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Topic: 2 = 1 how is this solved ? (Read 986 times) |
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atgdax
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2 = 1 how is this solved ?
« on: Nov 22nd, 2006, 5:59am » |
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"Proof" that 2 = 1:
a = b
a2 = ab
a2 - b2 = ab-b2
(a-b)(a+b) = b(a-b)
a+b = b
b+b = b
2b = b
2 = 1
can any one explain ?
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towr
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Re: 2 = 1 how is this solved ?
« Reply #1 on: Nov 22nd, 2006, 6:02am » |
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the problem is the step from the 4th to 5th line
0*x = 0*y does not imply x=y
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Icarus
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Re: 2 = 1 how is this solved ?
« Reply #2 on: Nov 22nd, 2006, 7:04am » |
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Yes. The correct conclusion from line 4 is:
(a-b)(a+b) = b(a-b)
either a+b = b or a - b = 0.
Since the latter was assumed at the beginning, the whole statement is true even though the first part, a+b = b, is false (unless a = 0).
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smart_one
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Re: 2 = 1 how is this solved ?
« Reply #3 on: May 17th, 2007, 7:22am » |
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a=b
(a+b)(a-b)
a-b= a-a= 0
so 0*x=0 and 0*y=0
if we simplify 0
then it will become x=y which are not equal
and that wat leads us to 2=1 as x=y 
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Michael Dagg
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Re: 2 = 1 how is this solved ?
« Reply #4 on: May 18th, 2007, 5:49pm » |
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> if we simplify 0 then it will become x=y which are not equal
Please enlighten us on how one might simplify zero.
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Michael Dagg
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JiNbOtAk
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Re: 2 = 1 how is this solved ?
« Reply #5 on: May 18th, 2007, 6:07pm » |
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0*2=0 --- (1)
0*1=0 --- (2)
Equating (1) and (2)
0*2 = 0*1
The wrong method would be to simplify 0, ie cancelling 0 from both sides, which gives
2=1, which obviously is wrong.
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Michael Dagg
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Re: 2 = 1 how is this solved ?
« Reply #6 on: May 18th, 2007, 6:20pm » |
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> The wrong method would be to simplify 0, ie cancelling 0 from both sides, which gives ...
That's what I thought. Surely you know that it is
not possible to cancel 0 this way?
Alternatively, it is call division by zero.
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Michael Dagg
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Michael Dagg
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Re: 2 = 1 how is this solved ?
« Reply #7 on: May 18th, 2007, 6:38pm » |
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That is, dividing by zero is erroneous. But, what Towr
pointing out in lines 4 and 5 is "less" erroneous.
But, since you want to divide by zero why wait until
later. Go ahead and do it and get yet another
result:
a = b
a - b = 0
(a - b)^(-1) * (a -b) = (a - b)^(-1) * 0
1 = 0
Now, we have
2 = 1 and 1 = 0.
Since there seems to be multiplication allowed here
lets pick an integer n and volia!
2n = n and n = 0.
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Michael Dagg
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JiNbOtAk
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Re: 2 = 1 how is this solved ?
« Reply #8 on: May 18th, 2007, 8:11pm » |
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Quote: Now, we have
2 = 1 and 1 = 0.
Since there seems to be multiplication allowed here
lets pick an integer n and volia!
2n = n and n = 0.
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So it follows that all variables is equal to zero ? 
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Michael Dagg
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Re: 2 = 1 how is this solved ?
« Reply #9 on: May 18th, 2007, 8:47pm » |
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> So it follows that all variables is equal to zero ?
Yes, it is called nonsense.
Problems like these are nonsense. The surprising thing
that happens during a discussion of these kinds of
investigations is that nonsense is further perpetuated
(i.e. simplifying zero is a nice example) which is supposed
to lead to a conclusion that there is was an error but the
method of reference outlining the error is in error itself.
It is rather easy to get tripped up on this kind of thing
because it looks good.
There is a section in the College Math Journal called "Flaws,
Fallacies and Flim-Flam" that make up a collection of
these kinds of things -- some more advanced and difficult
to detect, that is, if you slip up on the flow of the presentation.
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| « Last Edit: May 18th, 2007, 8:51pm by Michael Dagg » |
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Michael Dagg
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Icarus
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Re: 2 = 1 how is this solved ?
« Reply #10 on: May 18th, 2007, 9:58pm » |
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A favorite of mine was a nice little geometric proof that all obtuse angles are equal to a right angle. I recall posting it once, but I can't find it now. When presented right, the error can be very subtle.
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And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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Michael Dagg
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Re: 2 = 1 algebra & Calculus 1 = 0  2eq1.JPG
« Reply #11 on: May 19th, 2007, 9:36am » |
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Observe
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Regards,
Michael Dagg
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Johan_Gunardi
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Re: 2 = 1 how is this solved ?
« Reply #12 on: May 20th, 2007, 8:49pm » |
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5*0=6*0, so is 5=6?
You can't divide any numbers by 0, it's infinite.
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