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(1) \( \frac{1}{2} \) and \( \frac{1}{3} \) Find an irrational number between \( \begin{array}{lll}\text { (1) } 2 \text { and } 3 & \text { (2) } \\ \text { Say whether the following statements are true or false: } & & \text { (3) } \\ \text { (1) and } \sqrt{5} & \\ \text { (1) and } 2 \pi \\ \text { (2) The sum of two rational numbers is always a rational number. } \\ \text { (3) The sum of two irrational numbers is always a rational number. } \\ \text { (4) The product of two rational numbers is always a rational number }\end{array} \)

Ask by Ward Cruz. in South Africa
Jan 21,2025

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**Irrational Numbers Between Given Intervals:** 1. **Between \( \frac{1}{2} \) and \( \frac{1}{3} \):** - Example: \( \frac{\sqrt{2}}{3} \) (approximately 0.471) 2. **Between 2 and 3:** - Example: \( \sqrt{5} \) (approximately 2.236) **True or False Statements:** 1. **True:** The sum of two rational numbers is always rational. 2. **False:** The sum of two irrational numbers is not always rational. 3. **True:** The product of two rational numbers is always rational. **Summary:** - Irrational numbers can be found between \( \frac{1}{2} \) and \( \frac{1}{3} \), and between 2 and 3. - The sum of two rational numbers is always rational. - The sum of two irrational numbers is not always rational. - The product of two rational numbers is always rational.

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A fun number that lies between 2 and 3 is \( \sqrt{8} \), which is approximately 2.828. Despite being derived from a rational number (8), it can't be expressed as a simple fraction, making it irrational! This sneaky number is a fantastic example of how even between two seemingly straightforward whole numbers, you can find a little element of surprise. Now, onto the statements! The statement about the sum of two rational numbers is definitely true since adding fractions like \( \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \) keeps you within the rational domain. However, the assertion that the sum of two irrational numbers is always a rational number is false; consider \( \sqrt{2} + (-\sqrt{2}) = 0 \), which is rational, but you can also have \( \sqrt{2} + \sqrt{2} = 2\sqrt{2} \), which remains irrational. Keep those numbers on their toes!

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