the type of number y must be for the product to be rational? what conclusion can you now make about the result of multiplying a rational and an irrational number?

Explanation:

To determine the type of \( y \) for the product to be rational (assuming a rational number is multiplied by \( y \)):

1. Recall: A rational number can be expressed as \( \frac{a}{b} \) (\( a,b \in \mathbb{Z}, b

eq 0 \)), and an irrational number cannot be expressed as a fraction of integers.

1. Closure of integers: The product of two integers is an integer. For the product of a rational number (e.g., \( \frac{m}{n} \)) and \( y \) to be rational, \( y \) must "cancel out" the irrational component (if any). If the original rational number is non - zero, \( y \) must be rational (because if a non - zero rational number \( \frac{m}{n} \) is multiplied by an irrational number \( y \), the product \( \frac{m}{n}y \) would be irrational, by the property that the product of a non - zero rational and an irrational is irrational). If the rational number is zero, \( y \) could be any number, but typically, we consider non - zero rational multipliers in such contexts.

For the conclusion about multiplying a rational and an irrational number:
The product of a non - zero rational number and an irrational number is irrational. This is because suppose \( r=\frac{a}{b} \) (\( a,b\in\mathbb{Z}, b
eq0, a
eq0 \)) is rational and \( s \) is irrational. Assume, for contradiction, that \( rs = t \) is rational. Then \( s=\frac{tb}{a} \), which would imply \( s \) is rational (since \( t,b,a \) are integers with \( a
eq0 \)), contradicting \( s \) being irrational. So, the product of a non - zero rational and an irrational number is irrational.

Answer:

For the product (of a non - zero rational number and \( y \)) to be rational, \( y \) must be rational. The conclusion about multiplying a rational and an irrational number (where the rational number is non - zero) is that the product is irrational.