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November 14th, 2013, 09:41 PM   #1
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what is the square root of +1 ?

I have a question,



because squaring on both sides gives the identity 1=1 .

However, if you use imaginary number i which is defined by


you will get


Because by definition of i


We will arrive at the result +1 = -1 , is there perhaps an error somewhere?
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November 14th, 2013, 10:45 PM   #2
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Re: what is the square root of +1 ?

Quote:
Originally Posted by TwoTwo
I have a question,

Actually, by definition:



That's why you have to write:



Quote:
Originally Posted by TwoTwo
However, if you use imaginary number i which is defined by
This is wrong, and you've shown exactly why it's wrong. i is defined as:



So:



With complex numbers there is no attempt to define square root (or the nth root) as any particular root. Unlike real numbers, there is no clear concept of the "positive one". So, for example:



And the nth root will always be n complex numbers.
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November 15th, 2013, 12:47 AM   #3
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That's not right. By convention, ?(-1) = i, not -i.
By convention, if z is a complex number with a non-zero imaginary part, ?z has an imaginary part with the same sign as the imaginary part of z.

The equation (?xy) = (?x)(?y) holds for all positive values of x and y.
As already shown, it doesn't hold for x = y = -1.
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November 15th, 2013, 02:19 AM   #4
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Re: what is the square root of +1 ?

I must admit that's news to me. To take the argument set out by the OP:



This must break down somewhere. I guess you have to say that, with that convention, for complex numbers:

?
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November 15th, 2013, 03:56 AM   #5
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Re: what is the square root of +1 ?

Quote:
Originally Posted by Pero
I must admit that's news to me. To take the argument set out by the OP:



This must break down somewhere. I guess you have to say that, with that convention, for complex numbers:

?

Look at my first post, 1 has two roots



so perhaps

holds for negative values of z and w too.

Look at how Wolfram Alpha calculates square roots (also known as 2nd roots) of 1:
http://www.wolframalpha.com/input/?i=sqrt1
You can see two real roots +1 (real root, principal root) and -1 (real root)
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November 15th, 2013, 05:04 PM   #6
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Sometimes it works for complex numbers and sometimes it doesn't.
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November 15th, 2013, 10:14 PM   #7
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Re: what is the square root of +1 ?



So everything is right as rain again.

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November 16th, 2013, 02:21 AM   #8
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Re: what is the square root of +1 ?

Quote:
Originally Posted by agentredlum


So everything is right as rain again.


Ok, very good. Consider now



but what is



Is

Or is
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November 16th, 2013, 05:07 AM   #9
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The rule that (z^m)^n = z^(mn) always works if z > 0 and both m and n are real, but only sometimes works otherwise.
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November 16th, 2013, 10:59 AM   #10
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Re:

Quote:
Originally Posted by skipjack
Sometimes it works for complex numbers and sometimes it doesn't.
This is a strange situation, and I try to understand what is the problem.
Again I write the same formulas but now a little differently. How to prove that

holds even when both z and w are negative:



Because


And this time the paradox +1 = -1 is hidden.
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