Question

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

Answer 1

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

Explanation:

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+...))))))`