# 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 $\color{#D61F06}{\textbf{false}}$.

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 rational

Reply: Yes, but $i$ isn’t, so $1+i$ is complex.

Rebuttal:

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**See Also**