Being a sixteen year old who has no above-average skills or talents, I’m wondering if this is legal. How can you multiply by sqrt(1) on one side and divide by it on the other and call it a simple operation? Squaring both sides altogether’d give 1=1/1, and multiplying by inverses of eachother might be like the first step, but it still seems wrong for some reason…
(I.E., show me a proof that says this is okay so I can just write it on the top of tests and argue any answer is right, so long as I have sig figs)
Erm, sorry the
sqrt(1) = -1 / sqrt(1)…Multiply sqrts together
1 = -1 / 1…Simply square roots.
is supposed to be independently bolded instead of the whole thing.
Cabbage, ultrafilter- Shush shush, shush.
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The error is indeed that the square root function is not single-valued, this is true. The error is NOT introduced at the point you specify. The error is introduced when the sqrts begin to SIMPLIFY- because each square root simplifies to TWO values, +/- 1 in this case.
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No. Cabbage. Wrongo. You may always distribute a square root over a fraction. The error creeps in because you’re committing the same error the proof is; you’re not accounting for the second result of the square root function.
The correct answer to the little counter example you spat out is:
-/+ i = -/+ i (or something that means this- my notion is rusty.) There certainly is no contradiction here.
- ** Iacob**, I’ll look up the proof for you, dont know it off the top of my head. But I think it’ll be useful to clam up these mathematical blowhards.
-C
Oops. I bolded the wrong equation. This is the one I meant:
-1 = -1
1/-1 = -1/1…Equivilant to the above statement
sqrt(1/-1) = sqrt(-1/1)…Take the square root of both sides
sqrt(1)/sqrt(-1) = sqrt(-1)/sqrt(1)…Distribute the root
sqrt(1) = sqrt(-1)sqrt(-1)/sqrt(1)…Multiply both sides by sqrt(-1)
sqrt(1) = -1 / sqrt(1)…Multiply sqrts together
1 = -1 / 1…Simply square roots.
1 = -1 …Simply further.
I give up.
-C
Maximum C, it’s not uncommon at all for the square root to be defined as a function, in the following manner:
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For nonnegative x, sqrt(x) = the nonnegative number whose square is x.
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sqrt(-1) = i.
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For nonnegative x, sqrt(-x) = i * sqrt(x).
This is a very standard definition of sqrt. Use whatever definition you want, but I’ll stand by this definition and my original claim.
Well gosh!
I never expected my most successful thread to be about maths…go figure.
Hope you’re having fun with the old “-1=1” proofs, if anyone wants to prove the Riemann-Lebsgue Lemma for me I’d appreciate it because there’s a good chance I’ll have to do it in an exam in about 36 hours…
God i love engineering finals.
Seriously.
You learn this crap and the last week are told, “Now you know which building to walk into to pay someone to do it for you. Because math majors are poor.” and you can then promptly forget it.
This reminds me.
Second year calculus was the only time in my high school career that the teacher told us that our material had absolutely no practical use.
But by then, the class was filled with math geeks like myself, so we didn’t really care that it was completely useless. 
What are we talking about here? I thought that calculus was invented so people to get some good answers to physics problems.
Those who think of math as useless probably don’t go into technical fields where they use it, and lack the mental dexterity to appreciate the art for what it is.
And they be tripp’n, yo. Stay outta my hood, dig.
-C
my favorite is this:
x=y
xx=yy **************square both sides
-xx=-yy *************multiply by -1
xy-xx=xy-yy *********add (xy)
(y-x)x=(x-y)y ********undistribute
-(x-y)x=(x-y)y ********multiply LHS parens by -1
-x=y ********divide by (x-y)
**-(x-y)x=(x-y)y ******multiply LHS parens by -1
There error is there, correct?
nope. -(x-y)x does equal (y-x)x, they both evaluate to -xx+xy.
Dividing by zero baby (x-y)
Actually, that problem isn’t so tough, as long as you follow along with an imaginary x and y value (which, of course, are the same). Once you do this, it’s pretty straightforward to find where the error lies.
Yes. but you’re doing something to one side of the equation that you’re not doing to the other, which I had beaten into my skill is a Bad Thing™.
Q.E.D., note that he also rearranged what was inside the parentheses. The explanation was a little unclear, but certainly (x-y) = -(y-x)…
Why should I respect the math? The math does not respect me. I never did anything to the math, and yet it is constantly bothering me. Why can’t it take it’s silly exponents and exes and whys and negatives and positives and squares and pies off somewhere by itself and leave me the hell alone? Beastly, tricksy stuff.
[quote]
quote:
Originally posted by cmosdes
Assume x > 0
x*x = x + x + x + x + x + x + x + … x times
d(x*x)/dx = d(x + x + x + x + …)/dx
2x = 1 + 1 + 1 + 1 + 1 + … x times
2x = x
2 = 1
Yes, this one is clever too. A variant on the “subtly dividing by zero” school; sneaking in an equation that forbids certain operations. I wont point out the answer, though- for the sake of other’s enjoyment.
[\quote]
I’m not sure where the “subtly dividing by zero” parts comes in. My understand of the fault with this lies in the fact that differentiating a sum is not (necessarily) the same as summing the differentiation.
i = x i = x
d(sum(x[sub]i[/sub]))/dx != sum(d(x[sub]i[/sub]/dx)
i = 0 i = 0
What equation did I sneak in?