How do you evaluate $\sqrt { - 328}$ ?

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Answer 1 · 2018-01-25T20:58:24

$sqrt(-328) = 2sqrt(82)i = 2i(9+1/(18+1/(18+1/(18+1/(18+1/(18+...))))))$

Note that $18^2 = 324$ , while:

$328 = 2^3 * 41$

Hence we find:

$sqrt(-328) = sqrt(2^2*82)i = 2sqrt(82)i$

is an irrational multiple of $i$ , a little larger than $18i$

Since $82 = 9^2+1$ we find that $sqrt(82)$ has a simple continued fraction expansion:

$sqrt(82) = [9; bar(18)] = 9+1/(18+1/(18+1/(18+1/(18+1/(18+...)))))$

So we could write:

$sqrt(-328) = 2i(9+1/(18+1/(18+1/(18+1/(18+1/(18+...))))))$

How do you evaluate \sqrt { - 328}\sqrt { - 328}?