On Monday, June 26, 2023 at 2:29:24 AM UTC-7, patdolan wrote:
> On Sunday, June 25, 2023 at 10:55:32 AM UTC-7, Trevor Lange wrote:
> > On Sunday, June 25, 2023 at 12:54:57 AM UTC-7, patdolan wrote:
> > > > > > This from the dolt who claims the relation (-1)^2 = (1)^2 rigorously implies -1 = 1.
> > > > >
> > > > > That is not my claim at all. My claim is that i^4 = i^8 and the law of exponent rigorously implies -1 = 1.
> > > >
> > > > LOL. That's what I said. Since i^2 = -1, your claim is that the relation (-1)^2 = (1)^2 implies -1 = 1. Wow. There's even a section in the Wikipedia page on Math Fallacies on this kind of idiocy.
> > > > >
> > > > > i^4 = i^8
> > > > > sqrt[ i^4 ] = sqrt[ i^8 ]
> > > >
> > > > LOL. Nope. There you are, as I said, assuming that the relation (-1)^2 = (1)^2 implies -1 = 1.
> > >
> > > Nowhere do I assume or type this.
> > You assumed it (and typed it) in the quote above. Again, since i^2 = -1 and (-1)^2 = 1 unequivocally, your first equality is simply (-1)^2 = (1)^2, and sure enough, the values of the left and right sides are both 1, so you are asserting that 1=1. An auspicious start!
> >
> > But then you assume that this implies sqrt(1) = sqrt(1), which is not unequivocally true, because the expression "sqrt(1)" signifies a number whose square is 1, and there are two such numbers, namely +1 and -1. If you specify that you are referring to +1 on both sides, or that you are referring to -1 on both sides, then equality holds and your second assertion amounts to, again, 1=1. So you're typing the same trivial thing repeatedly. ("All work and no play makes jack a dull boy.") But of course you're not trying to just assert 1=1, you are trying to assert -1=1, which obviously doesn't validly follow.
> >
> > Look, consider the equality sin(0)=sin(pi). Now we take the inverse sine of both sides, and assert that 0 = pi. Well, that's precisely isomorphic to what you're doing, trying to infer from a relation of the form f(x)=f(y) that x=y. If the inverse of f is a multi-valued function, such as inverse sine or square root, that is an obviously false inference. (Now, for multi-valued functions, people sometimes adopt a convention of defining the "principal value", but in the grown-up world the principal value of a function is distinct from the function.)
> > > My valid derivation of a contradiction cleverly exploits the cyclic nature of the powers
> > > of i and the law of exponents.
> > There's nothing clever about what you're doing, it's blatently invalid reasoning, as explained above. What you're describing is standard fallacy, common among slow-witted school children.
> >
> > > Long ago an ad hoc rule was added to arithmetic ... That law expressly
> > > forbad moving a negative sign under a radical sign...
> >
> > Moving a negative sign from outside to inside a radical sign is not an "ad hoc rule", it is an obviously fallacious mistake. Again, this is a very common befuddlement for slow-witted 4th grade school children. The values of -sqrt(1) are +1 and -1, whereas the values of sqrt(-1) are +i and -i. So, "bringing a negative sign inside a radical" is simply a trivial mistake. It's the same as thinking that -cos(x) equals cos(-x)... duh.
>
> [Okay, you've exploded my fallacious belief that bringing a negative sign inside a radical was ever anything other than a juvenile misconception, revealing a complete lack of understanding of 4th grade arithmetic, but, astonishingly undeterred, let me ask about] the ad hoc and arbitrary rule against factoring a positive number into two negative numbers whilst under the radical sign.

There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1, and the expression sqrt((-1)(-1)) = sqrt(-1)sqrt(-1) also represents either of two values, +1 and -1. Moreover, arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied with precision and clarity.

> Did you just accept it ... when in college the professor drew the yellow
> chalk coordinate system on top of the white chalk coordinate system and
> all of a sudden you believed in relativity?

The illustration of the fallacy of your claim is not a demonstration of special relativity, it is simply exploding your fallacious logic. If you want someone to teach you special relativity, you're going about it the wrong way. The point of the illustration is that, regardless of whether the trajectory of the ball is described in terms of the white coordinate system or the yellow coordinate system, it still goes into the cup. Yes, the coordinates of the ball at the end of its journey are different, but the coordinates of the cup are also different, and they match, so the ball goes into the cup. Your belief that, when described in terms of one system it goes into the cup, but when described in terms of the other system it must physically karoom into the sand trap is just bizzare.

On Monday, 26 June 2023 at 15:56:59 UTC+2, Trevor Lange wrote:

> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1

You're an idiot, but that was known before.
And sqrt function is a function. Do you know what
a function is?

On Monday, June 26, 2023 at 6:56:59 AM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 2:29:24 AM UTC-7, patdolan wrote:
> > On Sunday, June 25, 2023 at 10:55:32 AM UTC-7, Trevor Lange wrote:
> > > On Sunday, June 25, 2023 at 12:54:57 AM UTC-7, patdolan wrote:
> > > > > > > This from the dolt who claims the relation (-1)^2 = (1)^2 rigorously implies -1 = 1.
> > > > > >
> > > > > > That is not my claim at all. My claim is that i^4 = i^8 and the law of exponent rigorously implies -1 = 1.
> > > > >
> > > > > LOL. That's what I said. Since i^2 = -1, your claim is that the relation (-1)^2 = (1)^2 implies -1 = 1. Wow. There's even a section in the Wikipedia page on Math Fallacies on this kind of idiocy.
> > > > > >
> > > > > > i^4 = i^8
> > > > > > sqrt[ i^4 ] = sqrt[ i^8 ]
> > > > >
> > > > > LOL. Nope. There you are, as I said, assuming that the relation (-1)^2 = (1)^2 implies -1 = 1.
> > > >
> > > > Nowhere do I assume or type this.
> > > You assumed it (and typed it) in the quote above. Again, since i^2 = -1 and (-1)^2 = 1 unequivocally, your first equality is simply (-1)^2 = (1)^2, and sure enough, the values of the left and right sides are both 1, so you are asserting that 1=1. An auspicious start!
> > >
> > > But then you assume that this implies sqrt(1) = sqrt(1), which is not unequivocally true, because the expression "sqrt(1)" signifies a number whose square is 1, and there are two such numbers, namely +1 and -1. If you specify that you are referring to +1 on both sides, or that you are referring to -1 on both sides, then equality holds and your second assertion amounts to, again, 1=1. So you're typing the same trivial thing repeatedly. ("All work and no play makes jack a dull boy.") But of course you're not trying to just assert 1=1, you are trying to assert -1=1, which obviously doesn't validly follow.
> > >
> > > Look, consider the equality sin(0)=sin(pi). Now we take the inverse sine of both sides, and assert that 0 = pi. Well, that's precisely isomorphic to what you're doing, trying to infer from a relation of the form f(x)=f(y) that x=y. If the inverse of f is a multi-valued function, such as inverse sine or square root, that is an obviously false inference. (Now, for multi-valued functions, people sometimes adopt a convention of defining the "principal value", but in the grown-up world the principal value of a function is distinct from the function.)
> > > > My valid derivation of a contradiction cleverly exploits the cyclic nature of the powers
> > > > of i and the law of exponents.
> > > There's nothing clever about what you're doing, it's blatently invalid reasoning, as explained above. What you're describing is standard fallacy, common among slow-witted school children.
> > >
> > > > Long ago an ad hoc rule was added to arithmetic ... That law expressly
> > > > forbad moving a negative sign under a radical sign...
> > >
> > > Moving a negative sign from outside to inside a radical sign is not an "ad hoc rule", it is an obviously fallacious mistake. Again, this is a very common befuddlement for slow-witted 4th grade school children. The values of -sqrt(1) are +1 and -1, whereas the values of sqrt(-1) are +i and -i. So, "bringing a negative sign inside a radical" is simply a trivial mistake.. It's the same as thinking that -cos(x) equals cos(-x)... duh.
> >
> > [Okay, you've exploded my fallacious belief that bringing a negative sign inside a radical was ever anything other than a juvenile misconception, revealing a complete lack of understanding of 4th grade arithmetic, but, astonishingly undeterred, let me ask about] the ad hoc and arbitrary rule against factoring a positive number into two negative numbers whilst under the radical sign.
>
> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1, and the expression sqrt((-1)(-1)) = sqrt(-1)sqrt(-1) also represents either of two values, +1 and -1. Moreover, arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied with precision and clarity.

Now let me get this straight, Legion. Is it your testimony to this forum that it would never even enter the mind of any normal person to decompose a positive number into two negative factors whilst that positive number is under a radical? It is just an a prior fact of common sense put into our DNA at the moment of our conception that positive numbers are never to be factored into negative numbers whilst under the radical? Is that your position?

>
> > Did you just accept it ... when in college the professor drew the yellow
> > chalk coordinate system on top of the white chalk coordinate system and
> > all of a sudden you believed in relativity?
>
> The illustration of the fallacy of your claim is not a demonstration of special relativity, it is simply exploding your fallacious logic. If you want someone to teach you special relativity, you're going about it the wrong way. The point of the illustration is that, regardless of whether the trajectory of the ball is described in terms of the white coordinate system or the yellow coordinate system, it still goes into the cup. Yes, the coordinates of the ball at the end of its journey are different, but the coordinates of the cup are also different, and they match, so the ball goes into the cup. Your belief that, when described in terms of one system it goes into the cup, but when described in terms of the other system it must physically karoom into the sand trap is just bizzare.

On 6/26/2023 10:20 AM, Maciej Wozniak wrote:
> On Monday, 26 June 2023 at 15:56:59 UTC+2, Trevor Lange wrote:
>
>
>> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1
>
> You're an idiot, but that was known before.
> And sqrt function is a function. Do you know what
> a function is?

Since it has two values over its domain, it is not a simple function
like y=x^2 is. Only simple (single valued) functions could be used like
that.

Otherwise we could do this:

4=4
sqrt(4)=sqrt(4)
-2=2.

Dolan's "proof" is exactly the same thing.

On Monday, June 26, 2023 at 11:02:45 AM UTC-7, Volney wrote:
> On 6/26/2023 10:20 AM, Maciej Wozniak wrote:
> > On Monday, 26 June 2023 at 15:56:59 UTC+2, Trevor Lange wrote:
> >
> >
> >> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1
> >
> > You're an idiot, but that was known before.
> > And sqrt function is a function. Do you know what
> > a function is?
> Since it has two values over its domain, it is not a simple function
> like y=x^2 is. Only simple (single valued) functions could be used like
> that.
>
> Otherwise we could do this:
>
> 4=4
> sqrt(4)=sqrt(4)
> -2=2.
>
> Dolan's "proof" is exactly the same thing.
You've take liberties with your choice of roots, Volrony. And ignored your own observations about multi-valued functions. Carefully retract your steps while I correct your proof

4 = 4
sqrt(4) = sqrt(4)
+/-(2) = +/-(2)
Nothing wrong here, unlike your confusion conclusion.

You will recall that the conclusion to my proof is
-/+(1) = +/-(1)

very, very different results.

On Monday, 26 June 2023 at 20:02:45 UTC+2, Volney wrote:
> On 6/26/2023 10:20 AM, Maciej Wozniak wrote:
> > On Monday, 26 June 2023 at 15:56:59 UTC+2, Trevor Lange wrote:
> >
> >
> >> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1
> >
> > You're an idiot, but that was known before.
> > And sqrt function is a function. Do you know what
> > a function is?
> Since it has two values over its domain

But it doesn't, stupid Mike.

On Monday, June 26, 2023 at 9:33:16 AM UTC-7, patdolan wrote:
> > > [Okay, you've exploded my fallacious belief that bringing a negative sign inside a radical was ever anything other than a juvenile misconception, revealing a complete lack of understanding of 4th grade arithmetic, but, astonishingly undeterred, let me ask about] the ad hoc and arbitrary rule against factoring a positive number into two negative numbers whilst under the radical sign.
> >
> > There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1, and the expression sqrt((-1)(-1)) = sqrt(-1)sqrt(-1) also represents either of two values, +1 and -1. Moreover, arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied with precision and clarity.
>
> [Are you saying] it would never even enter the mind of any normal person to decompose a positive number into two negative factors whilst that positive number is under a radical?

You grossly mis-read. I explained to you that there is no rule against factoring a number into two negative factors under a radical sign. I even gave you an example: The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1. Again, there is no harm whatsoever in factoring numbers under a radical sign.

Moreover, I acquainted you with the fact that arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied to quantitative reasoning. Again, your claim amounts to asserting that the relation (-1)^2 = (1)^2 implies -1=1, which of course is absurd.

On Monday, 26 June 2023 at 20:46:11 UTC+2, Trevor Lange wrote:
> On Monday, June 26, 2023 at 9:33:16 AM UTC-7, patdolan wrote:
> > > > [Okay, you've exploded my fallacious belief that bringing a negative sign inside a radical was ever anything other than a juvenile misconception, revealing a complete lack of understanding of 4th grade arithmetic, but, astonishingly undeterred, let me ask about] the ad hoc and arbitrary rule against factoring a positive number into two negative numbers whilst under the radical sign.
> > >
> > > There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1, and the expression sqrt((-1)(-1)) = sqrt(-1)sqrt(-1) also represents either of two values, +1 and -1. Moreover, arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied with precision and clarity.
> >
> > [Are you saying] it would never even enter the mind of any normal person to decompose a positive number into two negative factors whilst that positive number is under a radical?
>
> You grossly mis-read. I explained to you that there is no rule against factoring a number into two negative factors under a radical sign. I even gave you an example: The expression sqrt(1) represents either of two values, +1 and -1,

I explained you you're a mathematically illiterate idiot.

On Monday, June 26, 2023 at 11:46:11 AM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 9:33:16 AM UTC-7, patdolan wrote:
> > > > [Okay, you've exploded my fallacious belief that bringing a negative sign inside a radical was ever anything other than a juvenile misconception, revealing a complete lack of understanding of 4th grade arithmetic, but, astonishingly undeterred, let me ask about] the ad hoc and arbitrary rule against factoring a positive number into two negative numbers whilst under the radical sign.
> > >
> > > There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1, and the expression sqrt((-1)(-1)) = sqrt(-1)sqrt(-1) also represents either of two values, +1 and -1. Moreover, arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied with precision and clarity.
> >
> > [Are you saying] it would never even enter the mind of any normal person to decompose a positive number into two negative factors whilst that positive number is under a radical?
>
> You grossly mis-read. I explained to you that there is no rule against factoring a number into two negative factors under a radical sign. I even gave you an example: The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1)

Very well Legion. Then please evaluate for this forum the expression you have typed: sqrt(-1)sqrt(-1). AND DON'T mix branches of the solutions--that is to say DON'T MIX ROOTS. Be consistent in your rooting. Show your work..

also each represent either of two values, +1 and -1. Again, there is no harm whatsoever in factoring numbers under a radical sign.
>
> Moreover, I acquainted you with the fact that arithmetic is not a collection of arbitrary "rules", it is nothing but rationality applied to quantitative reasoning. Again, your claim amounts to asserting that the relation (-1)^2 = (1)^2 implies -1=1, which of course is absurd.

On 6/26/2023 2:12 PM, patdolan wrote:
> On Monday, June 26, 2023 at 11:02:45 AM UTC-7, Volney wrote:
>> On 6/26/2023 10:20 AM, Maciej Wozniak wrote:
>>> On Monday, 26 June 2023 at 15:56:59 UTC+2, Trevor Lange wrote:
>>>
>>>
>>>> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1
>>>
>>> You're an idiot, but that was known before.
>>> And sqrt function is a function. Do you know what
>>> a function is?
>> Since it has two values over its domain, it is not a simple function
>> like y=x^2 is. Only simple (single valued) functions could be used like
>> that.
>>
>> Otherwise we could do this:
>>
>> 4=4
>> sqrt(4)=sqrt(4)
>> -2=2.
>>
>> Dolan's "proof" is exactly the same thing.
> You've take liberties with your choice of roots, Volrony. And ignored your own observations about multi-valued functions. Carefully retract your steps while I correct your proof
>
> 4 = 4
> sqrt(4) = sqrt(4)
> +/-(2) = +/-(2)
> Nothing wrong here, unlike your confusion conclusion.
>
> You will recall that the conclusion to my proof is
> -/+(1) = +/-(1)
>
> very, very different results.

Nope. You are still implicitly trying to make the square root a simple
function Let's look again:

> i^4 = i^8
> sqrt[ i^4 ] = sqrt[ i^8 ]

You use the principle (positive) square root here implicitly.

> Again, since i^2 = -1

That's the definition of i, yes.

> and (-1)^2 = 1

Yes

> your first equality is simply (-1)^2 = (1)^2

Yes

> did you ever bother to ask yourself why the no-negative-sign-under-the-radical rule was there in the first place?

What is this "no-negative-sign-under-the-radical rule" ? Something you
made up? An old explanation by a teacher not to take the square root of
a negative number (pre complex numbers) ? A way for you to hide your use
of the principle square root and hoping we won't catch you?

Your next step where "(-1)^2 = (1)^2 rigorously implies -1 = 1" (quoting
yourself) is using the PRINCIPLE (positive) value of the square root
function, trying to use it as a simple function when it is not valid to
do so.

On Monday, 26 June 2023 at 21:33:26 UTC+2, Volney wrote:
> On 6/26/2023 2:12 PM, patdolan wrote:
> > On Monday, June 26, 2023 at 11:02:45 AM UTC-7, Volney wrote:
> >> On 6/26/2023 10:20 AM, Maciej Wozniak wrote:
> >>> On Monday, 26 June 2023 at 15:56:59 UTC+2, Trevor Lange wrote:
> >>>
> >>>
> >>>> There is no such "rule". the expression sqrt(1) represents either of two values, +1 and -1
> >>>
> >>> You're an idiot, but that was known before.
> >>> And sqrt function is a function. Do you know what
> >>> a function is?
> >> Since it has two values over its domain, it is not a simple function
> >> like y=x^2 is. Only simple (single valued) functions could be used like
> >> that.
> >>
> >> Otherwise we could do this:
> >>
> >> 4=4
> >> sqrt(4)=sqrt(4)
> >> -2=2.
> >>
> >> Dolan's "proof" is exactly the same thing.
> > You've take liberties with your choice of roots, Volrony. And ignored your own observations about multi-valued functions. Carefully retract your steps while I correct your proof
> >
> > 4 = 4
> > sqrt(4) = sqrt(4)
> > +/-(2) = +/-(2)
> > Nothing wrong here, unlike your confusion conclusion.
> >
> > You will recall that the conclusion to my proof is
> > -/+(1) = +/-(1)
> >
> > very, very different results.
> Nope. You are still implicitly trying to make the square root a simple
> function

No, stupid Mike, he is just trying to make sqrt simple function
a simple function, which it is.

On Monday, June 26, 2023 at 12:17:02 PM UTC-7, patdolan wrote:
> > You grossly mis-read. I explained that there is no rule against factoring a
> > number into two negative factors under a radical sign. I even gave you an
> > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1)
>
> [Thank you... and accept my apology... I did indeed completely mos-read your message].
> Now, if you would be so kind, please explain to me how to evaluate the expression
> sqrt(-1)sqrt(-1).

Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).

> Don't mix branches of the solutions... Be consistent in your rooting.

These trivial grade school operations of arithmetic on complex numbers are perfectly consistent. Your crackpotism is in claiming that the relation (-1)^2 = (1)^2 implies -1 = 1, but there is no such implication. You've simply committed a commonplace grade school blunder, as explained above.

On Monday, June 26, 2023 at 4:24:47 PM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 12:17:02 PM UTC-7, patdolan wrote:
> > > You grossly mis-read. I explained that there is no rule against factoring a
> > > number into two negative factors under a radical sign. I even gave you an
> > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1)
> >
> > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > Now, if you would be so kind, please explain to me how to evaluate the expression
> > sqrt(-1)sqrt(-1).
>
> Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
>
> > Don't mix branches of the solutions... Be consistent in your rooting.
>
> These trivial grade school operations of arithmetic on complex numbers are perfectly consistent. Your crackpotism is in claiming that the relation (-1)^2 = (1)^2 implies -1 = 1, but there is no such implication. You've simply committed a commonplace grade school blunder, as explained above.

I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1. This is indeed new ground for arithmetic. But let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1. That resulted from your evaluation of sqrt(-1)sqrt(-1). Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1) to arrive at the expression sqrt( -1 x -1). Do you agree with this? If not, why not?

On Monday, June 26, 2023 at 6:12:17 PM UTC-7, patdolan wrote:
> > > > You grossly mis-read. I explained that there is no rule against factoring a
> > > > number into two negative factors under a radical sign. I even gave you an
> > > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent
> > > > either of two values, +1 and -1.
> > >
> > > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > > Now, if you would be so kind, please explain to me how to evaluate the expression
> > > sqrt(-1)sqrt(-1).
> >
> > Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
>
> I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1.

Again, you grossly mis-read. As it says just up above: "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." Do you see the difference between what was explained to you and what you tried to recite back?

> This is indeed new ground for arithmetic.

Nope, this is simple grade school arithmetic.

> Let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1.

Ah, so now you intend to focus your formidable brain power on the (to you) astonishing statement that i^2 = -1, which you attribute to me as some kind of new and novel proposition. LOL.

> That resulted from your evaluation of sqrt(-1)sqrt(-1).

Again, no. The fact that i^2 = -1 did not result from "my" evaluation sqrt(-1)sqrt(-1). The imaginary number i is defined as the number such that i^2 = -1. Sheesh.

> Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1)
> to arrive at the expression sqrt( -1 x -1).

Again, as patiently explained to you two posts ago, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." So, now you embarking on a search to discover the two values that correspond to sqrt[(-1)(-1)], after being told multiple times, and shown, what they are? Have you suffered a severe head injury?

On Monday, June 26, 2023 at 6:37:29 PM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 6:12:17 PM UTC-7, patdolan wrote:
> > > > > You grossly mis-read. I explained that there is no rule against factoring a
> > > > > number into two negative factors under a radical sign. I even gave you an
> > > > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent
> > > > > either of two values, +1 and -1.
> > > >
> > > > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > > > Now, if you would be so kind, please explain to me how to evaluate the expression
> > > > sqrt(-1)sqrt(-1).
> > >
> > > Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1.. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
> >
> > I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1.
> Again, you grossly mis-read. As it says just up above: "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." Do you see the difference between what was explained to you and what you tried to recite back?
> > This is indeed new ground for arithmetic.
> Nope, this is simple grade school arithmetic.
>
> > Let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1.
>
> Ah, so now you intend to focus your formidable brain power on the (to you) astonishing statement that i^2 = -1, which you attribute to me as some kind of new and novel proposition. LOL.
> > That resulted from your evaluation of sqrt(-1)sqrt(-1).
> Again, no. The fact that i^2 = -1 did not result from "my" evaluation sqrt(-1)sqrt(-1). The imaginary number i is defined as the number such that i^2 = -1. Sheesh.
> > Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1)
> > to arrive at the expression sqrt( -1 x -1).
> Again, as patiently explained to you two posts ago, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." So, now you embarking on a search to discover the two values that correspond to sqrt[(-1)(-1)], after being told multiple times, and shown, what they are? Have you suffered a severe head injury?
Just answer the question Legion, yes or no. Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ? If not, why not?

On Monday, June 26, 2023 at 6:41:55 PM UTC-7, patdolan wrote:
> On Monday, June 26, 2023 at 6:37:29 PM UTC-7, Trevor Lange wrote:
> > On Monday, June 26, 2023 at 6:12:17 PM UTC-7, patdolan wrote:
> > > > > > You grossly mis-read. I explained that there is no rule against factoring a
> > > > > > number into two negative factors under a radical sign. I even gave you an
> > > > > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > > > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent
> > > > > > either of two values, +1 and -1.
> > > > >
> > > > > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > > > > Now, if you would be so kind, please explain to me how to evaluate the expression
> > > > > sqrt(-1)sqrt(-1).
> > > >
> > > > Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
> > >
> > > I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1.
> > Again, you grossly mis-read. As it says just up above: "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." Do you see the difference between what was explained to you and what you tried to recite back?
> > > This is indeed new ground for arithmetic.
> > Nope, this is simple grade school arithmetic.
> >
> > > Let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1.
> >
> > Ah, so now you intend to focus your formidable brain power on the (to you) astonishing statement that i^2 = -1, which you attribute to me as some kind of new and novel proposition. LOL.
> > > That resulted from your evaluation of sqrt(-1)sqrt(-1).
> > Again, no. The fact that i^2 = -1 did not result from "my" evaluation sqrt(-1)sqrt(-1). The imaginary number i is defined as the number such that i^2 = -1. Sheesh.
> > > Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1)
> > > to arrive at the expression sqrt( -1 x -1).
> > Again, as patiently explained to you two posts ago, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." So, now you embarking on a search to discover the two values that correspond to sqrt[(-1)(-1)], after being told multiple times, and shown, what they are? Have you suffered a severe head injury?
> Just answer the question Legion, yes or no. Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ? If not, why not?
I'm going to cut you some slack Legion. I makes no difference to the contradiction into which I am leading you. To wit, CAN sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ? That should make you happier. If not, why not?

On Monday, June 26, 2023 at 6:41:55 PM UTC-7, patdolan wrote:
> On Monday, June 26, 2023 at 6:37:29 PM UTC-7, Trevor Lange wrote:
> > On Monday, June 26, 2023 at 6:12:17 PM UTC-7, patdolan wrote:
> > > > > > You grossly mis-read. I explained that there is no rule against factoring a
> > > > > > number into two negative factors under a radical sign. I even gave you an
> > > > > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > > > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent
> > > > > > either of two values, +1 and -1.
> > > > >
> > > > > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > > > > Now, if you would be so kind, please explain to me how to evaluate the expression
> > > > > sqrt(-1)sqrt(-1).
> > > >
> > > > Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
> > >
> > > I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1.
> > Again, you grossly mis-read. As it says just up above: "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." Do you see the difference between what was explained to you and what you tried to recite back?
> > > This is indeed new ground for arithmetic.
> > Nope, this is simple grade school arithmetic.
> >
> > > Let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1.
> >
> > Ah, so now you intend to focus your formidable brain power on the (to you) astonishing statement that i^2 = -1, which you attribute to me as some kind of new and novel proposition. LOL.
> > > That resulted from your evaluation of sqrt(-1)sqrt(-1).
> > Again, no. The fact that i^2 = -1 did not result from "my" evaluation sqrt(-1)sqrt(-1). The imaginary number i is defined as the number such that i^2 = -1. Sheesh.
> > > Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1)
> > > to arrive at the expression sqrt( -1 x -1).
> > Again, as patiently explained to you two posts ago, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." So, now you embarking on a search to discover the two values that correspond to sqrt[(-1)(-1)], after being told multiple times, and shown, what they are? Have you suffered a severe head injury?
>
> Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?

The squares of those two expressions are equal. Again, as patiently explained to you four times now, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."

I say again, the standard newbie fallacy to which you have succumbed is that if the squares of two expressions are equal to each other, then the two expressions are equal to each other. In other words, you are claiming that the relation (-1)^2 = (1)^2 implies -1 = 1, which it does not. A complex number has two square roots, and three cube roots, and so on.

On Monday, June 26, 2023 at 7:30:21 PM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 6:41:55 PM UTC-7, patdolan wrote:
> > On Monday, June 26, 2023 at 6:37:29 PM UTC-7, Trevor Lange wrote:
> > > On Monday, June 26, 2023 at 6:12:17 PM UTC-7, patdolan wrote:
> > > > > > > You grossly mis-read. I explained that there is no rule against factoring a
> > > > > > > number into two negative factors under a radical sign. I even gave you an
> > > > > > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > > > > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent
> > > > > > > either of two values, +1 and -1.
> > > > > >
> > > > > > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > > > > > Now, if you would be so kind, please explain to me how to evaluate the expression
> > > > > > sqrt(-1)sqrt(-1).
> > > > >
> > > > > Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
> > > >
> > > > I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1.
> > > Again, you grossly mis-read. As it says just up above: "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." Do you see the difference between what was explained to you and what you tried to recite back?
> > > > This is indeed new ground for arithmetic.
> > > Nope, this is simple grade school arithmetic.
> > >
> > > > Let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1.
> > >
> > > Ah, so now you intend to focus your formidable brain power on the (to you) astonishing statement that i^2 = -1, which you attribute to me as some kind of new and novel proposition. LOL.
> > > > That resulted from your evaluation of sqrt(-1)sqrt(-1).
> > > Again, no. The fact that i^2 = -1 did not result from "my" evaluation sqrt(-1)sqrt(-1). The imaginary number i is defined as the number such that i^2 = -1. Sheesh.
> > > > Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1)
> > > > to arrive at the expression sqrt( -1 x -1).
> > > Again, as patiently explained to you two posts ago, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." So, now you embarking on a search to discover the two values that correspond to sqrt[(-1)(-1)], after being told multiple times, and shown, what they are? Have you suffered a severe head injury?
> >
> > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> The squares of those two expressions are equal.
But are the two expression equal or not? Is the equation is false? If so, then why?

Forget what I am claiming or have claimed. Now we are focusing on what YOU are claiming.

Again, as patiently explained to you four times now, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
>
> I say again, the standard newbie fallacy to which you have succumbed is that if the squares of two expressions are equal to each other, then the two expressions are equal to each other. In other words, you are claiming that the relation (-1)^2 = (1)^2 implies -1 = 1, which it does not. A complex number has two square roots, and three cube roots, and so on.

On Monday, June 26, 2023 at 7:48:18 PM UTC-7, patdolan wrote:
> On Monday, June 26, 2023 at 7:30:21 PM UTC-7, Trevor Lange wrote:
> > On Monday, June 26, 2023 at 6:41:55 PM UTC-7, patdolan wrote:
> > > On Monday, June 26, 2023 at 6:37:29 PM UTC-7, Trevor Lange wrote:
> > > > On Monday, June 26, 2023 at 6:12:17 PM UTC-7, patdolan wrote:
> > > > > > > > You grossly mis-read. I explained that there is no rule against factoring a
> > > > > > > > number into two negative factors under a radical sign. I even gave you an
> > > > > > > > example: The expression sqrt(1) represents either of two values, +1 and -1,
> > > > > > > > and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent
> > > > > > > > either of two values, +1 and -1.
> > > > > > >
> > > > > > > [Thank you... and accept my apology... I did indeed completely mos-read your message].
> > > > > > > Now, if you would be so kind, please explain to me how to evaluate the expression
> > > > > > > sqrt(-1)sqrt(-1).
> > > > > >
> > > > > > Again? Sheesh. As explained before, and not referring to conventional "principal values", the expression sqrt(-1) represents either +i or -i, and this applies to both factors, so the product has four possible values, namely (+i)(+i) = -1, (-i)(+i) = +1, (+i)(-i) = +1, and (-i)(-i) = -1. Naturally there are only two distinct values, namely +1 and -1, which of course are the values represented by sqrt(1).
> > > > >
> > > > > I am intrigued by your conclusion that sqrt(-1)sqrt(-1) = +1.
> > > > Again, you grossly mis-read. As it says just up above: "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." Do you see the difference between what was explained to you and what you tried to recite back?
> > > > > This is indeed new ground for arithmetic.
> > > > Nope, this is simple grade school arithmetic.
> > > >
> > > > > Let us concentrate on your computation/evaluation resulting in (+i)(+i) = -1.
> > > >
> > > > Ah, so now you intend to focus your formidable brain power on the (to you) astonishing statement that i^2 = -1, which you attribute to me as some kind of new and novel proposition. LOL.
> > > > > That resulted from your evaluation of sqrt(-1)sqrt(-1).
> > > > Again, no. The fact that i^2 = -1 did not result from "my" evaluation sqrt(-1)sqrt(-1). The imaginary number i is defined as the number such that i^2 = -1. Sheesh.
> > > > > Let us apply the rules of arithmetic to your expression sqrt(-1)sqrt(-1)
> > > > > to arrive at the expression sqrt( -1 x -1).
> > > > Again, as patiently explained to you two posts ago, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1." So, now you embarking on a search to discover the two values that correspond to sqrt[(-1)(-1)], after being told multiple times, and shown, what they are? Have you suffered a severe head injury?
> > >
> > > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> > The squares of those two expressions are equal.
> But are the two expression equal or not? Is the equation is false? If so, then why?
>
> Forget what I am claiming or have claimed. Now we are focusing on what YOU are claiming.
> Again, as patiently explained to you four times now, "The expression sqrt(1) represents either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
> >
> > I say again, the standard newbie fallacy to which you have succumbed is that if the squares of two expressions are equal to each other, then the two expressions are equal to each other. In other words, you are claiming that the relation (-1)^2 = (1)^2 implies -1 = 1, which it does not. A complex number has two square roots, and three cube roots, and so on.

Can sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ? If not, why not?

On Monday, June 26, 2023 at 7:48:18 PM UTC-7, patdolan wrote:
> > > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> >
> > The squares of those two expressions are equal. Again, as patiently
> > explained to you four times now, "The expression sqrt(1) represents
> > either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and
> > sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
>
> But are the two expression equal or not?

The quantity on each side is under-specified, because it represents any value whose square is 1, which is a set of numbers that includes +1 and -1. The two square roots of 1 are not equal to each other, but of course each root is equal to itself. So, if you ask whether a given square root of 1 is equal to itself, the answer is obviously yes, but if you ask if one square root of 1 is equal to the other square root of 1, the answer is obviously no.

Of course, you aren't asking either of those rational questions. What you are asking is: If I don't specify which root I'm talking about on the left side (so it could be +1 or -1), and I don't specify which root I'm talking about on the right ride (so it could be +1 or -1), can I validly claim that the two sides are equal? The answer to this is obviously no. You have not specified the values, so you cannot assert equality. That would only be justified if the relation (-1)^2 = (1)^2 implied -1 = 1... which it obviously does not.

It might help you to consider 7th roots. Each non-zero complex number has seven distinct 7th roots. It is not valid to say that any 7th root of 1 equals any other 7th root of 1. In general we have rj = q^j r0 for j=0 to 6. It is incorrect to claim that r3 = r5, but it is correct to say that r2 = r2. This takes the average 4th grade child about 2 minutes to understand. How many decades have you been struggling with it?

On Monday, June 26, 2023 at 9:12:00 PM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 7:48:18 PM UTC-7, patdolan wrote:
> > > > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> > >
> > > The squares of those two expressions are equal. Again, as patiently
> > > explained to you four times now, "The expression sqrt(1) represents
> > > either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and
> > > sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
> >
> > But are the two expression equal or not?
> The quantity on each side is under-specified, because it represents any value whose square is 1, which is a set of numbers that includes +1 and -1. The two square roots of 1 are not equal to each other, but of course each root is equal to itself. So, if you ask whether a given square root of 1 is equal to itself, the answer is obviously yes, but if you ask if one square root of 1 is equal to the other square root of 1, the answer is obviously no.
>
> Of course, you aren't asking either of those rational questions. What you are asking is: If I don't specify which root I'm talking about on the left side (so it could be +1 or -1), and I don't specify which root I'm talking about on the right ride (so it could be +1 or -1), can I validly claim that the two sides are equal? The answer to this is obviously no. You have not specified the values, so you cannot assert equality. That would only be justified if the relation (-1)^2 = (1)^2 implied -1 = 1... which it obviously does not.
>
> It might help you to consider 7th roots. Each non-zero complex number has seven distinct 7th roots. It is not valid to say that any 7th root of 1 equals any other 7th root of 1. In general we have rj = q^j r0 for j=0 to 6. It is incorrect to claim that r3 = r5, but it is correct to say that r2 = r2. This takes the average 4th grade child about 2 minutes to understand. How many decades have you been struggling with it?

On Monday, June 26, 2023 at 9:12:00 PM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 7:48:18 PM UTC-7, patdolan wrote:
> > > > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> > >
> > > The squares of those two expressions are equal. Again, as patiently
> > > explained to you four times now, "The expression sqrt(1) represents
> > > either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and
> > > sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
> >
> > But are the two expression equal or not?
> The quantity on each side is under-specified, because it represents any value whose square is 1, which is a set of numbers that includes +1 and -1. The two square roots of 1 are not equal to each other, but of course each root is equal to itself. So, if you ask whether a given square root of 1 is equal to itself, the answer is obviously yes, but if you ask if one square root of 1 is equal to the other square root of 1, the answer is obviously no.
>
> Of course, you aren't asking either of those rational questions. What you are asking is: If I don't specify which root I'm talking about on the left side (so it could be +1 or -1), and I don't specify which root I'm talking about on the right ride (so it could be +1 or -1), can I validly claim that the two sides are equal? The answer to this is obviously no. You have not specified the values, so you cannot assert equality. That would only be justified if the relation (-1)^2 = (1)^2 implied -1 = 1... which it obviously does not.
>
> It might help you to consider 7th roots. Each non-zero complex number has seven distinct 7th roots. It is not valid to say that any 7th root of 1 equals any other 7th root of 1. In general we have rj = q^j r0 for j=0 to 6. It is incorrect to claim that r3 = r5, but it is correct to say that r2 = r2. This takes the average 4th grade child about 2 minutes to understand. How many decades have you been struggling with it?

I'll take your answer to mean "Yes, sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) can be a true equations." Do you disagree with this, Legion?

On Monday, June 26, 2023 at 9:33:53 PM UTC-7, patdolan wrote:
> On Monday, June 26, 2023 at 9:12:00 PM UTC-7, Trevor Lange wrote:
> > On Monday, June 26, 2023 at 7:48:18 PM UTC-7, patdolan wrote:
> > > > > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> > > >
> > > > The squares of those two expressions are equal. Again, as patiently
> > > > explained to you four times now, "The expression sqrt(1) represents
> > > > either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and
> > > > sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
> > >
> > > But are the two expression equal or not?
> > The quantity on each side is under-specified, because it represents any value whose square is 1, which is a set of numbers that includes +1 and -1.. The two square roots of 1 are not equal to each other, but of course each root is equal to itself. So, if you ask whether a given square root of 1 is equal to itself, the answer is obviously yes, but if you ask if one square root of 1 is equal to the other square root of 1, the answer is obviously no.
> >
> > Of course, you aren't asking either of those rational questions. What you are asking is: If I don't specify which root I'm talking about on the left side (so it could be +1 or -1), and I don't specify which root I'm talking about on the right ride (so it could be +1 or -1), can I validly claim that the two sides are equal? The answer to this is obviously no. You have not specified the values, so you cannot assert equality. That would only be justified if the relation (-1)^2 = (1)^2 implied -1 = 1... which it obviously does not.
> >
> > It might help you to consider 7th roots. Each non-zero complex number has seven distinct 7th roots. It is not valid to say that any 7th root of 1 equals any other 7th root of 1. In general we have rj = q^j r0 for j=0 to 6. It is incorrect to claim that r3 = r5, but it is correct to say that r2 = r2. This takes the average 4th grade child about 2 minutes to understand. How many decades have you been struggling with it?
>
> I'll take your answer to mean "Yes, sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) can be a true equations."

As always, you're mistaken. A mathematical equation of the form x=y does not signify that x can or could or might equal y, it signifies that x equals y. Duh. In contrast, the nth root of a complex number is an n-valued mapping, so taking the nth root of both sides of the tautological 1 = 1 does not yield a scalar numerical equality. The expression (1+i)^(1/7) does not represent a single numerical value. Ironically, the key to the adult understanding of this is via Riemann surfaces (named for the same person whose differential geometry you also don't understand).

On Monday, June 26, 2023 at 10:54:24 PM UTC-7, Trevor Lange wrote:
> On Monday, June 26, 2023 at 9:33:53 PM UTC-7, patdolan wrote:
> > On Monday, June 26, 2023 at 9:12:00 PM UTC-7, Trevor Lange wrote:
> > > On Monday, June 26, 2023 at 7:48:18 PM UTC-7, patdolan wrote:
> > > > > > Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
> > > > >
> > > > > The squares of those two expressions are equal. Again, as patiently
> > > > > explained to you four times now, "The expression sqrt(1) represents
> > > > > either of two values, +1 and -1, and the expressions sqrt[(-1)(-1)] and
> > > > > sqrt(-1)sqrt(-1) also each represent either of two values, +1 and -1."
> > > >
> > > > But are the two expression equal or not?
> > > The quantity on each side is under-specified, because it represents any value whose square is 1, which is a set of numbers that includes +1 and -1. The two square roots of 1 are not equal to each other, but of course each root is equal to itself. So, if you ask whether a given square root of 1 is equal to itself, the answer is obviously yes, but if you ask if one square root of 1 is equal to the other square root of 1, the answer is obviously no.
> > >
> > > Of course, you aren't asking either of those rational questions. What you are asking is: If I don't specify which root I'm talking about on the left side (so it could be +1 or -1), and I don't specify which root I'm talking about on the right ride (so it could be +1 or -1), can I validly claim that the two sides are equal? The answer to this is obviously no. You have not specified the values, so you cannot assert equality. That would only be justified if the relation (-1)^2 = (1)^2 implied -1 = 1... which it obviously does not.
> > >
> > > It might help you to consider 7th roots. Each non-zero complex number has seven distinct 7th roots. It is not valid to say that any 7th root of 1 equals any other 7th root of 1. In general we have rj = q^j r0 for j=0 to 6. It is incorrect to claim that r3 = r5, but it is correct to say that r2 = r2. This takes the average 4th grade child about 2 minutes to understand. How many decades have you been struggling with it?
> >
> > I'll take your answer to mean "Yes, sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) can be a true equations."
> As always, you're mistaken. A mathematical equation of the form x=y does not signify that x can or could or might equal y, it signifies that x equals y. Duh. In contrast, the nth root of a complex number is an n-valued mapping, so taking the nth root of both sides of the tautological 1 = 1 does not yield a scalar numerical equality. The expression (1+i)^(1/7) does not represent a single numerical value. Ironically, the key to the adult understanding of this is via Riemann surfaces (named for the same person whose differential geometry you also don't understand).

Did you ever think you would have beclowned and buffoon yourself as you are doing now, Legion?

Let's help Legion out.

Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?

a) yes
b) no
c) yes and no
d) maybe
e) maybe not
f) sometimes
g) sometimes not
h) undecidable
i) don't know

I open this question to all members of this forum

Le 27/06/2023 à 16:53, demented idiot patdolan wrote:
....
> Did you ever think you would have beclowned and buffoon yourself as you are doing now, Legion?
>
> Let's help Legion out.
>
> Does sqrt[(-1)(-1)] = sqrt(-1)sqrt(-1) ?
>
> a) yes
> b) no
> c) yes and no
> d) maybe
> e) maybe not
> f) sometimes
> g) sometimes not
> h) undecidable
> i) don't know
>
> I open this question to all members of this forum

Assume sqrt to be the single value function returning
the principal value the square root :

(see https://en.wikipedia.org/wiki/Principal_value)

sqrt(-1*-1) = sqrt(1) = 1

as -1 = exp(i*pi) ( phi = pi \in ]-pi, pi] )

sqrt(-1)*sqrt(-1) = exp(i * pi/2) * exp(i * pi/2)
= ext(i * pi) = - 1

so no, in term of principal values.

so what, pathetic, idiotic, demented, self-immolising
asshole Pat Dolan?