In an earlier thread "Simple Math Dilemma" (April 13) Chuck wrote:

Quote:I was surprised by the comment "The TI-8X calculators give the third-glance result (-1)" with no mention of the capability to yield the so-called "second glance" result. The actual situation is

One thing led to another, and it got me thinking about the value of (-1)^(2/6).At first glance I "want" it to be +1, since we have "even" powers, (sort of).

On second glance it could be 1/2 + sqrt(3)/2 i, (the principal sixth root of a negative is complex; and then squared).

But, on third glance, basic algebra says to reduce all rational exponents to lowest terms before evaluating, which gives us -1. Hmmmm.

The TI-8X calculators give the third-glance result (-1) and produces a graph from -inf to +inf (the typical cube root function).

Mathematica gives the second-glance result (complex value), and a graph from 0 to +inf accordingly.

Back On-track, slightly: my HP-15 gives -1 (as expected). Haven't checked the others. my 28S gives the complex answer (as does my Casio) and graphs for only x>0.

I've yet to find one that gives +1 (bummer,even powers and all).

What do other calculators (hp or non-hp) give,and what "should" it be?

I now can only presume TI's are for the less mathematically mature student, and HP's are for the more mathematically advanced. (Ouch!)

But I wait your opinions.

1. The TI-80, TI-81 and TI-82 can only give the "third glance" result since those machines do not have a built-in complex number capability. A program which would yield the "second glance" result may be able to be written for one or more of those machines but I am not aware of one.

2. The TI-83, TI-84, TI-85, TI-86 and TI-89 give the "third glance" result if the -1 is entered as a real number. Those machines will give the "second glance " result if the user enters the -1 as a complex number; i.e., as (-1 + 0i) with the TI-83, TI-84 and TI-89, or as (-1,0) with the TI-85 and 86. All that is required is that the user understand how the machines operate..

"What do other calculators (hp or non hp) give ...?" With my HP-41's the keyboard sequence

1 CHS ENTER 2 ENTER 6 / Shift y^x yields DATA ERROR

where Appendix E (Messages and Errors) says that "The HP-41 attempted to perform a meaningless operation." where y<0 and x is non-integer. If the Math Pac module is installed then the sequence

0 ENTER 1 CHS ENTER 2 ENTER 6 / XEQ ALPHA Z Shift ENTER N ALPHA

yields the "second glance" result even though the instructions say that this method will "Raise z to an integer power." I have tried other non-integer powers and get resultst which is the same as that received from my HP-28S, TI-83, etc. The sequence includes sixteen keystrokes.

On my TI-59 the sequence ( 1 +/- ) y^x ( 2 / 6 ) = yields a flashing one error indication. The fourth error condition on page B-1 of the manual is "Raising a negative number to any power (or root). The power (Ior root) of the absolute value of the number is flashed. With the Master Library module installed the sequence

2nd Pgm 0 4 2 / 6 = A 0 A 1 +/-2nd A 0 2nd A D

yields the "second glance" result after niineteen keystrokes versus the sixteen keystrokes with the HP-41C with the Math Pac installed. My friend Richard Nelson would tell me that is one more demonstration of the keystroke efficiency of RPN.

The TI-59 with the Master Library module will permit the use of complex exponents. My HP-41C with the Math Pac module does not seem to permit the use of complex exponents -- but, maybe I just haven't figured out how to do that. I would expect that some other HP-41 module or stand-alone program will permit the use of complex exponents. I just don't have such a module or program in my collection.

Finally, I note that all the machines I tested can be induced to yield positive one for the problem if the user first squares the -1 value and then raises the intermediate result to the 1/6 power. Of course, that isn't consisten with the problem as written if one interprets parentheses in the standard manner. Positive one can be obtained if a different problem were proposed; i.e., (( -1)^2))(1/6) . Isn't it wonderful what a few added parentheses can achieve?