Why won't the prime graph correctly?

[Nigel (UK)]

(05-23-2019 09:14 AM)Aries Wrote:  Quite the contrary, teachers know very well what they are teaching, calculators are stupid and/or badly programmed
Best,

Aries

I suppose different people have different opinions. Here are what the four calculators with me at school (I teach Physics) today say:

• DM42: doing 1 +/- ENTER 3 1/x y^x gives the first complex cube root of -1, rather than -1.
• WP34S: doing 1 +/- ENTER 3 1/x y^x gives DOMAIN ERROR.
• Casio Classwiz fx-85GTX: doing (-1)^(1/3) gives -1.
• HP Prime: (-1)^(1/3) gives an error in Home mode (only integral powers of negative numbers allowed) and a complex result in CAS mode.
  

I like all of these calculators and I think they behave well.

On a calculator that can handle complex numbers, the complex result is what I would expect. If a calculator tells me that \((-1)^{1/3}\) is \(-1\), I don't know what result it will give me for \((-1\pm{\bf i}\epsilon)^{1/3}\). The cut in the complex plane must be somewhere unusual.

On a calculator that handles only real numbers, I prefer an error. The problem is that when the power is a real number - for example, \(-0.4\) - a calculator that tries to give a real answer must first turn the power into a rational number, express it in its lowest terms, and then throw an error only if the denominator is even. The Casio appears to do this - \((-1)^{0.4}\) gives 1, powers of 0.401 up to 0.407 give Math Error, and a power of 0.408 gives -1. It's logical, it's consistent, but is it sensible?

I think that to put the Prime's behaviour down to being "badly programmed" is to miss the point. There is a conceptual difference between a fractional power and an integral nth root: whether to ignore this difference, or if not, how to address it, aren't trivial questions. I think the Prime gets it right by having two separate functions, each behaving as I'd expect. If I were a maths teacher teaching fractional powers at a basic level, I might prefer the behaviour of the Casio. I'm surprised to hear that the TI NSpire appears to follow the Casio - is there a complex mode that needs to be turned on, and if so, does this change the behaviour?

Nigel (UK)