(n+1)^2 = n^2 + 2n + 1
or, (n+1)^2 - (2n+1) = n^2
subtracting n(2n+1),
or, (n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)
or, (n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2
or, (n - 1/2)^2 = (n + 1/2)^2
or, (n - 1/2) = (n + 1/2)
or, -1/2 = +1/2
or, 1 = 0
or, 2 = 1 !!
Second Way
4*4 = 4 + 4 + 4 + 4
:. 4*4= 4 + 4 +… 4 times.
:. x*x = x + x + … x times.
Differentiating,
2*x = 1 + 1 + 1 … x times.
:. 2*x = x
Hence 2 = 1!!!
Third Way
Every1 here knows the pretty little Binomial thorem going as :
(a+b)n = an + n*b*an-1 + ... + n*bn-1*a + bn
Observing we find that leaving the first and last term, all other terms are a multiple of n.
For some wierd results, let us put n = 0 ;
LHS = (a+b)0 = 1
RHS = a0 + 0 ... + 0 + b0
= 1 + 0 ... 0 + 1
= 2
:. since LHS = RHS
2 = 1
Fourth Way
- 1 = - 1
or, sqrt(-1) = sqrt(-1)
or, sqrt(-1/1) = sqrt(1/-1)
or, sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)
or, i/1 = 1/i
or, i2 = 1
or, -1 = 1
or, 2 = 0
or, 1 = 0
or, 2 =1
Fifth Way
The very interesting infinitie series :
0 = 0 + 0 + 0 + 0 ...
For those who don't find it so interesting, follow me.
0 = ( 1 - 1 )
:. 0 = ( 1-1 ) + ( 1-1 ) + ( 1-1 ) ...
Now all of you would have paid attention in primary school when we were taught the associative law of addition.
Interestingly applying that here,
or, 0 = 1 + (-1+1) + (-1+1) ...
or, o = 1 + 0 + 0 + 0 + 0 + ...
or, 0 = 1
or, 1 = 2
Sixth Way
- 20 = - 20
or, 25 - 45 = 16 - 36
or, 52 - 5*9 = 42 - 4*9
or, 52 - 5*9 + 81/4 = 42 - 4*9 + 89/4
or, (5 - 9/2)2 = (4 - 9/2)2
or, (5 - 9/2) = (4 - 9/2)
or, 5 = 4
or, 2 = 1
The question is, Can you find another way of proving it?
6 comments:
Uh ... I am like the idiot here. But just one contradiction.
In proof 5, when you prove 0=1, that is fine, but then you can't add 1 to both sides, as that proves 1=2, so infact, you have to add 1 to one side, and 2 to the other side, or vice versa. But, I hope you get my point.
You can't add 1 on both sides if it later proves 1=2.
I too have to say something about proof 5. When you are applying the law of addition, then you take out 1 but the last no. that would be left in the series would be -1 alone and it wouldn't have 1 to go with it. That 1 would be the 1 that is taken out after applying the law so that equals 0. So, according to me it is wrong.
I m BACK
Shubham,
I think you are in class 9 and you aren't equipped with the properties of "infinity"
The proof simply uses the property
"infinity=infinity+1"
That "..." in the first statement meant series going to infinite.
Your reason would have been correct if the series was going till n terms
You'll know more about infinity and its different behavior once you reach class 11.
I hope I made sense
Other moderators, please correct me if I am wrong
This is a stupid proof, but here goes:
x - x = 0
Or, 1(x - x) = 2(x - x)
Or, 1 = 2
=P
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