(a)      For each of the following statements, either prove it is true or give counter example to show it is false;

i)             The product of two rational numbers is always rational

ii)            The product of two rational numbers is always irrational.

iii)           The product of a non-zero rational and an irrational is always irrational.

iv)           The product of two irrational numbers is always rational(a)      For each of the following statements, either prove it is true or give counter example to show it is false;

i)             The product of two rational numbers is always rational

ii)            The product of two rational numbers is always irrational.

iii)           The product of a non-zero rational and an irrational is always irrational.

iv)           The product of two irrational numbers is always rational

Let's analyze each of the given statements and determine if they are true or false, providing proofs or counterexamples as needed.

### i) The product of two rational numbers is always rational.

**Proof:**

- **Definition of a rational number:** A rational number can be expressed as the ratio of two integers, i.e., \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).
  
- **Let \( \frac{a}{b} \) and \( \frac{c}{d} \) be two rational numbers:**
  \[
  \text{Product} = \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
  \]
  
- **Since \( ac \) and \( bd \) are integers and \( bd \neq 0 \),** \( \frac{ac}{bd} \) is a ratio of two integers, hence a rational number.

**Conclusion:** The statement is **true**.

### ii) The product of two rational numbers is always irrational.

**Counterexample:**

- Consider the two rational numbers \( \frac{1}{2} \) and \( \frac{1}{3} \):
  \[
  \text{Product} = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}
  \]
  
- **Since \( \frac{1}{6} \) is a rational number**, this shows that the product of two rational numbers is not always irrational.

**Conclusion:** The statement is **false**.

### iii) The product of a non-zero rational number and an irrational number is always irrational.

**Proof:**

- **Definition of a non-zero rational number:** Let the non-zero rational number be \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).
  
- **Let \( x \) be an irrational number:** By definition, \( x \) cannot be expressed as a ratio of two integers.
  
- **Consider the product \( \frac{a}{b} \times x \):** 
  \[
  \text{Product} = \frac{a \times x}{b}
  \]
  
- **If this product were rational, \( x \) would have to be rational** (since \( \frac{a}{b} \) is rational), contradicting the fact that \( x \) is irrational.

**Conclusion:** The statement is **true**.

### iv) The product of two irrational numbers is always rational.

**Counterexample:**

- Consider two irrational numbers \( \sqrt{2} \) and \( \sqrt{2} \):
  \[
  \text{Product} = \sqrt{2} \times \sqrt{2} = 2
  \]
  
- **Since 2 is a rational number,** this shows that the product of two irrational numbers can be rational.

**Conclusion:** The statement is **false**.

### Summary:
1. **(i) True**: The product of two rational numbers is always rational.
2. **(ii) False**: The product of two rational numbers is not always irrational.
3. **(iii) True**: The product of a non-zero rational number and an irrational number is always irrational.
4. **(iv) False**: The product of two irrational numbers is not always rational.

Would you like further details or have any other questions?

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Here are 5 related questions:

1. Can the sum of two irrational numbers be rational?
2. Can the sum of a rational number and an irrational number ever be rational?
3. What is an example of two irrational numbers whose product is irrational?
4. Can the difference of two irrational numbers be irrational?
5. How does the multiplication of complex numbers compare with that of real numbers in terms of rationality?

**Tip:** When working with rational and irrational numbers, understanding their definitions and properties helps in constructing proofs and counterexamples effectively.

Explore the properties of rational and irrational numbers in mathematics with proofs and counterexamples. Learn about the nature of their products and theorems involved. Suitable for high school students studying number theory.

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