Wikipedia:Reference desk/Archives/Mathematics/2020 January 16

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January 16

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I am not sure nor complete about, but have a question.

imaginary root of imaginary ,

as real root of imaginary would be for imaginary,

and imaginary would be for "-" , and plotted as another dimentional ,

imaginary root of imaginary may be for "-" root of imaginary , and plotted as another than real root of imaginary ...

Wow! sushi (talk) 01:53, 16 January 2020 (UTC)

I found your question very hard to understand. I think you are asking that "If the square root of a negative real number is an imaginary number which requires a 'second dimension'. Then does the square root of some imaginary number require a 3rd dimension?" If that is your question the answer is "no".

On a sheet of paper, draw the real axis from 0 to some positive number (maybe 10) and extend it the same distance the other way to maybe -10. Then at right angles draw the imaginary axis from +10i through zero to -10i. Now mark +4 (at +4+0i) and consider its square root using rotations. If we treat the vector from zero to +10 as direction 0° or as direction 360° or 360*N°, we can convert +4+0i to a polar coordinate of distance 4 from zero in direction 0 or 360 (or 720). The two square roots of this value are at polar coordinates of radius (positive_square_root of original_radius(4)) and half the bearing. Thus (4,0°) has square root (2,0°) which is +2+0i while (4,360°) has square root at (2,180°) which is -2+0i.

Now if we do the same for 0+4i we have polar coordinate (4,90°) and (4,450°) ((360° more than the first case)). The "positive" square root of (4,90°) is (2,45°) which is at sqrt(2)+sqrt(2)i and the "negative root" of (4,450°) is at (2,225°) which is at -sqrt(2)-sqrt(2)i.

Of course, I may have misunderstood your question. -- SGBailey (talk) 09:51, 16 January 2020 (UTC)

Thank you. Thank you.

Wow! sushi (talk) 05:47, 18 January 2020 (UTC)

As I understood the user asked about ${\sqrt[{i}]{i}}$. Ruslik_Zero 16:51, 16 January 2020 (UTC)

I get ${\sqrt[{i}]{i}}=i^{1/i}=i^{-i}=1/{i^{i}}$. Amusingly, the principal value of $i^{i}$ is a real number, around 0.207, see i^i. So ${\sqrt[{i}]{i}}$ is around 4.8. In general you can define exponentiation in terms of the exp function, $a^{b}=\exp {(b\ln {a})}$. 2601:648:8202:96B0:0:0:0:DF95 (talk) 20:25, 16 January 2020 (UTC)

Thank you. And my question for now , is to , how to find ...

----------------------------wander for now is whether or not another coordinate space is reasonable.

Wow! sushi (talk) 05:47, 18 January 2020 (UTC)

Quaternion may interest you. —Tamfang (talk) 06:06, 18 January 2020 (UTC)

In Quaternion , is the picture part saying it most concisely?

Wow! sushi (talk) 01:43, 19 January 2020 (UTC)

I found , in picture , "i" leads to "k" , and "j" leads to "-k"...

Wow! sushi (talk) 05:00, 21 January 2020 (UTC)