third root of -8...

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toml_12953
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Re: third root of -8...

Post by toml_12953 »

mcc wrote:
Sat Jan 30, 2021 4:59 pm
Hi,

All credits goes to Scott Collins - he found this. See his video
https://www.youtube.com/watch?v=iNR7kcA08As

In short:

-2 * -2 * -2 = -8

On many calculators cuberoot( -8 ) gives -2...as above.

On the DM42 with CRES enabled this gives:
1.0 + i1.73205...
which is one correct answer.
On Prime you get 1+1.73205...*i

Then if you raise Ans to the third power you get

-8.00000000000001-6.422811...e-12 * i
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Thomas Okken
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Re: third root of -8...

Post by Thomas Okken »

ManuMa wrote:
Sun Jan 31, 2021 9:07 pm
Hi Thomas,
Sorry, but in my monitor, the square root symbol appears almost exactly the same as the division one, and that confused me. In fact I had to apply a very big zoom before realizing that you in fact wrote 1+i sqrt(3), and not 1+i/3, as I first read.
Oh! I see. That's not good, I wasn't aware that that symbol gets rendered so poorly on some systems. Guess I should stick to writing sqrt() or using MathJax or something like that!
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Re: third root of -8...

Post by Thomas Okken »

This is what it looks like in Firefox on my Mac:
Screen Shot 2021-01-31 at 4.42.42 PM.png
Screen Shot 2021-01-31 at 4.42.42 PM.png (20.55 KiB) Viewed 290 times
HPMike
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Re: third root of -8...

Post by HPMike »

Both -2 and 1+i sqrt(3) are valid roots. In fact, there are three valid roots, the third being 1-i sqrt(3). If you look at this in polar form, you see that all three numbers have the same magnitude of 2, and with phase angles of 0, 60 and -60 degrees respectively. So, in reality, the calculator is giving only one of the three valid roots. The real function xth root of y gives the one real root, and the complex function y to the xth power gives one of the two complex roots. Of course, a real number is just a special case of a complex number in which the complex part is zero. As long as magnitude is the only needed final result, it should not matter which one of the roots are used in the overall calculation.
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HPMike
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Re: third root of -8...

Post by HPMike »

Thomas Okken wrote:
Sun Jan 31, 2021 10:44 pm
This is what it looks like in Firefox on my Mac:

Screen Shot 2021-01-31 at 4.42.42 PM.png
Yeah, that's much better than on my 25" FHD monitor in Chrome. I also made the same mistake and first though you had 1/3 instead of sqrt(3).
DM15L, S/N 00548. DM42, SN: 00159. DM41X, SN: 00973.
richmit
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Re: third root of -8...

Post by richmit »

On an algebraic calculator like the TI-89 or the Casio Classpad, you can type in (-8)^(1/3). In "complex" mode you get the complex root, and in "real" mode you get the real one. In this case the calculator can "see" the entire equation, and notice the special case of a rational power with an odd denominator.

On an RPN calculator without an Nth root function, the software has no opportunity to notice you are raising to a rational power with an odd denominator. It just sees you raising -8 to the power of .33333333 (the *truncated* result of taking the reciprocal of 3) -- which is *NOT* the same thing as 1/3.

Now it could do some wacky stuff like try and recognize the truncated real decimal representation as a rational number, but that sort of magic is why my Casio fx-5800p thinks 80143857/25510582 is Pi. I think I would rather not have the magic. ;)

What you can do is code up an Nth root program that notices the special cases, and tries to deliver a real number when it can.
pipedwho
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Re: third root of -8...

Post by pipedwho »

richmit wrote:
Thu Feb 18, 2021 8:40 pm
On an algebraic calculator like the TI-89 or the Casio Classpad, you can type in (-8)^(1/3). In "complex" mode you get the complex root, and in "real" mode you get the real one. In this case the calculator can "see" the entire equation, and notice the special case of a rational power with an odd denominator.

On an RPN calculator without an Nth root function, the software has no opportunity to notice you are raising to a rational power with an odd denominator. It just sees you raising -8 to the power of .33333333 (the *truncated* result of taking the reciprocal of 3) -- which is *NOT* the same thing as 1/3.

Now it could do some wacky stuff like try and recognize the truncated real decimal representation as a rational number, but that sort of magic is why my Casio fx-5800p thinks 80143857/25510582 is Pi. I think I would rather not have the magic. ;)

What you can do is code up an Nth root program that notices the special cases, and tries to deliver a real number when it can.
The complex result is more clear when viewed as a polar result. ie. (-1)^(1/3) is shown as 2 @60 degrees, whereas in rectangular it 1, i1.73... So all three roots are valid, but in real terms, only one of them is expected (-2,0), while the other two are unwanted spurious (1,i1.73..) and (1,-i1.73..).

I agree that an nth Root function would be a worthwhile addition to the DM42, and if not there, then definitely the DM43/WP43S.
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Walter
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Re: third root of -8...

Post by Walter »

pipedwho wrote:
Fri Feb 19, 2021 5:54 am
I agree that an nth Root function would be a worthwhile addition to the DM42, and if not there, then definitely the DM43/WP43S.
It's in the WP43S: just press g[EXP] and you'll see it. And you can try it on your PC or your DM4x.
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