Simple Math Dilemma Revisited Message #1 Posted by Palmer O. Hanson, Jr. on 21 Apr 2011, 9:51 p.m.
In an earlier thread "Simple Math Dilemma" (April 13) Chuck wrote:
Quote:
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 TI8X calculators give the thirdglance result (1) and produces a graph from inf to +inf (the typical cube root function).
Mathematica gives the secondglance result (complex value), and a graph from 0 to +inf accordingly.
Back Ontrack, slightly: my HP15 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 nonhp) 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.
I was surprised by the comment "The TI8X calculators give the thirdglance result (1)" with no mention of the capability to yield the socalled "second glance" result. The actual situation is
1. The TI80, TI81 and TI82 can only give the "third glance" result since those machines do not have a builtin 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 TI83, TI84, TI85, TI86 and TI89 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 TI83, TI84 and TI89, or as (1,0) with the TI85 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 HP41'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 HP41 attempted to perform a meaningless operation." where y<0 and x is noninteger. 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 noninteger powers and get resultst which is the same as that received from my HP28S, TI83, etc. The sequence includes sixteen keystrokes.
On my TI59 the sequence ( 1 +/ ) y^x ( 2 / 6 ) = yields a flashing one error indication. The fourth error condition on page B1 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 HP41C with the Math Pac installed. My friend Richard Nelson would tell me that is one more demonstration of the keystroke efficiency of RPN.
The TI59 with the Master Library module will permit the use of complex exponents. My HP41C 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 HP41 module or standalone 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?
