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        riddles >> medium >> Medium: 2 = 1
        
(Message started by: ootte on Jul 24^th, 2002, 4:05pm)

Title: Medium: 2 = 1
Post by ootte on Jul 24^th, 2002, 4:05pm

"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.

--
Oliver

Title: Re: Medium: 2 = 1
Post by Charlie Dobbie on Jul 27^th, 2002, 3:33am

Sorry, nope. The last step doesn't show b = 1, it shows dividing through by b. Keep trying!

Title: Re: Medium: 2 = 1
Post by ootte on Jul 27^th, 2002, 3:55am

on 07/27/02 at 03:33:23, Charlie Dobbie wrote:

Sorry, nope. The last step doesn't show b = 1, it shows dividing through by b. Keep trying!

Uh man. Ok, take the first step a = b.
One must still divide by (a-b), which is zero.
q.e.d.

--
Oliver

Title: Re: Medium: 2 = 1
Post by mook on Aug 2^nd, 2002, 2:38pm

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.

Title: Re: Medium: 2 = 1
Post by Eric Yeh on Aug 3^rd, 2002, 7:20am

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

Title: Re: Medium: 2 = 1
Post by James Huckaby on Aug 26^th, 2002, 7:14am

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

Title: Re: Medium: 2 = 1
Post by arvind mayank on Oct 7^th, 2002, 12:43pm

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.

Title: Re: Medium: 2 = 1
Post by todd on Nov 20^th, 2002, 8:15pm

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.