From Random Hugs to Greeting Conversations...

[llearch n'n'daCorna]

"1=(-1)*(-1)=sqr((-1)*(-1))=sqr(-1)*sqr(-1)=i*i=i^2=-1"
Or:
1
=(-1)*(-1)
=sqr((-1)*(-1)) -- b
=sqr(-1)*sqr(-1)
=i*i  -- a
=i^2
=-1

a) and here is where it all goes wrong. The square root of -i is both positive and negative - technically, i and -i. Which makes for some interesting problems resolving this...
b) Or possibly here. Certainly, messing with squares and square roots, you need to remain aware that there are two possible square roots to every number...

[Nerdjob]

"1=(-1)*(-1)=sqr((-1)*(-1))=sqr(-1)*sqr(-1)=i*i=i^2=-1"
Or:
1
=(-1)*(-1)
=sqr((-1)*(-1)) -- b
=sqr(-1)*sqr(-1)
=i*i  -- a
=i^2
=-1

a) and here is where it all goes wrong. The square root of -i is both positive and negative - technically, i and -i. Which makes for some interesting problems resolving this...
b) Or possibly here. Certainly, messing with squares and square roots, you need to remain aware that there are two possible square roots to every number...

Somewhat off-topic for the thread, but okay...

According to my professors at college, the line right after (b) is where things really go wrong. You know that quote saying "the sound you are hearing is a paradigm shifting without a clutch"? Well, that's what the example is supposed to illustrate: real numbers are one-dimensional, while complex numbers are two-dimensional. And you either work with one or the other. The guy doing the 'proof', that 1=-1, shifts from one paradigm to the other, with no regard for context or proper application, he just manipulates symbols.

For those of you who are totally confuzzled about what 'complex numbers' are, the Wikipedia article at http://en.wikipedia.org/wiki/Complex_number provides a concise treatment of the subject that will probably leave you even more confuzzled. Suffice it to say that "i" is the customary notation for the other-dimensional equivalent of "1", and that complex number theory has important applications in electrical engineering. It allows inductors and capacitors to be plugged into the same equations as resistors, using Ohm's and Kirchhoff's laws.

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...*must stop typing . . . Bode plots without using a calculator . . . ARGH!*...
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Okay, I'm a nerd. I guess this proves it beyond any reasonable doubt. Maybe I should start thinking about a different sig. 

[llearch n'n'daCorna]

No, I kinda liked it. At least I stopped to think about it.

In return, try this.

[Nerdjob]

No, I kinda liked it. At least I stopped to think about it.

In return, try this.

"Evil Inc., how may I harm you?" http://evil-comic.com/archive/20060513.html

I think that about does it for the introduction thing. 

[llearch n'n'daCorna]

Good point, Gabi.

Splitsville!

[Gabi]

Thanks, llearch! Now both this discussion and random huggles have their  own places.

From what I've been taught, the square root is defined as the positive value, so the square root of -1 is i by definition. But it is not true that sqr((-1)*(-1))=sqr(-1)*sqr(-1). The distributive property of power over product does not hold over complex numbers.

As for the "paradox department", that's only a paradox if they've never told the truth before. Otherwise it's a lie. The liar's paradox is "this is a lie", which can't be true nor false.

[superluser]

From what I've been taught, the square root is defined as the positive value, so the square root of -1 is i by definition. But it is not true that sqr((-1)*(-1))=sqr(-1)*sqr(-1). The distributive property of power over product does not hold over complex numbers.

To keep a square root function a function (and not a relation), it is defined as the positive root, but it's understood to be a convention, and not the mathematical truth.

Other than that, I'd like to point out that while imaginary and complex numbers are undefined in the set of real numbers (and trying to treat them as such leads to problems), generally even if the solution to a problem requires using complex numbers at some point in the computation, the solution is still valid.

Contrast that to dividing by zero, where an undefined number can (though I could certainly craft cases where it does not) invalidate the solution.

[Gabi]

Imaginary numbers themselves are not a mathematical truth, if such a thing exists. They're a patch created to solve a set of problems.

Then again, numbers in general are mental constructs. Mathematics are a way to keep abstract thoughts organized, and a tool to deal with many kinds of problems, some derived from real life, some others from mathematics even.

My point is: all definitions regarding mathematics are arbitrary and have been made to serve a purpose, not because they are essentially true.

[Reese Tora]

Imaginary numbers themselves are not a mathematical truth, if such a thing exists. They're a patch created to solve a set of problems.

Then again, numbers in general are mental constructs. Mathematics are a way to keep abstract thoughts organized, and a tool to deal with many kinds of problems, some derived from real life, some others from mathematics even.

My point is: all definitions regarding mathematics are arbitrary and have been made to serve a purpose, not because they are essentially true.

Hmm, numbers are arbitrary, but mathmatical formulae aren't.

Imaginarry unmebrs aren't a patch so mucha s what we've anmed what they do.  They work, though I believe they were invented before what they work with was around to demonstrate them.

A formula doesn't generally change, the constants that the formula uses get refined, and variables in the formula are found to be the results of further variables interacting, btu the base formul remains the same.

For instance, there's a theory of motion that, if you set the speed related variables to 0, simplifies to an older verion of that formula that had been in use for hundreds of years. (I want to say it's the formula for acceleration, but I can't quite remember which formula it is.)

http://www.xkcd.com/435/

[Bagi23]

"1=(-1)*(-1)=sqr((-1)*(-1))=sqr(-1)*sqr(-1)=i*i=i^2=-1"
Or:
1
=(-1)*(-1)
=sqr((-1)*(-1)) -- b
=sqr(-1)*sqr(-1)
=i*i  -- a
=i^2
=-1

a) and here is where it all goes wrong. The square root of -i is both positive and negative - technically, i and -i. Which makes for some interesting problems resolving this...
b) Or possibly here. Certainly, messing with squares and square roots, you need to remain aware that there are two possible square roots to every number...

that r because math r evil! be ignorant! like me!!!

[llearch n'n'daCorna]

Bagi, read the rules. Over a month is considered to be necroing a dead thread.