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## Main Question or Discussion Point

[tex] -1 = \sqrt[3]{-1} = -1 ^ \frac{1}{3} = -1 ^ \frac{2}{6} = ((-1)^2)^\frac{1}{6} = 1^\frac{1}{6} = 1 [/tex]

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[tex] -1 = \sqrt[3]{-1} = -1 ^ \frac{1}{3} = -1 ^ \frac{2}{6} = ((-1)^2)^\frac{1}{6} = 1^\frac{1}{6} = 1 [/tex]

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matt grime

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i really want that FAQ.

mirror question: what makes you think it is good logic?

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matt grime

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why would you need to explain it using elementary mathematics? rule: you cannot replace fractional powers with other (equivalent in Q) fractions. reason: cos it buggers up the maths. you don't need any more justification than that.

there are six numbers that raised to the sixth power give one. -1 is one of them, so it's as good as it can be, why even go to this length? square -1 then square root it, 1=-1 apparently, this is just the same.

there are six numbers that raised to the sixth power give one. -1 is one of them, so it's as good as it can be, why even go to this length? square -1 then square root it, 1=-1 apparently, this is just the same.

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matt grime

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Yeah, like me. I am seventeen years old, I have had some maths (over here in holland, i am in fifth grade VWO), but I can't see a logical or mathematical error in this one, other than it obviously being false.dimachka said:

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matt grime

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sqauring takes the reals to the positive reals, and we can square root only positive reals to obtain reals, and for convenience of the two numbers that sqaure to give x we take the positive square root for all positive x.

so it is no surprise that squaring and square rooting 1 and -1 gives the same answer. by design we cannot end up with a negative square root, and that is our choice.

the cuberooting and taking sixth roots is just the same 'trick'

it is not deep! it is not hiding some great mystery.

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yes i feel you are correct matt and this question can just be explained by explaining why [itex] -1 = (-1)^\frac{2}{2} = (-1^2)^2 = 1^2 = 1 [/itex] is wrong. But i dont exactly see how this can be explained through [itex] \sqrt{1} = 1 & \mbox{ or } -1 [/itex] Care to explain?

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matt grime

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What? I thought you were supposed to be teaching this? non-bijective functions cannot be inverted. taking the power 1/2 is not the inverse function of the map R --> R, x--->x^2. That map has no inverse. End. Fin. Nuff said.

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Probably I am just too stubborn too see it, as usual. For some reason I tend to forget what I am doing, and then I don't understand a thing anymore.

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matt grime

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see what? do you understand the idea of bijective function? of inverse functions? that is the reason why this happens; you're looking far too hard for something that isn't there.nazgjunk said:

Probably I am just too stubborn too see it, as usual. For some reason I tend to forget what I am doing, and then I don't understand a thing anymore.

if you square to different numbers you can get the same answer thus there is no way to undo the operation of sqauring on all numbers: the square root cannot know, if you squared -1, that you want it to give the negative square root. Look at it this way. i have a number, its square is 1, what was the number? see, can't tell me. that is all that's going on.

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matt grime

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dimachka said:

that would depend upon the person you're taking to obviously. however, that is for you to decide in teaching it since only you know to whom you are explaining it.

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matt grime

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but that isn't accurate. it is prefectly possible to have none reversible implications and still have a sound argument, indeed that is the point of implication.nazgjunk said:

the difference here is that you are *claiming* that the steps are reversible implicitly, when they aren't. the point is yo'ure just making a false and unkustified claim that people don't notice is false.

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matt grime

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dimachka said:

i want to amend my opinion: that is false.

it is not the algebriac manipulations need to be reverisible, just that each logical step is actually sound to begin with. it so happens that becuase you cannot invert squaring that you have made a logically inconsistent deduction.

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Geez, this is getting me depressed again.

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matt grime

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I think it is that each algebraic manipulation must be reversible, i think that is the crux of algebra. In this case, the reason the operation of squaring is not reversible is precisely because you cannot invert to get a unique answer, but instead have two answers.matt grime said:it is not the algebriac manipulations need to be reverisible, just that each logical step is actually sound to begin with. it so happens that becuase you cannot invert squaring that you have made a logically inconsistent deduction.

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I can indeed come up with x=-1 and y=1, which proves you are right on this one. But to my relatively low-educated head it feels logical though, that if x^2=y^2 then x=y.matt grime said:

This is getting me even more depressed...

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matt grime

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It depends upon what you're claiming.dimachka said:I think it is that each algebraic manipulation must be reversible, i think that is the crux of algebra. In this case, the reason the operation of squaring is not reversible is precisely because you cannot invert to get a unique answer, but instead have two answers.

for instance the proposition, for x and y in R

x=y implies x^2=y^2

is true.

It is not a reversible algebraic manipulation.

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matt grime

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So despite the fact you can prove it is false you think it ought to be true? Then that is psychological issue not a mathematical one: in face of evidence to the contrary you wish to believe something is true.nazgjunk said:I can indeed come up with x=-1 and y=1, which proves you are right on this one. But to my relatively low-educated head it feels logical though, that if x^2=y^2 then x=y.

This is getting me even more depressed...

Now, if i'd said, x and y are positive real numbers and x^2=y^2 then i can prove that x=y. in general i can prove that x=y or -y, since if x^2=y^2 then (x-y)(x+y)=0 and then either x-y=0 or x+y=0. Note that if x and y are constrained to both be positive then the only way for two positive numbers to add to zero is if they're both zero, so either x-y=0 and x=y or x=y=0 and again x=y.

(note i am just for the sake of argument taking 0 to be positive)

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ahh, i see what you are saying, so not necessarily reversible, just only allowed to broaden scope, rather than narrow your scope. I can't exactly see how to make a more concrete explanation of this.matt grime said:It depends upon what you're claiming.

for instance the proposition, for x and y in R

x=y implies x^2=y^2

is true.

It is not a reversible algebraic manipulation.

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No, by god. Why did I ever think I was too stubborn? You are far worse. I said it "felt" right, which doesn't mean I believe it. In Dutch class, I feel that it should be answer A, but with some reason I have to admit it is B. OK, this is a crappy example, but I hope you get the point.matt grime said:So despite the fact you can prove it is false you think it ought to be true? Then that is psychological issue not a mathematical one: in face of evidence to the contrary you wish to believe something is true.

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