Post here to share useful tips and tricks, to ask questions about using your DM42 or to report software-related problems
jant
Posts: 9
Joined: Sun Aug 16, 2020 10:07 am
Location: Germany

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?
salvomic
Posts: 153
Joined: Sat Dec 30, 2017 10:09 am
Location: Ragusa, Sicily
Contact:

### Re: Question about a result

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
∫aL√0mic (IT9CLU) :: DM42 (SN: 00881), DM41X (SN 00523), DM16, HP Prime, 50g, 41CX, 42s, 71b, 15C, 12C, 35s, WP34s -- Free42
jant
Posts: 9
Joined: Sun Aug 16, 2020 10:07 am
Location: Germany

### Re: Question about a result

Thank you very much, I will look into that!
I have to say, I'm not very familiar with complex numbers in general..
salvomic
Posts: 153
Joined: Sat Dec 30, 2017 10:09 am
Location: Ragusa, Sicily
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### Re: Question about a result

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
Posts: 24
Joined: Sat Dec 28, 2019 4:50 am
Location: Bavaria, Germany

### Re: Question about a result

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?
Walter
Posts: 1745
Joined: Tue May 02, 2017 11:13 am
Location: Close to FRA, Germany

### Re: Question about a result

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
Posts: 838
Joined: Tue May 02, 2017 5:48 pm
Location: United States
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### Re: Question about a result

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
Posts: 32
Joined: Sun Jan 26, 2020 5:21 pm
Location: Lund/Sweden

### Re: Question about a result

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]
akaTB
Posts: 570
Joined: Tue May 02, 2017 1:56 pm
Location: Milan, Italy

### Re: Question about a result

Even the HP6s (with [xth_root_y] key) returns -1.0457879178
Greetings,
Massimo
ajcaton
-+×÷ left is right and right is wrong Casted in gold
rawi
Posts: 24
Joined: Sat Dec 28, 2019 4:50 am
Location: Bavaria, Germany

### Re: Question about a result

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.