Wolfram Alpha: all 2nd roots of 1

[AllyScientific]

I think a lot a users have vague concepts about the roots of unity.
I try to post a link to WolframAlpha, which calculates all the second roots
of unity

http://www.wolframalpha.com/input/?i=sqrt(1)

There you can see the input \sqrt{1} and the plot of all roots in the complex
plane.
The roots are lying on the unit circle \ e^{i\alpha} = cos\alpha+i\dot{}sin\alpha

There are two real roots as you can see on the plot:
\sqrt{1} = +1 (principal root)
\sqrt{1} = -1

Wikipedia says that there exists a mathematical fallacy of the following kind:
1= \sqrt{1} = \sqrt{(-1)\dot{}(-1)} = \sqrt{-1}\dot{}\sqrt{-1} = i\dot{}i = -1
the fallacy is that the rule \sqrt{x\dot{}y} = \sqrt{x}\dot{}\sqrt{y} is not valid here according to Wikipedia:
http://en.wikipedia.org/wiki/Mathematical_fallacy#Positive_and_negative_roots

WolframAlpha implies no error. Which one should we trust? My guess is
Wikipedia is just wrong and WolframAlpha is correct.

 

[Number Nine]

You've already started this argument in another thread; there's no reason to make another one. Several forum members have already explained quite clearly why you've misunderstood the Wikipedia entry, and why your post in incorrect. I honestly can't imagine why you think the Wikipedia and Wolfram articles contradict each other in any way.

 

[pwsnafu]

The Alpha page you cite lists the input as $\sqrt{1}$ and the result as $1$. Not 1 and -1. The "All 2nd roots of unity" section is extra information in case you wanted it. Nowhere on the page does Alpha say that $\sqrt{1} = -1$.

 

[jbriggs444]

You've already started this argument in another thread; there's no reason to make another one.

The other thread ("Marilyn vos Savant: -1x-1=+1, instead of -1") was closed because this argument was off-topic there. The suggestion was made that another thread be created.

 

[D H]

I think a lot a users have vague concepts about the roots of unity.
I try to post a link to WolframAlpha, which calculates all the second roots
of unity

http://www.wolframalpha.com/input/?i=sqrt(1)

There you can see the input \sqrt{1} and the plot of all roots in the complex
plane.
The roots are lying on the unit circle \ e^{i\alpha} = cos\alpha+i\dot{}sin\alpha

You missed something important: What does Wolfram Alpha report as the value of √1? That's a rhetorical question. The answer is that it reports that √1 is 1. Not -1. Just 1.

Another rhetorical question: Why? The answer to this question is that Mathematica's power operator ^ uses the principal value. When confronted with an expression of the form z^w (i.e., z^w), Mathematica will choose a single value for that expression, the principal value. I suggest you read about branch cuts.

Wikipedia says that there exists a mathematical fallacy of the following kind:
1= \sqrt{1} = \sqrt{(-1)\dot{}(-1)} = \sqrt{-1}\dot{}\sqrt{-1} = i\dot{}i = -1
the fallacy is that the rule \sqrt{x\dot{}y} = \sqrt{x}\dot{}\sqrt{y} is not valid here according to Wikipedia:
http://en.wikipedia.org/wiki/Mathematical_fallacy#Positive_and_negative_roots

Correct. Every statement that attempts to show 1=-1 is somehow erroneous.

Sometimes it's a simple error. Have you ever gone through some painstakingly complicated mathematics only to arrive at 1=-1? You have not shown that there is something wrong with mathematics. You've just made a mistake somewhere.

Other times the error is deceptively hidden. That is the point of this particular wikipedia article. You appear to have missed that point. Any derivation that ends up with something along the lines of 1=-1 is erroneous. The trick in each of these deceptive "proofs" is to find the place where the jokester made the mistake.

WolframAlpha implies no error.

Sure it does. Let's see what WolframAlpha thinks of that supposed proof that 1=-1, one step at a time:

• $1=\sqrt{1}$
  No problem here.
• $\sqrt{1} = \sqrt{(-1)(-1)}$
  No problem here, either.
• $\sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1}$
  Oops. Big problem here.
  

 

[AllyScientific]

Correct. Every statement that attempts to show 1=-1 is somehow erroneous.

Sometimes it's a simple error. Have you ever gone through some painstakingly complicated mathematics only to arrive at 1=-1? You have not shown that there is something wrong with mathematics. You've just made a mistake somewhere.

Other times the error is deceptively hidden. That is the point of this particular wikipedia article. You appear to have missed that point. Any derivation that ends up with something along the lines of 1=-1 is erroneous. The trick in each of these deceptive "proofs" is to find the place where the jokester made the mistake.

I don't think anyone can say for certain that proving true the statement 1=-1 will contain
an error. It is only a priori hypothesis that the attempt is doomed to failure. It does not prevent someone from finding a proof some day, whatever the result of the proof will be.
Of course what I have written here are subject to falsification, and you are free to prove them wrong if you can, I just want to know where the error is, you have not convinced me yet.
I have always been suspicious of such a priori hypotheses, for example Gödel's incompleteness theorem, which says something similar, how you must believe that some mathematical
statements are false although at the same time these are not possible to prove false.
In this case mathematics would be just a religion, and we should just believe something,
and never even try to prove it. At least we should try to prove the things we claim to be true, no matter how impossible or hopeless the task may seem, we can be certain only after we try, not before. In summary, if you are right in your a priori hypothesis, and of course you may be,
who knows, but to me it sounds only a religious satement and theology to me at the moment.

 

[D H]

Ally, please stop dragging threads off-topic. The mis-proof that 1=-1 cited in the opening post has absolutely nothing to do with Gödel's incompleteness theorems. Nothing. It's a mis-proof, and that's all it is.

 

[AllyScientific]

If it is a mis-proof, it must be proved to be such, just saying it is does not make it a mis-proof.
I am myself also trying to prove -1 = 1 false, if I can. We should
abandon theology and aim to scientific methodology if possible. In that case,
the only possibility is to study the problem objectively without a priori bias or strong subjective
opinions that the statement must be false because I feel so, and leave all options open. This means we all should consider seriously the possibility that
-1 = 1 may also be true. I admit people have opinions, hardly anyone is free of them, not even scientists, but can you trust them ?
You may be right about Gödel, his theorems are maybe too philosophical to deal with my problem.
And I don't even know if anyone has understood them completely, not even himself, his
theorems may be incomplete or inconsistent, or both.

 

[pwsnafu]

You may be right about Gödel, his theorems are maybe too philosophical to deal with my problem.
And I don't even know if anyone has understood them completely, not even himself, his
theorems may be incomplete or inconsistent, or both.

Theorems are neither incomplete nor inconsistent. Those terms are for axiomatic systems.

 

[D H]

If it is a mis-proof, it must be proved to be such, just saying it is does not make it a mis-proof.

It's right there in that wikipedia article you quoted. $\sqrt{xy} = \sqrt{x}\sqrt{y}$ is not a valid operation in general. If you don't like that, it's the last entry in post #5.

 

[arildno]

Ally:
You seem not to understand that the operations "multiplication/root extractions between complex numbers" NECESSARILY are different from "multiplication/root extractions between real numbers" because "complex numbers" are different types of beasts than "real numbers".

And, how do they differ, exactly?

Real numbers can be regarded as a DISTINCT SUBSET of the complex numbers, so that if we ONLY work with them, then the multiplication/root extraction operations valid for complex numbers in general simplify into those multiplication/root extraction operations YOU are familiar with.

This means, for example, that simplifying relationships that occur within the realm of real numbers cannot automatically be regarded as valid for the complex numbers in general.

But, that misapprehension is the one you, and Van Savant are labouring under.