Let us assume that √3 is a rational number.then, as we know a rational number should be in the form of p/qwhere p and q are co- prime number.So,√3 = p/q { where p and q are co- prime}√3q = pNow, by squaring both the sidewe get,(√3q)² = p²3q² = p² …….. ( i )So,if 3 is the factor of p²then, 3 is also a factor of p ….. ( ii )=> Let p = 3m { where m is any integer }squaring both sidesp² = (3m)²p² = 9m²putting the value of p² in equation ( i )3q² = p²3q² = 9m²q² = 3m²So,if 3 is factor of q²then, 3 is also factor of qSince3 is factor of p & q bothSo, our assumption that p & q are co- prime is wronghence,. √3 is an irrational number
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