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Prove that 1/√2 is irrational. |
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Answer» Let us assume 2\u200b1\u200b is rational. So we can write this number as 2\u200b1\u200b=ba\u200b ---- (1) Here, a and b are two co-prime numbers and b is not equal to zero. Simplify the equation (1) multiply by 2\u200bboth sides, we get 1=ba2\u200b\u200bNow, divide by b, we get b=a2\u200b or ab\u200b=2\u200bHere, a and b are integers so, ab\u200b is a rational number, so 2\u200b should be a rational number. But 2\u200b is a irrational number, so it is contradictory. Therefore, 2\u200b1\u200b is irrational number Let us assume\xa02\u200b1\u200b\xa0is rational.So we can write this number as2\u200b1\u200b=ba\u200b\xa0---- (1)Here,\xa0a\xa0and\xa0b\xa0are two co-prime numbers and\xa0b\xa0is not equal to zero.Simplify the equation (1) multiply by\xa02\u200bboth sides, we get1=ba2\u200b\u200bNow, divide by\xa0b, we getb=a2\u200b\xa0or\xa0ab\u200b=2\u200bHere,\xa0a\xa0and\xa0b\xa0are integers so,\xa0ab\u200b\xa0is a rational number,\xa0so\xa02\u200b\xa0should be a rational number.But\xa02\u200b\xa0is a irrational number, so it is contradictory.Therefore,\xa02\u200b1\u200b\xa0is irrational number |
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