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| 1. |
Given that root 2 is irrational prove that 5 + 3 root 2 is an irrational number |
| Answer» Let 5\xa0+ 3{tex}\\sqrt{2}{/tex}\xa0be a rational number.{tex}\\Rightarrow{/tex}\xa05\xa0+ 3{tex}\\sqrt{2}{/tex}\xa0=\xa0{tex}\\frac{p}{q}{/tex}, p, q\xa0{tex}\\in{/tex}\xa0I, q\xa0{tex}\\ne{/tex}\xa00{tex}\\Rightarrow{/tex}\xa03{tex}\\sqrt{2}{/tex}\xa0=\xa0{tex}\\frac{p}{q}{/tex}\xa0- 5{tex}\\Rightarrow \\sqrt{2}{/tex}\xa0=\xa0{tex}\\frac{p - 5q}{3q}{/tex}{tex}\\frac{p - 5q}{3q}{/tex}\xa0is rational\xa0{tex}\\Rightarrow{/tex}\xa0{tex}\\sqrt{2}{/tex}\xa0is rational number which is a contradiction.5 + 3{tex}\\sqrt{2}{/tex} is irrational number. | |