Re: Cube root of negative number (HP 50G) Message #3 Posted by Hal Bitton in Boise on 18 Sept 2008, 3:37 a.m., in response to message #1 by macky
Hi Macky,
Of course you realize that according to DeMoivre's theorum, there are 3 cube roots of a negative number (or any number, for that matter), which can be nicely represented as complex numbers in polar form, spaced 120 degrees apart. If the original radicand was real, one of the cube roots will also be real, with that real root being positive (lying at 0 degrees) for a positive real radicand, and negative (lying at 180 degrees) for a negative real radicand. The three cube roots of 125 would therefor be:
5 at 60 degrees
5 at 180 degrees (or 5)
5 at 300 degrees
If in rectangular mode, these results would display as:
(2.5, 4.33)
(5, 0)
(2.5, 4.33)
It seems the calculator is under no obligation to return the real root when such a query is put to it, but rather, it returns the first root going CCW from the origin, which is 5 at 60 degrees for the above problem. All 4 of my HP's with complex number capability (50G, 48GX, 42S, 15C) gave me this result. Interestingly, I was unable to get 5 as a displayed result at any time.
I wrote a short RPL program for my 50G a few months back that will return all the nth roots of any number (real or complex). I would be glad to post it if you think you may find it useful. Just let me know.
Hope this helps,
Best regards, Hal
