(f) Find an imational number between (1) 2 and 3 (2) \( \sqrt{2} \) and \( \sqrt{3} \) (3) \( \pi \) and \( 2 \pi \) (g) Say whether the following statements are true or false: (1) The sum of two rational numbers is always a rational number. (2) The sum of two irrational numbers is always a rational number. (3) The product of two rational numbers is always a rational number. (4) The product of two irrational numbers is always an irrational number. (5) The product of two real numbers is always a real number. (6) When a real number is divided by a real number the result is always a real number. (7) The sum of a rational number and an irrational number is always an irrational number (8) The product of a rational number and an irrational number is always an irrational number: (Be carefull) (h) Between which two consecutive integers do the following irrational numbers lie? (3) \( \sqrt{97} \)

### Problem (f): Find an Irrational Number Between Specified Intervals **(1) Between 2 and 3** One simple irrational number between 2 and 3 is √5. - **Explanation:** √5 ≈ 2.23607, which clearly lies between 2 and 3. Moreover, since 5 is not a perfect square, √5 is irrational. **(2) Between \( \sqrt{2} \) and \( \sqrt{3} \)** An example of an irrational number between \( \sqrt{2} \) and \( \sqrt{3} \) is \( \sqrt{2.5} \). - **Explanation:** - \( \sqrt{2} \) ≈ 1.4142 - \( \sqrt{3} \) ≈ 1.7320 - \( \sqrt{2.5} \) ≈ 1.5811, which lies between them. Additionally, 2.5 is not a perfect square, so \( \sqrt{2.5} \) is irrational. **(3) Between \( \pi \) and \( 2\pi \)** One such number is \( \frac{3\pi}{2} \). - **Explanation:** - \( \pi \) ≈ 3.1416 - \( 2\pi \) ≈ 6.2832 - \( \frac{3\pi}{2} \) ≈ 4.7124, which lies between \( \pi \) and \( 2\pi \). Since \( \pi \) is irrational and scaling by a rational number preserves irrationality (unless scaled by zero, which it is not here), \( \frac{3\pi}{2} \) is irrational. --- ### Problem (g): Determine Whether Statements Are True or False **(1) The sum of two rational numbers is always a rational number.** - **Answer:** **True** - **Reasoning:** Let \( r_1 \) and \( r_2 \) be rational numbers. Then, \( r_1 + r_2 \) is also rational because the sum of fractions (or integers) is rational. **(2) The sum of two irrational numbers is always a rational number.** - **Answer:** **False** - **Reasoning:** While some pairs of irrational numbers can sum to a rational number (e.g., \( \sqrt{2} + (-\sqrt{2}) = 0 \)), many pairs sum to an irrational number (e.g., \( \sqrt{2} + \sqrt{3} \) is irrational). Therefore, it's not always true. **(3) The product of two rational numbers is always a rational number.** - **Answer:** **True** - **Reasoning:** Let \( r_1 \) and \( r_2 \) be rational numbers. Their product \( r_1 \times r_2 \) is also rational because the product of fractions (or integers) is rational. **(4) The product of two irrational numbers is always an irrational number.** - **Answer:** **False** - **Reasoning:** Some products of irrational numbers can be rational. For example, \( \sqrt{2} \times \sqrt{2} = 2 \), which is rational. However, \( \sqrt{2} \times \sqrt{3} \) is irrational. **(5) The product of two real numbers is always a real number.** - **Answer:** **True** - **Reasoning:** The set of real numbers is closed under multiplication, meaning the product of any two real numbers is also a real number. **(6) When a real number is divided by a real number, the result is always a real number.** - **Answer:** **False** - **Reasoning:** Division by zero is undefined. If the denominator is zero, the result is not a real number. **(7) The sum of a rational number and an irrational number is always an irrational number.** - **Answer:** **True** - **Reasoning:** Suppose \( r \) is rational and \( x \) is irrational. If \( r + x \) were rational, then \( x = (r + x) - r \) would also be rational (since the difference of two rational numbers is rational), which contradicts the assumption that \( x \) is irrational. **(8) The product of a rational number and an irrational number is always an irrational number.** - **Answer:** **False** - **Reasoning:** If the rational number is zero, the product is zero, which is rational (e.g., \( 0 \times \sqrt{2} = 0 \)). Therefore, the statement is not always true. --- ### Problem (h): Between Which Two Consecutive Integers Does \( \sqrt{97} \) Lie? **Answer:** Between **9** and **10**. - **Explanation:** - \( 9^2 = 81 \) - \( 10^2 = 100 \) - Since \( 81 < 97 < 100 \), taking square roots, we have \( 9 < \sqrt{97} < 10 \). Therefore, \( \sqrt{97} \) lies between the consecutive integers 9 and 10.