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If 'N' is an irrational number. Is '2N' rational or irrational? why?

If 'N' is irrational, then so is '2N'.

'N' is irrational. If '2N' were rational, then '2N' can be expressed as a ratio of integers: 2N=i/j. By dividing both sides by 2, N=i/(2j). Since 'j' is an integer, so is 2j. That would mean i/(2j)=N is a ratio of integers, a rational number. We already know N is not rational. As a result, our original assumption that 2N is rational cannot be true: 2N is irrational.

Dr. Mellendorf
Illinois Central College

Irrational. Here is why: N is irrational. Assume, 2*N is rational. This means: R= 2*N=p/q where 'p' and 'q' are integers. But then R/2 = p/2q is also rational. But R/2 = N would have to be rational, which contradicts the stipulation that N is irrational. QED by contradiction.

Vince Calder

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