Can you pick out what's wrong here?

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Bullislander05
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Joined: Sat Mar 08, 2008 1:37 am UTC

Can you pick out what's wrong here?

Hey guys. First post and all, so I figure I'll start it off with a bang.

This isn't a trick question (I read the rules. Don't wanna be stepping on anyone's toes right off the bat. ), it's a quiz of your knowledge of mathematics rules. I'm going to show you an expression, and ask you to show me the part of it that is incorrect and tell me why it's incorrect.

So, defining i = √-1, the following should hold true.

i2 = ii = √-1•√-1 = √(-1•-1) = √1 = 1

But, in reality, i2 = -1.

Which step is illegal? By the way, steps are defined to begin with the first = sign, so just tell me the number of the = sign between the last true step and the first false step.

Sorry it's not a crazy think until your brains melt to the floor question. Heh, I'm no expert at creating these things. In fact, this is something a friend of mine just showed me a few days ago.

(For the record, since none of you know me, here's a little personal info: I'm currently a Junior in high school. My classes consist of 4 AP courses and Spanish IV (Just before AP Spanish). I'm taking Calculus-based physics, BC Calculus (Calc 2 I guess), AP English 11, and AP United States History. I have a wonderful girlfriend, plenty of great friends, and I have a never ending thirst for knowledge. I plan on doing something related to Computers when I get older, but nothing is set in stone yet.)

And that's about it. Thanks for reading. Good luck getting the puzzle right, as well! I hope it's not too unorthodox to post both a puzzle and a "hello" in the same time.

-Bull

parallax
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Re: Can you pick out what's wrong here?

Spoiler:
Well, precisely which step is wrong depends on your definition of the square root. Step 2 is wrong because the square root is multi-valued. If you have a single-valued square root with an appropriate branch cut, then step 3 is no longer correct. Steps 1, 4 and 5 are always correct with standard definitions, but it should be noted that -1 is a square root of 1.
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Bullislander05
Posts: 25
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Re: Can you pick out what's wrong here?

Spoiler:
Hmm. I don't quite understand what you're saying regarding the definition of a square root. Could you explain a bit better, or in more detail at least?

skeptical scientist
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Re: Can you pick out what's wrong here?

Spoiler:
In calculus and precalculus classes, or any time one only deals with real numbers, the square root is defined as a function defined on non-negative real numbers. Under the definition, one always has √a√b=√(ab) when a and b are in the domain of √.

When you start talking about complex numbers, you need to extend the definition of √ - the most common way to do this is take your definition as √(reit)=(√r)eit/2, where reit is the unique way of expressing your complex number with r a nonnegative real number, and t in the interval [0, 2pi). If this is your definition, then it's no longer true that √a√b=√(ab) in general, and specifically, √-1√-1 is not equal to √(-1*-1). There are other ways of extending this however - as mentioned, one of these ways is to make √ a multivalued (or set-valued) function, i.e. √x is the set of all y such that y2=x. If this is your definition, then it's incorrect to say that i=√-1 (and hence also to say that i*i=√-1*√-1), since √-1 is the set {i, -i}.
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Robin S
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Re: Can you pick out what's wrong here?

The
Spoiler:
third
equals sign is incorrect. Do I have to say why?
skeptical scientist wrote:
Spoiler:
reit is the unique way of expressing your complex number with r a nonnegative real number, and t in the interval [0, 2pi).
Spoiler:
Many people I know, however, take t to be in [-π,π) or (-π,π], which complicates things a bit.
This is a placeholder until I think of something more creative to put here.

Bullislander05
Posts: 25
Joined: Sat Mar 08, 2008 1:37 am UTC

Re: Can you pick out what's wrong here?

Spoiler:
Correct. It is the 3rd. Thanks for clarifying what he meant skeptical scientist. I was just a bit lost reading it literally the first time. I have nothing to give but "brownie points" to you guys for getting it correct, though.

skeptical scientist
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Re: Can you pick out what's wrong here?

Robin S wrote:
Spoiler:
Many people I know, however, take t to be in [-π,π) or (-π,π], which complicates things a bit.

Spoiler:
Using (-π,π] is also common, but I've never seen [-π,π) used.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

parallax
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Re: Can you pick out what's wrong here?

Would this be a good place to post similar puzzles? My favorite was always:

2πi = ln(e2πi)=ln(1)=0
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jestingrabbit
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Re: Can you pick out what's wrong here?

I don't think it would be a good place to post "fun with branchcuts" type stuff. That's much more maths than anything else. I think there are also several preexisting threads in the maths forum that deal with this stuff too.
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Skateside
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Re: Can you pick out what's wrong here?

It's far easier to thing of these as indices:

i = √-1 = -11/2
i2 = i * i = -11/2 * -11/2 = -11 = -1

Spoiler:
Therefore, I'd have to say your third step is the wrong one
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Spoiler:
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