Question

are the rational numbers closed under multiplication? justify your reasoning.

no, the product of any two rational numbers will not also be a rational number.
no, the product of any two rational numbers will also be a rational number.
yes, the product of any two rational numbers will not also be a rational number.
yes, the product of any two rational numbers will also be a rational number.

question 9

are irrational numbers closed under multiplication? justify your reasoning.

Explanation:

Define closure under multiplication

The Closure Property states that a set is closed under an operation if performing that operation on any two elements in the set always produces an element that is also in that set. For Closure Under Multiplication, multiplying any two numbers from the set must result in a number within the same set.

Analyze rational numbers under multiplication

Let \(a\) and \(b\) be any two Rational Numbers. By definition, they can be written as fractions of integers:

$$a = \frac{p}{q}, \quad b = \frac{r}{s}$$

where \(p, q, r, s\) are integers and \(q, s
eq 0\).
Their product is:

$$a \cdot b = \frac{p \cdot r}{q \cdot s}$$

Since the product of two integers is always an integer, \(p \cdot r\) and \(q \cdot s\) are integers, and \(q \cdot s
eq 0\). Thus, the product is always a rational number.
Therefore, rational numbers are closed under multiplication.

Analyze irrational numbers under multiplication

Let \(x\) and \(y\) be Irrational Numbers. Consider the counterexample:

$$x = \sqrt{2}, \quad y = \sqrt{2}$$

Both are irrational. Their product is:

$$x \cdot y = \sqrt{2} \cdot \sqrt{2} = 2$$

Since \(2\) is a rational number, the product of two irrational numbers is not always irrational.
Therefore, irrational numbers are not closed under multiplication.

Answer:

Question 1

• (A) No, the product of any two rational numbers will not also be a rational number.
• (B) No, the product of any two rational numbers will also be a rational number.
• (C) Yes, the product of any two rational numbers will not also be a rational number.
• (D) Yes, the product of any two rational numbers will also be a rational number. (Correct answer)

Question 2

No, irrational numbers are not closed under multiplication. For example, multiplying the irrational number \(\sqrt{2}\) by itself yields \(\sqrt{2} \cdot \sqrt{2} = 2\), which is a rational number.