What is wrong with this? 1=-1?

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VDOgamez
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What is wrong with this? 1=-1?

Somehow, I accidentally showed that 1=-1. Here. -1=-1^1=-1^(2/2)=(-1^2)^(1/2)=1^(1/2)=1. What did I do wrong to get this?

t0rajir0u
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Re: What is wrong with this? 1=-1?

[imath](a^b)^c = a^{bc}[/imath] isn't true in general when (edit:) [imath]b, c[/imath] aren't both integers (misread!). Any number has two square roots (with multiplicity - I know how you xkcd kids love to nitpick!), and what you did amounts to picking the wrong square root.
Last edited by t0rajir0u on Fri Feb 20, 2009 5:42 pm UTC, edited 1 time in total.

Token
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Re: What is wrong with this? 1=-1?

t0rajir0u wrote:[imath](a^b)^c = a^{bc}[/imath] isn't true in general when [imath]b[/imath] isn't an integer.

In this case, [imath]b[/imath] is an integer.
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jestingrabbit
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Re: What is wrong with this? 1=-1?

Token wrote:
t0rajir0u wrote:[imath](a^b)^c = a^{bc}[/imath] isn't true in general when [imath]b[/imath] isn't an integer.

In this case, [imath]b[/imath] is an integer.

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Token
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Re: What is wrong with this? 1=-1?

And loving it.
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Re: What is wrong with this? 1=-1?

Token wrote:
t0rajir0u wrote:[imath](a^b)^c = a^{bc}[/imath] isn't true in general when [imath]b[/imath] isn't an integer.

In this case, [imath]b[/imath] is an integer.

But C isn't. The fun thing about [imath]a^{bc}[/imath] is that b and c are interchangeable. You have [imath](1^2)^\frac{1}{2}=(-1^2)^\frac{1}{2}[/imath], which would be incredibly obvious if you wrote your proof in proper form. You start with an equality and modify only one side until you get to what you wanted to prove.
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Token
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Re: What is wrong with this? 1=-1?

headprogrammingczar wrote:The fun thing about [imath]a^{bc}[/imath] is that b and c are interchangeable.

But not in [imath](a^b)^c[/imath] (well, not unambiguously, anyway). Clearly you don't fully understand the spirit of pedantry here.
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TheQntty
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Re: What is wrong with this? 1=-1?

(-1^2)^(1/2)

He's where you went wrong. [imath]\sqrt{-1^2} = i \not= -1[/imath]

The mistake you made is thinking that [imath](-k)^n = -k^n[/imath] which isn't true because of order of operations. If you fix that, it would still be wrong because (as others said) [imath]\sqrt{n^2} = |n|[/imath]

a Person
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Re: What is wrong with this? 1=-1?

i^4=1
4th root of 1 is 1
Therefore, i = 1

Sir_Elderberry
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Re: What is wrong with this? 1=-1?

i^4=1
4th root of 1 is 1
Therefore, i = 1

Fourth root of 1 has multiple solutions.
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Blatm
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Re: What is wrong with this? 1=-1?

x4 - 1= 0 has 4 solutions: 1, i, -1, and -i. Just because two numbers are solutions to the same equation doesn't mean they are equal (you wouldn't claim that 0 = 2pi just because their sines are equal). In this particular case, the problem is that over the reals, the square root is defined to be positive because it's easier to deal with functions that aren't multivalued. Over the complex numbers, roots are multivalued, and can be treated as the inverse function to xn - c = 0 (this is how they are defined, actually).

Edit: Ninja'd, again.

cspirou
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Re: What is wrong with this? 1=-1?

t0rajir0u wrote:[imath](a^b)^c = a^{bc}[/imath] isn't true in general when (edit:) [imath]b, c[/imath] aren't both integers (misread!). Any number has two square roots (with multiplicity - I know how you xkcd kids love to nitpick!), and what you did amounts to picking the wrong square root.

Yes it is. Show me something where it's not true.

VDOgamez wrote:Somehow, I accidentally showed that 1=-1. Here. -1=-1^1=-1^(2/2)=(-1^2)^(1/2)=1^(1/2)=1. What did I do wrong to get this?

It's the last step where you take the root of 1. There's two answers. 1 and -1. One of them is valid.

antonfire
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Re: What is wrong with this? 1=-1?

cspirou wrote:
t0rajir0u wrote:[imath](a^b)^c = a^{bc}[/imath] isn't true in general when (edit:) [imath]b, c[/imath] aren't both integers (misread!). Any number has two square roots (with multiplicity - I know how you xkcd kids love to nitpick!), and what you did amounts to picking the wrong square root.
Yes it is. Show me something where it's not true.

((-1)2)1/2 = 11/2 = 1
(-1)1/2 * 2 = (-1)1 = -1

Sure, you can claim that "11/2 can be either 1 or -1", but (-1)1 certainly can't. So even if you think of ab as a multi-valued function, it doesn't quite work out.
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TheQntty
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Re: What is wrong with this? 1=-1?

In most cases [imath]x^{1/2}[/imath] is taken to mean the positive square root of x.

For skeptics
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Re: What is wrong with this? 1=-1?

t0rajr0u got this one, but later posters seem to have been confused by the fact that you didn't put parentheses around your -1, in which case, the error is more obvious.

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Re: What is wrong with this? 1=-1?

Blatm wrote:...you wouldn't claim that 0 = 2pi just because their sines are equal...

[imath]\sin0=\sin2\pi[/imath]
[imath]\sin^{-1}(\sin0)=\sin^{-1}(\sin2\pi)[/imath]
[imath]0=2\pi[/imath]

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Re: What is wrong with this? 1=-1?

I guess it's a good thing that sine and cosine aren't injective.
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Re: What is wrong with this? 1=-1?

There seem to be about three discusions in various threads about the merits (and lack thereof) of multi-valued functions. Possibly because almost every false proof uses them.
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cspirou
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Re: What is wrong with this? 1=-1?

antonfire wrote:
cspirou wrote:
t0rajir0u wrote:[imath](a^b)^c = a^{bc}[/imath] isn't true in general when (edit:) [imath]b, c[/imath] aren't both integers (misread!). Any number has two square roots (with multiplicity - I know how you xkcd kids love to nitpick!), and what you did amounts to picking the wrong square root.
Yes it is. Show me something where it's not true.

((-1)2)1/2 = 11/2 = 1
(-1)1/2 * 2 = (-1)1 = -1

Sure, you can claim that "11/2 can be either 1 or -1", but (-1)1 certainly can't. So even if you think of ab as a multi-valued function, it doesn't quite work out.

But you are ignoring a possible answer. Do you only assume it to be valid when all values should work out? As long as I see at least one case where it's true I still consider it to be true in general. Or maybe we should only consider the absolute value of these and that will satisfy things.

antonfire
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Re: What is wrong with this? 1=-1?

So, you'd discard the transitivity of equality, would you? Have fun with that.
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Re: What is wrong with this? 1=-1?

antonfire wrote:So, you'd discard the transitivity of equality, would you? Have fun with that.
After about a seconds thought, I've decided that that'd not be fun.

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Re: What is wrong with this? 1=-1?

cspirou wrote:As long as I see at least one case where it's true I still consider it to be true in general.

Then you're completely wrong about the meaning of "in general"...
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cameronv
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Re: What is wrong with this? 1=-1?

Nothing is wrong with it by Inverse mathematics, infinity sequence has the differential of Absolute 1 as 1+ , -1, +1, representing the absolute proprtion of infinity. the three differentials represent two exverse+ dimensions and 1 inverse- dimension ==I absolute which has no plus or minus signs appendaged to it , but is 010 absolute.

Many equations in linear math are symmetrical/correct but not proprtionate/right. NEWTONS laws were Absolute ! proprtional . Likewise +1---------------(-1)--------------+1= Absolute 0- 1- 0 Einsteins is not , neither according to his theory is C a absolute Constant , it is E constant(c2), the vector 19 speed reference of the space medium(only a cone 19 degrees of inverse mathematics can handle such a speed . Mathematically I have almost proved that there are absolute minumum K constants for mass and Energy, and Maximum K constants for Mass and energy and both these are absolute

If more equations are proportional and more we realize the differential of 1, the more we will understand inverse mathematics and ultimately incorporate it into main mathematics. Inverse simply means curved conal off linear, and our space is all inverse(curved inwards), Infinity sequence is vital to inverse mathematics, and if you behave this non mathematician that developed inverse mathematics for medical analysis will tell more, may be an equation to calculate the square of mass which Einstein did not realize existed," it is the maximun compaction of mass by Gravity , and Maximum speed constant is the maximum compaction of space time and distance"

Brilliant 1=-1, linear is prop0rtional equal to the curved (by vector19 only)

"GOD DEALS HIS DICE AT VECTOR INVERSE(Curve)19"

auteur52
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Re: What is wrong with this? 1=-1?

I kind of missed this guy.

cameronv
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Re: What is wrong with this? 1=-1?

Actually this Brilliant Fellow has just helped prove the fallacy of Einsteins theorem by his mathematics and by inverse mathematics, might as well say it , cant just wound your king Einstein. Einstein was using an inverse constant[curved] with a value of-1 and C2 thus is a +1 value (derived value of constant) rather than 0 1 0 Absolute . Actually the correct equation for mass and energy is E=M=1absolute , where M is the maximum squared of m, I predict that when mass inverses in the Black holes the speeds of energy would be much much more that of light.

REMEMBER when Absolute 1 is represented by an Absolute Constant it has zero on each side[infinity sequence every thing must turn to absolute zero before it approaches Absolute 1], I hope you all are not using -+ signs with Absolute 1. Going back to this brilliant fellow, unsquared mass density(compaction by gravity) m (+1) =C(-1), since C is inverse to E constant, but Einstein could not square mass, so he squared C and derived himself a Absolute Constant with a value of +1, the rest is history fellows, 100 years of mathematics progress!.

Curved constants are -value 1, E constant(1 absolute) neither + nor neg- value , will be parallel to every thing. So fellows I dare say the greatest tensor is +1= -1 or derived constant, but it needs another +1 as per above infinity sequence, to be Absolute 1. "even though it is mathematically equal, only in sequence , not by equation" There is your answer fellow. +1 is equal to -1 in a sequence such as infinity sequence or such.

Sir_Elderberry
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Re: What is wrong with this? 1=-1?

Well, not by equation, naturally, no, but I'm not sure I necessarily see the sequence.
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Re: What is wrong with this? 1=-1?

auteur52 wrote:I kind of missed this guy.

What is it? Is it human?
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Blatm
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Re: What is wrong with this? 1=-1?

He has 26 posts, but I can only find 2 of them. I really want to hear more of what he has to say.

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Re: What is wrong with this? 1=-1?

Is cameronv aiming for a Time Cube award?

What would Baron Harkonnen do?

cameronv
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Re: What is wrong with this? 1=-1?

Some one asked! IF the MODERATOR permits I can post the equation for determination of K max, and K min propertionate constant for all mathematical constants, in inverse mathematics which really should be integrated with your mathematics Here is the sequence for absolute E , where T time , Speed and Distances are compacted into 1, why not! most of you can understand if you really love mathematics and slow down and think independant.Integrated and inverse mathematics are 1. This is at vector 19 below

<><><><><><><> same as 1.0.1.0.1.0.1.0.1.0. same as +1-1+1. 0.+1-1+1.0 same as ---Absolute linear

"No Mathematics is not elite or educated , it is for ALLthose that think slowly and carefully,and yes we could some day compact energy, and some day discover the mathematic sequence of nuclear energy so it is detctable more efficienctly.

GOOD bye on this Post!

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Re: What is wrong with this? 1=-1?

SlyReaper wrote:Is cameronv aiming for a Time Cube award?

We need a new Gene Ray, since the old one has gone into "Racial War" mode.
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Re: What is wrong with this? 1=-1?

No replacement would be quite the same.

Rilian
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Re: What is wrong with this? 1=-1?

Someone showed me something like this using i, and a complicated string of equalities and I pointed out that it's just a more complicated form of saying 1=sqrt(1)=-1, which anyone past 8th grade can tell is stupid and wrong.
And I'm -2.