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Topic: Medium: 2 = 1 (Read 1614 times) |
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ootte
Newbie
Posts: 19
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Medium: 2 = 1
« on: Jul 24th, 2002, 4:05pm » |
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"Proof" that 2 = 1:
a = b
a^2 = ab
a^2 - b^2 = ab-b^2
(a-b)(a+b) = b(a-b)
a+b = b
b+b = b
2b = b
2 = 1
This argument doesn't make sense. In the last step, you clearly see that b = 1. So, if taken the first step into consideration you see that a = b hence a = 1. In Step some divided by (a-b) = (1-1) = 0, and division by zero is not allowed. So 2 still doesn't equal 1.
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Oliver
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Charlie Dobbie
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Sorry, nope. The last step doesn't show b = 1, it shows dividing through by b. Keep trying!
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ootte
Newbie
Posts: 19
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Re: Medium: 2 = 1
« Reply #2 on: Jul 27th, 2002, 3:55am » |
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on Jul 27th, 2002, 3:33am, Charlie Dobbie wrote:| Sorry, nope. The last step doesn't show b = 1, it shows dividing through by b. Keep trying! |
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Uh man. Ok, take the first step a = b.
One must still divide by (a-b), which is zero.
q.e.d.
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Oliver
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mook
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the first line of the equation a=b, is a hypothesis. You can do whatever you want to do on both sides of the equal sign for eternity, why not multiply both sides by one then divide both sides by 50, in the end a will still equal b, but 1 will never equal 2.
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Eric Yeh
Senior Riddler
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Posts: 318
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Re: Medium: 2 = 1
« Reply #4 on: Aug 3rd, 2002, 7:20am » |
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Guys:
For a "senior" version of this puzzle, see the new thread Icreated under hard/"NEW PROBLEM: Picard's Theorem Proof that 0 = 1".
Enjoy!
Eric
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"It is better to have puzzled and failed than never to have puzzled at all."
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James Huckaby
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At the step where:
(a - b)(a + b) = b(a- b)
since a=b
this the equivalent of saying
0 (a+ b) = b* 0
This is still true but the following
step of:
a + b = b
is only true if a = b = 0.
You cannot go from
0 * 2 = 0 * 1
and get
2 = 1
This is simply bad math.
Evaluating 2b = b
to 2 =1 is erroneous for different reasons.
Properly you would evaluate that by subtracting
b from both sides to get:
b = 0
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arvind mayank
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simply
bcoz a = b
then a- b = 0
(a-b)(a+b) = b(a-b)
0x(a+b)=bx0 is never defined we cannot cancel 0 from both side.
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todd
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ok, I don't think y'all's whole "division by 0 isn't allowed" argument really gets at the point of the problem. see, what you've got is (a-b)*(a+b)=(a-b)*b -> (a-b)*(a+b-b)=0, an equation with two solutions, namely a=b and a=0. this is perfectly legal, but it's also the trick to the "proof."
on the second step, when they squared both sides (a*a=a*b), they really slipped one by you by making the equation quadratic. instead of a=b being the only root, now a=0, b=whatever also solves it. when they cancelled (a-b) from both sides, the problem isn't division by 0, but the fact that you just lost the root you started with. now your only root is a=0, but they just go on like a still equals b.
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