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This is part of a series on common misconceptions.
True or False?
Why some people say it's true: is rational, so must also be rational.
Why some people say it's false: is rational while is imaginary, so can't be rational.
The statement is .
Proof: Just being rational (and real) is not enough to proclaim rational, but it can proclaim imaginary instead. After all, the square root of (and all negatives) is imaginary, so if is rational, will not be unless as if is positive then is negative (zero is neither positive nor negative). So, in fact, we have real (rational) + imaginary (and neither rational nor irrational), which (unless the imaginary term is 0, or the real term is 0, or both) is complex — neither purely imaginary nor purely real — and cannot be rational.
Rebuttal: 1 is rational
Reply: Yes, but isn’t, so is complex.