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| 1. |
Show that √2 is irrational. |
| Answer» Ans.Assume{tex} \\sqrt{2}{/tex} is rational,i.e. it can be expressed as a rational fraction of the form {tex}\\frac{b}{a},{/tex}where a and b are two relatively prime integers.Now, since {tex}\\sqrt{2}=\\frac{b}{a}{/tex},squaring both sides,we have\xa0{tex} 2=\\frac{b^2}{a^2}{/tex}, or {tex}b^2=2a^2{/tex}.Since 2a2\xa0is even, b2\xa0must be even, and since b2\xa0is even, so is b.Let b=2c.We have 4c2=2a2\xa0and thus a2=2c2.Since 2c2\xa0is even, a2\xa0is even, and since a2\xa0is even, so is a.However, two even numbers cannot be Co-prime,so {tex}\\sqrt 2{/tex}\xa0cannot be expressed as a rational fraction;Hence {tex}\\sqrt 2{/tex}\xa0is irrational | |