is 1+i rational?
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This is part of a series on common misconceptions.
True or False?
\(1+i\) is rational.
Why some people say it's true: \(\sqrt { 1 } \) is rational, so \(\sqrt { -1 } \) must also be rational.
Why some people say it's false: \(1\) is rational while \(i\) is imaginary, so \(1+i\) can't be rational.
The statement is \( \color{red}{\textbf{false}}\).
Proof: Just \(\sqrt { 1 } \) being rational (and real) is not enough to proclaim \(i\) rational, but it can proclaim \(i\) imaginary instead. After all, the square root of \(-1\) (and all negatives) is imaginary, so if \(\sqrt{x}\) is rational, \(\sqrt{-x}\) will not be \(\big(\)unless \(x=0)\) as if \(x\) is positive then \(-x\) 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 rationalReply: Yes, but \(i\) isn’t, so \(1+i\) is complex.
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