Paul Alsing wrote:

> On Wednesday, November 24, 2021 at 7:23:34 PM UTC-8, Thomas 'PointedEars'
> Lahn wrote:
>> Paul Alsing wrote:
>>
>> > You sure wasted a lot of time over essentially nothing!
>
>> I did waste time as apparent *now*, because you want to stay ignorant,
>> and "right" instead of educated.
>
> Well, I have university degrees in math, physics, and astronomy, so I
> don't consider myself to be uneducated... even if it was over 50 years
> ago...
>
>> > i = the square root of -1... end of discussion.
>
>> Repeating nonsense does not make it true.
>
> If you think this is nonsense then I'm pretty sure I know who the
> uneducated guy here is. This very basic principle is in virtually every
> math textbook in print! Do you also think that math textbooks are just
> bullshit? I would wager that this is stated in *your* textbooks, too.
>
> Like I said before, if you are convinced that I am wrong and you are right
> about this really basic definition, then you have a whole lot of sources
> to set straight because there are hundreds of thousands of them out there
> who still state that i is defined to be the square root of -1...

Since you are fond of Web sources, you will find this interesting, which
confirms everything that I said (and I doubt you have even read carefully,
if at all):

<https://en.wikipedia.org/wiki/Imaginary_unit>

And since you have studied, you will know (or be able to find out) on which
principles Wikipedia is based; you will follow the references, and won’t
give me that laypersons’ “Wikipedia is not a (reliable) source” shit.

PointedEars
--
«Nec fasces, nec opes, sola artis sceptra perennant.»
(“Neither high office nor power, only the scepters of science survive.”)

—Tycho Brahe, astronomer (1546-1601): inscription at Hven

Thomas 'PointedEars' Lahn wrote:

> Paul Alsing wrote:
>> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
>> Lahn wrote:
>>> Paul Alsing wrote: '
>>> > "It (i) stands for the square root of negative one"
>>
>>> This is wrong.
>>> > ... and there are thousands more, but at this point in time, I'm
>>> > bored...
>>
>>> No, you are incompetent and can’t read.
>>
>> Here, watch this...
>>
>> https://www.youtube.com/watch?v=cUzklzVXJwo
>>
>> ... and go ahead and skip to about 17:40 and watch for just a little
>> while... where you will learn that Euler was the first to label "i = the
>> square root of -1"...
>
> Fascinating video (thank you), as previously with Veritasium – apropos,
> his video on gravity in general relativity is congenial¹ –, and I will
> check the claim there that actually Euler defined 𝕚 this way.

This claim is refuted here:

,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
|
| […]
| In this translation, I have sometimes written i where Euler has written
| √(−1). We must note, however, that this substitution is not completely
| unproblematical, and should not be done mechanically. Euler knew very
^^^^^^^^^^^^^^^
| well, and articulates in several places in this article, that √(−1) has a
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| dual nature, that every quantity has two square roots, −1 included. So
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| when Euler is referring to that dual nature, it can be odd, or even
| misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
| is close enough, although not identical to, our notion of i that it seems
| justified to use i, in order not to distract the contemporary reader. But
| the reader should note that, in this article, Euler never wrote i, but
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| always wrote √(−1).”
^^^^^^^^^^^^^^^^^^

PointedEars

Thomas 'PointedEars' Lahn wrote:

> Paul Alsing wrote:
>> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
>> Lahn wrote:
>>> Paul Alsing wrote: '
>>> > "It (i) stands for the square root of negative one"
>>
>>> This is wrong.
>>> > ... and there are thousands more, but at this point in time, I'm
>>> > bored...
>>
>>> No, you are incompetent and can’t read.
>>
>> Here, watch this...
>>
>> https://www.youtube.com/watch?v=cUzklzVXJwo
>>
>> ... and go ahead and skip to about 17:40 and watch for just a little
>> while... where you will learn that Euler was the first to label "i = the
>> square root of -1"...
>
> Fascinating video (thank you), as previously with Veritasium – apropos,
> his video on gravity in general relativity is congenial¹ –, and I will
> check the claim there that actually Euler defined 𝕚 this way.

This claim is refuted here:

,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
|
| […]
| In this translation, I have sometimes written i where Euler has written
| √(−1). We must note, however, that this substitution is not completely
| unproblematical, and should not be done mechanically. Euler knew very
^^^^^^^^^^^^^^^
| well, and articulates in several places in this article, that √(−1) has a
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| dual nature, that every quantity has two square roots, −1 included. So
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| when Euler is referring to that dual nature, it can be odd, or even
| misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
| is close enough, although not identical to, our notion of i that it seems
| justified to use i, in order not to distract the contemporary reader. But
| the reader should note that, in this article, Euler never wrote i, but
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
| always wrote √(−1).”
^^^^^^^^^^^^^^^^^^

This source is cited at the end of
<https://en.wikipedia.org/wiki/Imaginary_unit>.

--
PointedEars

Op 25-nov.-2021 om 03:11 schreef Thomas 'PointedEars' Lahn:
> Paul Alsing wrote:
>
>> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
>> Lahn wrote:
>>> Paul Alsing wrote: '
>>
>>>> "It (i) stands for the square root of negative one"
>>
>>> This is wrong.
>>>> ... and there are thousands more, but at this point in time, I'm
>>>> bored...
>>
>>> No, you are incompetent and can’t read.
>>
>> Here, watch this...
>>
>> https://www.youtube.com/watch?v=cUzklzVXJwo
>>
>> ... and go ahead and skip to about 17:40 and watch for just a little
>> while... where you will learn that Euler was the first to label "i = the
>> square root of -1"...
>
> Fascinating video (thank you), as previously with Veritasium – apropos, his
> video on gravity in general relativity is congenial¹ –, and I will check the
> claim there that actually Euler defined 𝕚 this way.
>
> But regardless whether and how *Euler* defined it *then*, that does not
> change the mathematical facts that we know *today*:
>
> √(−1) = ±𝕚,

If you write that, you should also write
√(1) = ± 1
which is wrong and stupid.

That is the reason why we never write √(−1) to begin with.
We write i.

Dirk Vdm

Paul Alsing wrote:
> On Wednesday, November 24, 2021 at 7:23:34 PM UTC-8, Thomas 'PointedEars' Lahn wrote:
>> Paul Alsing wrote:
>>
>>> You sure wasted a lot of time over essentially nothing!
>
>> I did waste time as apparent *now*, because you want to stay ignorant,
>> and "right" instead of educated.
>
> Well, I have university degrees in math, physics, and astronomy, so I don't consider myself to be uneducated... even if it was over 50 years ago...
>
>>> i = the square root of -1... end of discussion.
>
>> Repeating nonsense does not make it true.
>
> If you think this is nonsense then I'm pretty sure I know who the uneducated guy here is. This very basic principle is in virtually every math textbook in print! Do you also think that math textbooks are just bullshit? I would wager that this is stated in *your* textbooks, too.
>
> Like I said before, if you are convinced that I am wrong and you are right about this really basic definition, then you have a whole lot of sources to set straight because there are hundreds of thousands of them out there who still state that i is defined to be the square root of -1...

Defining i as the square root of -1 is definitely NOT a rigorous
definition.

i is the equivalence class of the polynomials X in the quotient
set of R[X] by the equivalence relation ~ defined as P ~ Q iff
P = Q modulo (X^2+1).

While C is defined as this set of equivalence class.

Michael Moroney wrote:

>> both the muon and the lab scientists? If none of you can produce a
>> calculation showing 4.4 microseconds from both points of view then I
>> guess my mission here is complete and goes down as yet another victory.
>>
> There he goes, little puff of smoke, declaring victory as the door hits
> him on the ass on the way out. Hurried little footfalls down the damp
> alley, around the dumpster and away.

no, you are stupid not understanding where relativity is wrong. Possibly
also, other case why.

El jueves, 25 de noviembre de 2021 a las 4:03:44 UTC-3, Thomas 'PointedEars' Lahn escribió:
> Thomas 'PointedEars' Lahn wrote:
>
> > Paul Alsing wrote:
> >> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
> >> Lahn wrote:
> >>> Paul Alsing wrote: '
> >>> > "It (i) stands for the square root of negative one"
> >>
> >>> This is wrong.
> >>> > ... and there are thousands more, but at this point in time, I'm
> >>> > bored...
> >>
> >>> No, you are incompetent and can’t read.
> >>
> >> Here, watch this...
> >>
> >> https://www.youtube.com/watch?v=cUzklzVXJwo
> >>
> >> ... and go ahead and skip to about 17:40 and watch for just a little
> >> while... where you will learn that Euler was the first to label "i = the
> >> square root of -1"...
> >
> > Fascinating video (thank you), as previously with Veritasium – apropos,
> > his video on gravity in general relativity is congenial¹ –, and I will
> > check the claim there that actually Euler defined 𝕚 this way..
> This claim is refuted here:
>
> ,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
> |
> | […]
> | In this translation, I have sometimes written i where Euler has written
> | √(−1). We must note, however, that this substitution is not completely
> | unproblematical, and should not be done mechanically. Euler knew very
> ^^^^^^^^^^^^^^^
> | well, and articulates in several places in this article, that √(−1) has a
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> | dual nature, that every quantity has two square roots, −1 included. So
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> | when Euler is referring to that dual nature, it can be odd, or even
> | misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
> | is close enough, although not identical to, our notion of i that it seems
> | justified to use i, in order not to distract the contemporary reader. But
> | the reader should note that, in this article, Euler never wrote i, but
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> | always wrote √(−1).”
> ^^^^^^^^^^^^^^^^^^
>
> This source is cited at the end of
> <https://en.wikipedia.org/wiki/Imaginary_unit>.
>
> --
> PointedEars

At the beginning of that same wiki article it reads:

"The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.

Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.

There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root)".

Paparios wrote:

> "The imaginary unit or unit imaginary number (i) is a solution to the
> quadratic equation x^2 + 1 = 0. Although there is no real number with
> this property, i can be used to extend the real numbers to what are
> called complex numbers, using addition and multiplication. A simple
> example of the use of i in a complex number is 2 + 3i.
>
> Imaginary numbers are an important mathematical concept, which extend
> the real number system R to the complex number system C , in which at
> least one root for every nonconstant polynomial exists (see Algebraic
> closure and Fundamental theorem of algebra). Here, the term "imaginary"
> is used because there is no real number having a negative square.

good point. It proves, one more time, that negative numbers are
fictitious, being something about how you move a zero on a scale.

Op 25-nov.-2021 om 16:46 schreef Paparios:
> El jueves, 25 de noviembre de 2021 a las 4:03:44 UTC-3, Thomas 'PointedEars' Lahn escribió:
>> Thomas 'PointedEars' Lahn wrote:
>>
>>> Paul Alsing wrote:
>>>> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
>>>> Lahn wrote:
>>>>> Paul Alsing wrote: '
>>>>>> "It (i) stands for the square root of negative one"
>>>>
>>>>> This is wrong.
>>>>>> ... and there are thousands more, but at this point in time, I'm
>>>>>> bored...
>>>>
>>>>> No, you are incompetent and can’t read.
>>>>
>>>> Here, watch this...
>>>>
>>>> https://www.youtube.com/watch?v=cUzklzVXJwo
>>>>
>>>> ... and go ahead and skip to about 17:40 and watch for just a little
>>>> while... where you will learn that Euler was the first to label "i = the
>>>> square root of -1"...
>>>
>>> Fascinating video (thank you), as previously with Veritasium – apropos,
>>> his video on gravity in general relativity is congenial¹ –, and I will
>>> check the claim there that actually Euler defined 𝕚 this way.
>> This claim is refuted here:
>>
>> ,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
>> |
>> | […]
>> | In this translation, I have sometimes written i where Euler has written
>> | √(−1). We must note, however, that this substitution is not completely
>> | unproblematical, and should not be done mechanically. Euler knew very
>> ^^^^^^^^^^^^^^^
>> | well, and articulates in several places in this article, that √(−1) has a
>> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>> | dual nature, that every quantity has two square roots, −1 included. So
>> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>> | when Euler is referring to that dual nature, it can be odd, or even
>> | misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
>> | is close enough, although not identical to, our notion of i that it seems
>> | justified to use i, in order not to distract the contemporary reader. But
>> | the reader should note that, in this article, Euler never wrote i, but
>> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>> | always wrote √(−1).”
>> ^^^^^^^^^^^^^^^^^^
>>
>> This source is cited at the end of
>> <https://en.wikipedia.org/wiki/Imaginary_unit>.
>>
>> --
>> PointedEars
>
> At the beginning of that same wiki article it reads:
>
> "The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
>
> Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
>
> There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root)".
>

Yes, there are two square roots, and the principal square root is
taken to be i, just like the priicipla square root of 9, is √9 = 3.

Look at how the notation √(−1) is carefully avoided in the article,
and at the warnings given there.
There is a good reason for writing i.
If one really insists on writing √(−1), then that's just i.
So writing
√(−1) = +/- i
is silly.

See also the articles
https://en.wikipedia.org/wiki/Square_root
https://en.wikipedia.org/wiki/Complex_number

And remember these, from Androcles's good old days?
https://home.deds.nl/~dvdm/dirk/Physics/Fumbles/SqrtAnswers.html
https://home.deds.nl/~dvdm/dirk/Physics/Fumbles/NegSqrt.html
https://home.deds.nl/~dvdm/dirk/Physics/Fumbles/SqrtRev.html
;-)

Dirk Vdm

We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent. The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)

p and ~p

As we all know from whence the time we were babes, from the above contradictory premise anything, anything at all, can be proved in the arithmetic.

Capisci?

Thursday, November 25, 2021 at 7:46:37 AM UTC-8, Paparios wrote:
> El jueves, 25 de noviembre de 2021 a las 4:03:44 UTC-3, Thomas 'PointedEars' Lahn escribió:
> > Thomas 'PointedEars' Lahn wrote:
> >
> > > Paul Alsing wrote:
> > >> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
> > >> Lahn wrote:
> > >>> Paul Alsing wrote: '
> > >>> > "It (i) stands for the square root of negative one"
> > >>
> > >>> This is wrong.
> > >>> > ... and there are thousands more, but at this point in time, I'm
> > >>> > bored...
> > >>
> > >>> No, you are incompetent and can’t read.
> > >>
> > >> Here, watch this...
> > >>
> > >> https://www.youtube.com/watch?v=cUzklzVXJwo
> > >>
> > >> ... and go ahead and skip to about 17:40 and watch for just a little
> > >> while... where you will learn that Euler was the first to label "i = the
> > >> square root of -1"...
> > >
> > > Fascinating video (thank you), as previously with Veritasium – apropos,
> > > his video on gravity in general relativity is congenial¹ –, and I will
> > > check the claim there that actually Euler defined 𝕚 this way.
> > This claim is refuted here:
> >
> > ,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
> > |
> > | […]
> > | In this translation, I have sometimes written i where Euler has written
> > | √(−1). We must note, however, that this substitution is not completely
> > | unproblematical, and should not be done mechanically. Euler knew very
> > ^^^^^^^^^^^^^^^
> > | well, and articulates in several places in this article, that √(−1) has a
> > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> > | dual nature, that every quantity has two square roots, −1 included. So
> > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> > | when Euler is referring to that dual nature, it can be odd, or even
> > | misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
> > | is close enough, although not identical to, our notion of i that it seems
> > | justified to use i, in order not to distract the contemporary reader. But
> > | the reader should note that, in this article, Euler never wrote i, but
> > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> > | always wrote √(−1).”
> > ^^^^^^^^^^^^^^^^^^
> >
> > This source is cited at the end of
> > <https://en.wikipedia.org/wiki/Imaginary_unit>.
> >
> > --
> > PointedEars
> At the beginning of that same wiki article it reads:
>
> "The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
>
> Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
>
> There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root)".

On Thursday, November 25, 2021 at 8:48:04 AM UTC-8, patdolan wrote:
> We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent. The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)
>
> p and ~p
>
> As we all know from whence the time we were babes, from the above contradictory premise anything, anything at all, can be proved in the arithmetic.
>
> Capisci?

Extra credit: Which of you foolish fools can detect and type out the contradictive premise in the relativity?

> Thursday, November 25, 2021 at 7:46:37 AM UTC-8, Paparios wrote:
> > El jueves, 25 de noviembre de 2021 a las 4:03:44 UTC-3, Thomas 'PointedEars' Lahn escribió:
> > > Thomas 'PointedEars' Lahn wrote:
> > >
> > > > Paul Alsing wrote:
> > > >> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
> > > >> Lahn wrote:
> > > >>> Paul Alsing wrote: '
> > > >>> > "It (i) stands for the square root of negative one"
> > > >>
> > > >>> This is wrong.
> > > >>> > ... and there are thousands more, but at this point in time, I'm
> > > >>> > bored...
> > > >>
> > > >>> No, you are incompetent and can’t read.
> > > >>
> > > >> Here, watch this...
> > > >>
> > > >> https://www.youtube.com/watch?v=cUzklzVXJwo
> > > >>
> > > >> ... and go ahead and skip to about 17:40 and watch for just a little
> > > >> while... where you will learn that Euler was the first to label "i = the
> > > >> square root of -1"...
> > > >
> > > > Fascinating video (thank you), as previously with Veritasium – apropos,
> > > > his video on gravity in general relativity is congenial¹ –, and I will
> > > > check the claim there that actually Euler defined 𝕚 this way.
> > > This claim is refuted here:
> > >
> > > ,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
> > > |
> > > | […]
> > > | In this translation, I have sometimes written i where Euler has written
> > > | √(−1). We must note, however, that this substitution is not completely
> > > | unproblematical, and should not be done mechanically. Euler knew very
> > > ^^^^^^^^^^^^^^^
> > > | well, and articulates in several places in this article, that √(−1) has a
> > > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> > > | dual nature, that every quantity has two square roots, −1 included. So
> > > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> > > | when Euler is referring to that dual nature, it can be odd, or even
> > > | misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
> > > | is close enough, although not identical to, our notion of i that it seems
> > > | justified to use i, in order not to distract the contemporary reader. But
> > > | the reader should note that, in this article, Euler never wrote i, but
> > > ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> > > | always wrote √(−1).”
> > > ^^^^^^^^^^^^^^^^^^
> > >
> > > This source is cited at the end of
> > > <https://en.wikipedia.org/wiki/Imaginary_unit>.
> > >
> > > --
> > > PointedEars
> > At the beginning of that same wiki article it reads:
> >
> > "The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
> >
> > Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
> >
> > There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root)".

El jueves, 25 de noviembre de 2021 a las 13:07:06 UTC-3, Dirk Van de moortel escribió:
> Op 25-nov.-2021 om 16:46 schreef Paparios:

> >
> > At the beginning of that same wiki article it reads:
> >
> > "The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
> >
> > Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
> >
> > There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root)".
> >
> Yes, there are two square roots, and the principal square root is
> taken to be i, just like the priicipla square root of 9, is √9 = 3.
>
> Look at how the notation √(−1) is carefully avoided in the article,
> and at the warnings given there.
> There is a good reason for writing i.
> If one really insists on writing √(−1), then that's just i.
> So writing
> √(−1) = +/- i
> is silly.
>

Well, if one reads "There are two complex square roots of −1, namely i and −i", is not the same as writing √(−1) = +/- i ?

In my field (electrical engineering) we use the notation "j", instead of "i", to differentiate "i" from the current i. "i" is just a geometrical notation to define a complex plane, such as 3+j4 identifies a phasor of magnitude 5 and angle arctan (4/3). The Euler equation is also fundamental (e^(jx)=cosx + jsinx).

patdolan wrote:
> We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent.

I guess you refer to Gödel incompeteness theorem. This is NOT what the
theorem states. It statest that if such a system is consistent THEN it
is incomplete.

> The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)

i is perfectly well defined as a set of polynomials, it is the
equivalence class of X in C = R[X]/(X^2+1).

"any [real] number multiplied by itself is necessarily non-negative"
does not contradict that in a different set than R (set of real numbers)
some element can have a square of -1 (considered as a representant of an
equivalence class of polynomials).

What about learning some math, Mr Dolan?

On Thursday, November 25, 2021 at 8:54:06 AM UTC-8, Python wrote:
> patdolan wrote:
> > We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent..
> I guess you refer to Gödel incompeteness theorem. This is NOT what the
> theorem states. It statest that if such a system is consistent THEN it
> is incomplete.
> > The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)
> i is perfectly well defined as a set of polynomials, it is the
> equivalence class of X in C = R[X]/(X^2+1).
>
> "any [real] number multiplied by itself is necessarily non-negative"
> does not contradict that in a different set than R (set of real numbers)
> some element can have a square of -1 (considered as a representant of an
> equivalence class of polynomials).
>
> What about learning some math, Mr Dolan?

You can type set theory until your fingertips drop off and your fangs fall out Python. But I have used well formed formula in athematic and exploited the i contradiction to prove that 1=-1 and that -1=1.

1^4 = i^4
sqrt[ 1^4 ] = sqrt[ i^4 ]
+/- 1 = +/-(i^2)
+/-1 = +/-(-1)
+/-1 = -/+1

I can also exploit i to prove anything I want in set theory.

Paparios wrote:

> El jueves, 25 de noviembre de 2021 a las 4:03:44 UTC-3, Thomas
> 'PointedEars' Lahn escribió:
>> <https://en.wikipedia.org/wiki/Imaginary_unit>.
>
> At the beginning of that same wiki article it reads:

It is a _Wikipedia_ article, not a “wiki article”. “Wiki” is a concept, not
a name.

> "The imaginary unit or unit imaginary number (i) is a solution to the
> quadratic equation x^2 + 1 = 0. Although there is no real number with this
> property, i can be used to extend the real numbers to what are called
> complex numbers, using addition and multiplication. A simple example of
> the use of i in a complex number is 2 + 3i.
>
> Imaginary numbers are an important mathematical concept, which extend the
> real number system R to the complex number system C , in which at least
> one root for every nonconstant polynomial exists (see Algebraic closure
> and Fundamental theorem of algebra). Here, the term "imaginary" is used
> because there is no real number having a negative square.
>
> There are two complex square roots of −1, namely i and −i, just as there
> are two complex square roots of every real number other than zero (which
> has one double square root)".

Yes. What is your point?

PointedEars
--
Heisenberg is out for a drive when he's stopped by a traffic cop.
The officer asks him "Do you know how fast you were going?"
Heisenberg replies "No, but I know where I am."
(from: WolframAlpha)

On Thursday, November 25, 2021 at 9:06:24 AM UTC-8, patdolan wrote:
> On Thursday, November 25, 2021 at 8:54:06 AM UTC-8, Python wrote:
> > patdolan wrote:
> > > We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent.
> > I guess you refer to Gödel incompeteness theorem. This is NOT what the
> > theorem states. It statest that if such a system is consistent THEN it
> > is incomplete.
> > > The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p.. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)
> > i is perfectly well defined as a set of polynomials, it is the
> > equivalence class of X in C = R[X]/(X^2+1).
> >
> > "any [real] number multiplied by itself is necessarily non-negative"
> > does not contradict that in a different set than R (set of real numbers)
> > some element can have a square of -1 (considered as a representant of an
> > equivalence class of polynomials).
> >
> > What about learning some math, Mr Dolan?
> You can type set theory until your fingertips drop off and your fangs fall out Python. But I have used well formed formula in athematic and exploited the i contradiction to prove that 1=-1 and that -1=1.
>
> 1^4 = i^4
> sqrt[ 1^4 ] = sqrt[ i^4 ]
> +/- 1 = +/-(i^2)
> +/-1 = +/-(-1)
> +/-1 = -/+1
>
> I can also exploit i to prove anything I want in set theory.

Another interestingly inconsistent concept to explore with online guru Prof.. Wildburger is that of infinity and infinite series. You can make those things do anything you want. This epiphany destroys the most hallowed relation in modern physics: Euler's equation, formula, etc. Euler is a compound fail because it combines both the contradiction i and the inconsistency of infinity and infinite series. Kiss quantum good-bye.

patdolan wrote:
> On Thursday, November 25, 2021 at 8:54:06 AM UTC-8, Python wrote:
>> patdolan wrote:
>>> We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent.
>> I guess you refer to Gödel incompeteness theorem. This is NOT what the
>> theorem states. It statest that if such a system is consistent THEN it
>> is incomplete.

well, at least you may have learn that...

>>> The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)
>> i is perfectly well defined as a set of polynomials, it is the
>> equivalence class of X in C = R[X]/(X^2+1).
>>
>> "any [real] number multiplied by itself is necessarily non-negative"
>> does not contradict that in a different set than R (set of real numbers)
>> some element can have a square of -1 (considered as a representant of an
>> equivalence class of polynomials).
>>
>> What about learning some math, Mr Dolan?
>
> You can type set theory until your fingertips drop off and your fangs fall out Python. But I have used well formed formula in athematic and exploited the i contradiction to prove that 1=-1 and that -1=1.
>
> 1^4 = i^4
> sqrt[ 1^4 ] = sqrt[ i^4 ]

when used as single-valued function sqrt(x) return the
principal branch of the multi-valued sqrt, i.e., it
implies:

1 = 1

and NOT this:

> +/- 1 = +/-(i^2)

(which is basically meaningless btw)

patdolan wrote:
> On Thursday, November 25, 2021 at 9:06:24 AM UTC-8, patdolan wrote:
>> On Thursday, November 25, 2021 at 8:54:06 AM UTC-8, Python wrote:
>>> patdolan wrote:
>>>> We all know from the time we were babies that any deductive system powerful enough to express basic arithmetic is both incomplete and inconsistent.
>>> I guess you refer to Gödel incompeteness theorem. This is NOT what the
>>> theorem states. It statest that if such a system is consistent THEN it
>>> is incomplete.
>>>> The reason i proves to be so important to modern mathematics, and the physics which employs it, is that it expresses a negation of a basic premise of the arithmetic, namely, that "any number multiplied by itself is necessarily non-negative". Let us express the statement in quotation marks as p. In the propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)
>>> i is perfectly well defined as a set of polynomials, it is the
>>> equivalence class of X in C = R[X]/(X^2+1).
>>>
>>> "any [real] number multiplied by itself is necessarily non-negative"
>>> does not contradict that in a different set than R (set of real numbers)
>>> some element can have a square of -1 (considered as a representant of an
>>> equivalence class of polynomials).
>>>
>>> What about learning some math, Mr Dolan?
>> You can type set theory until your fingertips drop off and your fangs fall out Python. But I have used well formed formula in athematic and exploited the i contradiction to prove that 1=-1 and that -1=1.
>>
>> 1^4 = i^4
>> sqrt[ 1^4 ] = sqrt[ i^4 ]
>> +/- 1 = +/-(i^2)
>> +/-1 = +/-(-1)
>> +/-1 = -/+1
>>
>> I can also exploit i to prove anything I want in set theory.
>
> Another interestingly inconsistent concept to explore with online guru Prof. Wildburger is that of infinity and infinite series. You can make those things do anything you want. This epiphany destroys the most hallowed relation in modern physics: Euler's equation, formula, etc. Euler is a compound fail because it combines both the contradiction i and the inconsistency of infinity and infinite series. Kiss quantum good-bye.
>

well... you are bad at physics and bad at math. Like most cranks.

Op 25-nov.-2021 om 17:53 schreef Paparios:
> El jueves, 25 de noviembre de 2021 a las 13:07:06 UTC-3, Dirk Van de moortel escribió:
>> Op 25-nov.-2021 om 16:46 schreef Paparios:
>
>>>
>>> At the beginning of that same wiki article it reads:
>>>
>>> "The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x^2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.
>>>
>>> Imaginary numbers are an important mathematical concept, which extend the real number system R to the complex number system C , in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square.
>>>
>>> There are two complex square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root)".
>>>
>> Yes, there are two square roots, and the principal square root is
>> taken to be i, just like the priicipla square root of 9, is √9 = 3.
>>
>> Look at how the notation √(−1) is carefully avoided in the article,
>> and at the warnings given there.
>> There is a good reason for writing i.
>> If one really insists on writing √(−1), then that's just i.
>> So writing
>> √(−1) = +/- i
>> is silly.
>>
>
> Well, if one reads "There are two complex square roots of −1, namely i and −i", is not the same as writing √(−1) = +/- i ?

And the same as writing √9 = +/- 3 ;-)
Nope. Extremely bad practice. Yikes.
Sure, "two complex square roots of -1..."
And one of them is taken as the principal root. See the
other articles.

Dirk Vdm

>
> In my field (electrical engineering) we use the notation "j", instead of "i", to differentiate "i" from the current i. "i" is just a geometrical notation to define a complex plane, such as 3+j4 identifies a phasor of magnitude 5 and angle arctan (4/3). The Euler equation is also fundamental (e^(jx)=cosx + jsinx).
>

patdolan <patdolan@comcast.net> wrote:
> We all know from the time we were babies that any deductive system
> powerful enough to express basic arithmetic is both incomplete and
> inconsistent. The reason i proves to be so important to modern
> mathematics, and the physics which employs it, is that it expresses a
> negation of a basic premise of the arithmetic, namely, that "any number
> multiplied by itself is necessarily non-negative".

Any real number.

> Let us express the statement in quotation marks as p. In the
> propositional calculus i is equivalent to the statement (and since Bombelli, the PREMISE)
>
> p and ~p
>
> As we all know from whence the time we were babes, from the above
> contradictory premise anything, anything at all, can be proved in the arithmetic.
>
> Capisci?
>
> Thursday, November 25, 2021 at 7:46:37 AM UTC-8, Paparios wrote:
>> El jueves, 25 de noviembre de 2021 a las 4:03:44 UTC-3, Thomas
>> 'PointedEars' Lahn escribió:
>>> Thomas 'PointedEars' Lahn wrote:
>>>
>>>> Paul Alsing wrote:
>>>>> On Saturday, November 20, 2021 at 7:47:06 PM UTC-8, Thomas 'PointedEars'
>>>>> Lahn wrote:
>>>>>> Paul Alsing wrote: '
>>>>>>> "It (i) stands for the square root of negative one"
>>>>>
>>>>>> This is wrong.
>>>>>>> ... and there are thousands more, but at this point in time, I'm
>>>>>>> bored...
>>>>>
>>>>>> No, you are incompetent and can’t read.
>>>>>
>>>>> Here, watch this...
>>>>>
>>>>> https://www.youtube.com/watch?v=cUzklzVXJwo
>>>>>
>>>>> ... and go ahead and skip to about 17:40 and watch for just a little
>>>>> while... where you will learn that Euler was the first to label "i = the
>>>>> square root of -1"...
>>>>
>>>> Fascinating video (thank you), as previously with Veritasium – apropos,
>>>> his video on gravity in general relativity is congenial¹ –, and I will
>>>> check the claim there that actually Euler defined 𝕚 this way.
>>> This claim is refuted here:
>>>
>>> ,-<https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-eulers-notation>
>>>
>>> |
>>> | […]
>>> | In this translation, I have sometimes written i where Euler has written
>>> | √(−1). We must note, however, that this substitution is not completely
>>> | unproblematical, and should not be done mechanically. Euler knew very
>>> ^^^^^^^^^^^^^^^
>>> | well, and articulates in several places in this article, that √(−1) has a
>>> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>> | dual nature, that every quantity has two square roots, −1 included. So
>>> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>> | when Euler is referring to that dual nature, it can be odd, or even
>>> | misleading, to replace √(−1) with i. At other times, Euler's use of √(−1)
>>> | is close enough, although not identical to, our notion of i that it seems
>>> | justified to use i, in order not to distract the contemporary reader. But
>>> | the reader should note that, in this article, Euler never wrote i, but
>>> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>>> | always wrote √(−1).”
>>> ^^^^^^^^^^^^^^^^^^
>>>
>>> This source is cited at the end of
>>> <https://en.wikipedia.org/wiki/Imaginary_unit>.
>>>
>>> --
>>> PointedEars
>> At the beginning of that same wiki article it reads:
>>
>> "The imaginary unit or unit imaginary number (i) is a solution to the
>> quadratic equation x^2 + 1 = 0. Although there is no real number with
>> this property, i can be used to extend the real numbers to what are
>> called complex numbers, using addition and multiplication. A simple
>> example of the use of i in a complex number is 2 + 3i.
>>
>> Imaginary numbers are an important mathematical concept, which extend
>> the real number system R to the complex number system C , in which at
>> least one root for every nonconstant polynomial exists (see Algebraic
>> closure and Fundamental theorem of algebra). Here, the term "imaginary"
>> is used because there is no real number having a negative square.
>>
>> There are two complex square roots of −1, namely i and −i, just as there
>> are two complex square roots of every real number other than zero (which
>> has one double square root)".
>

--
Odd Bodkin — Maker of fine toys, tools, tables

El jueves, 25 de noviembre de 2021 a las 14:24:26 UTC-3, Dirk Van de moortel escribió:
> Op 25-nov.-2021 om 17:53 schreef Paparios:

> > Well, if one reads "There are two complex square roots of −1, namely i and −i", is not the same as writing √(−1) = +/- i ?
> And the same as writing √9 = +/- 3 ;-)
> Nope. Extremely bad practice. Yikes.

Ok, but (-3)^2=(3^2)=9, so both -3 and +3 are square roots of the number 9.

> >

El jueves, 25 de noviembre de 2021 a las 14:13:49 UTC-3, Thomas 'PointedEars' Lahn escribió:
> Paparios wrote:

> > There are two complex square roots of −1, namely i and −i, just as there
> > are two complex square roots of every real number other than zero (which
> > has one double square root)".
> Yes. What is your point?

You wrote a few post above, in a response to Alsing:

"> i = the square root of -1... end of discussion.
Repeating nonsense does not make it true".

Creating a totally irrelevant discussion is complete nonsense!!!

From https://mathworld.wolfram.com/ImaginaryUnit.html

"The imaginary number i=sqrt(-1), i.e., the square root of -1. The imaginary unit is denoted and commonly referred to as "i." Although there are two possible square roots of any number, the square roots of a negative number cannot be distinguished until one of the two is defined as the imaginary unit, at which point +i and -i can then be distinguished. Since either choice is possible, there is no ambiguity in defining i as "the" square root of -1. In the Wolfram Language, the imaginary unit is implemented as I".

Paparios wrote:

> El jueves, 25 de noviembre de 2021 a las 14:13:49 UTC-3, Thomas
> 'PointedEars' Lahn escribió:
>> Paparios wrote:
>> > There are two complex square roots of −1, namely i and −i, just as
>> > there are two complex square roots of every real number other than zero
>> > (which has one double square root)".
>> Yes. What is your point?
>
> You wrote a few post above, in a response to Alsing:
>
> "> i = the square root of -1... end of discussion.
> Repeating nonsense does not make it true".

Yes. Because this statement is wrong, or, in more colorful language,
“nonsense”.
> Creating a totally irrelevant discussion is complete nonsense!!!

The discussion is not irrelevant. There is a difference between the
mathematical statements

i = √(−1)

and

i² = −1.

I have explained that difference, and why it is important. Obviously you
have not read or understood my explanation; therefore you have not
understood the difference and its importance. That is also why you are
pointlessly quoting a Wikipedia article that I cited which reiterates, and
thereby confirms, what I said.
> From https://mathworld.wolfram.com/ImaginaryUnit.html

We have discussed this already. You are not paying attention.

PointedEars
--
Q: Who's on the case when the electricity goes out?
A: Sherlock Ohms.

(from: WolframAlpha)

Op 25-nov.-2021 om 19:06 schreef Paparios:
> El jueves, 25 de noviembre de 2021 a las 14:24:26 UTC-3, Dirk Van de moortel escribió:
>> Op 25-nov.-2021 om 17:53 schreef Paparios:
>
>>> Well, if one reads "There are two complex square roots of −1, namely i and −i", is not the same as writing √(−1) = +/- i ?
>> And the same as writing √9 = +/- 3 ;-)
>> Nope. Extremely bad practice. Yikes.
>
> Ok, but (-3)^2=(3^2)=9, so both -3 and +3 are square roots of the number 9.

Yes, +3 is a square root of 9, and -3 is a square root of 9,
but +3 is the principle square root of 9, and is almost
always referred to as "THE square root" of 9.

Dirk Vdm

El jueves, 25 de noviembre de 2021 a las 15:45:22 UTC-3, Dirk Van de moortel escribió:
> Op 25-nov.-2021 om 19:06 schreef Paparios:
> > El jueves, 25 de noviembre de 2021 a las 14:24:26 UTC-3, Dirk Van de moortel escribió:
> >> Op 25-nov.-2021 om 17:53 schreef Paparios:
> >
> >>> Well, if one reads "There are two complex square roots of −1, namely i and −i", is not the same as writing √(−1) = +/- i ?
> >> And the same as writing √9 = +/- 3 ;-)
> >> Nope. Extremely bad practice. Yikes.
> >
> > Ok, but (-3)^2=(3^2)=9, so both -3 and +3 are square roots of the number 9.
> Yes, +3 is a square root of 9, and -3 is a square root of 9,
> but +3 is the principle square root of 9, and is almost
> always referred to as "THE square root" of 9.
>
> Dirk Vdm

Well,

In mathematics, a square root of a number x is a number y such that y^2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 4^2 = (−4)2 = 16. Every *nonnegative real number* x has a unique nonnegative square root, called the principal square root, which is denoted by √(x) where the symbol √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √(9)=3 because 3^2 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.

Every positive number x has two square roots: √(x) which is positive, and -√(x) which is negative. Together, these two roots are denoted as ± √(x). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive x, the principal square root can also be written in exponent notation, as x^1/2..

Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.