Q2. a). Find an irrational number that lies between $$ \begin{array}{llll} ext { i). } 2 ext { and } 3 & ext { (ii). } 19 ext { and } 19.01 & ext { (iii), } 4 ext { and }-2\end{array} $$ $$ \begin{array}{lll} ext { b). Prove that } 1+\sqrt{2} & ext { is irrational }\end{array} $$

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Q2. a). Find an irrational number that lies between $$ \begin{array}{llll}	ext { i). } 2 	ext { and } 3 & 	ext { (ii). } 19 	ext { and } 19.01 & 	ext { (iii), } 4 	ext { and }-2\end{array} $$ $$ \begin{array}{lll}	ext { b). Prove that } 1+\sqrt{2} & 	ext { is irrational }\end{array} $$

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### Q2. a) **i. Between 2 and 3:** $$ \sqrt{8} $$ lies between 2 and 3. **ii. Between 19 and 19.01:** $$ 19 + \frac{1}{100} \sqrt{2} $$ lies between 19 and 19.01. **iii. Between 4 and -2:** $$ \sqrt{2} $$ lies between 4 and -2. ### Q2. b) **Proof:** Assume $$ 1 + \sqrt{2} $$ is rational, then it can be expressed as a fraction. Rearranging the equation leads to a contradiction, proving $$ 1 + \sqrt{2} $$ is irrational.

Q2. a). Find an irrational number that lies between \( \begin{array}{llll}\text { i). } 2 \text { and } 3 & \text { (ii). } 19 \text { and } 19.01 & \text { (iii), } 4 \text { and }-2\end{array} \) \( \begin{array}{lll}\text { b). Prove that } 1+\sqrt{2} & \text { is irrational }\end{array} \)

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Q2. a). Find an irrational number that lies between \( \begin{array}{llll}\text { i). } 2 \text { and } 3 & \text { (ii). } 19 \text { and } 19.01 & \text { (iii), } 4 \text { and }-2\end{array} \) \( \begin{array}{lll}\text { b). Prove that } 1+\sqrt{2} & \text { is irrational }\end{array} \)

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