Question about a result

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jant
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Question about a result

Post by jant »

Hello everyone,

I have a question concerning real and complex results. Consider the following calculation:

\(\begin{eqnarray}
x^{11} &=& -\frac{18}{11}
\end{eqnarray}\)


Trying to calculate the result for x should give me the real number \(\begin{eqnarray}x \cong -1,046\end{eqnarray}\).
My DM42/Free42 gives me the complex result \(\begin{eqnarray}x \cong 1,0034 + 0,2946i\end{eqnarray}\).

When I disable compelx results the calculator shows me an error "Invalid Data". The same goes for (I think) any odd root for a negative number.
Can the DM42/Free42 give the real result for such a calculation anyhow?
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salvomic
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Re: Question about a result

Post by salvomic »

both result are correct: they are two of the 11 complex roots of the number 11√(-18/11).
See here and try with the program of DaveGary (or try in DM 41X with Math 1 pack module)

The 6th is real, the other are complex, DM42 and Free42 show the first, the Dave's program all.

Salvo
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jant
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Re: Question about a result

Post by jant »

Thank you very much, I will look into that!
I have to say, I'm not very familiar with complex numbers in general..
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salvomic
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Re: Question about a result

Post by salvomic »

jant wrote:
Sun Jan 17, 2021 3:17 pm
Thank you very much, I will look into that!
I have to say, I'm not very familiar with complex numbers in general..
no problem at all :)
enjoy with DM42!

Salvo

EDIT: a bit of theory of nth Root of a complex number is here
∫aL√0mic (IT9CLU) :: DM42 (SN: 00881), DM41X (SN 00523), DM16, HP Prime, 50g, 41CX, 42s, 71b, 15C, 12C, 35s, WP34s -- Free42
rawi
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Re: Question about a result

Post by rawi »

I think it strange that HP calculators and derivates are not able to find the real solution -1.0458. I tried with HP 41, DM 15, DM 41X and HP prime. They all deliver an error message. If I try it with HP prime in CAS mode it delivers the imaginary solution 1.0034 + 0.2946i.

On the other side TI 89 Titanium and TI Nspire CX CAS deliver the right real result -1.0458.

Does anybody know why there is this difference?
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Walter
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Re: Question about a result

Post by Walter »

18 [+/-] [ENTER] 11 [/] 11 [xth_root_y] returns -1.045787917714882796046987298089874 on the WP43S. Just try it yourself.

For comparison: 18 [+/-] [ENTER] 11 [/] 11 [1/x] [y^x] returns
1.003426158938366206140354971348624 + ix0.2946325039716886208438314513741246.

I guess that taking an odd (integer) root of negative number, the sign is separated, then the root is calculated of the absolute value, and the sign is added eventually.

(EDIT: corrected an error)
DM42 SN: 00041 β
WP 43S running on this device

HP-35, HP-45, ..., HP-35S, WP 34S, WP 31S, DM16L
Thomas Okken
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Re: Question about a result

Post by Thomas Okken »

rawi wrote:
Tue Jan 19, 2021 9:16 am
I think it strange that HP calculators and derivates are not able to find the real solution -1.0458.
Since what you're calculating is not actually the 11th root but the 1/11th power, the results you are getting are correct.

If a calculator has an "x-th root of y" button, it is not unreasonable to expect it to find the real 11th root of a negative number, but when calculating a power, the result is exp(x*ln(y)), and there is no ambiguity in what the result of that should be.
ecsfang
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Re: Question about a result

Post by ecsfang »

I think it strange that HP calculators and derivates are not able to find the real solution -1.0458. I tried with HP 41, DM 15, DM 41X and HP prime.
The HP35s does (since it also has a [xth_root_y] key) ...

18 [+/-] [ENTER] 11 [/] 11 [xth_root_y] returns -1.04578791771 on the HP35s.
[35/45/65/67/25/29C/31E/33E/41C|CV|CX/71B/10C/11C/15C/16C/32SII/42S/28S/48GX/49G/35S/DM41X(#00456)]
(7397)[134]
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akaTB
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Re: Question about a result

Post by akaTB »

Even the HP6s (with [xth_root_y] key) returns -1.0457879178
:mrgreen:
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rawi
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Re: Question about a result

Post by rawi »

Thank you Walter, Thomas Okken, ecsfang and akaTB for your answers. Now it became clearer to me.

I tried as well with the TI calculators with using exact result and they deliver (-11^(10/11)*3^(2/11)*2^(1/11))/11.
If you use the approx key they obviously calculate with CAS first this result which is approximated after that delivering -1.0458. The HPs (with the exception of the Prime) do not have CAS so they cannot use this.
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