Prove that 5-root 3 is irrational.
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Let us assume that 5 - root 3 is rational. Then it can be written in the form
5 - root3 = p/q
or 5 - p/q = root3
It implies root3 is a rational number [Since 5 - p/q are rationals]
But this contradicts to the fact that root 3 is irrational. Hence our supposition was wrong. Therefore 5 - root 3 is irrational.
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Let 's assume 3-root5 is rational
3 - root5 = a/b (a and b are co-prime, b is not equal to 0)
-root5 = (a/b) - 3
root5 = 3 - (a/b)
We know that, a, b and 3 are integers which means the RHS is rational. But LHS (Root 5) is irrational as it is given. Here we have LHS = RHS which is a contradiction. Therefore, our assumption is wrong.
Hence, 3 - root5 is irrational.