1=2 Can anyone prove this?

Question

Its a mathematical & logical question.

Answer 1

Here is the false proof that 1 = 2.

Let x = 1 and y = 1.

Then x = y.

If we multiply both sides by x, we get

x^2 = xy. Therefore

x^2 - xy = 0

Going back to x = y, if we square both sides of x = y, then

x^2 = y^2. Therefore,

x^2 - y^2 = 0

Now we have 2 things equated to 0. That means we can equate them to each other. That is

x^2 - xy = x^2 - y^2

Factor both sides,

x(x - y) = (x - y)(x + y)

Divide both sides by (x - y), and we have

x = x + y

Plug in x = 1 and y = 1 back, and we have

1 = 1 + 1
1 = 2.

This proof is false, and the logic fails at one point. In actuality, if there was an actual proof out there that 1 = 2, the mathematical foundation would crumble.

Answer 2

Here is a serious PROOF. THere is no trickery involved, here. In logic, the symbol = can stand for any equivalence relation under consideration, provided that it satisfies the logical precepts for an equivalence relation, namely:

(1) A = A for all A (Reflexive Property)
(2) If A = B, then B = A (Symmetric Property)
(3) If A = B, and B = C, then A = C. (Transitivity Property).

There are a variety of mathematical systems, in which 1=2 is a valid logical statement. Here are two familiar examples:

I. {The real numbers modulo 1}. Here, we equate each number with its fractional part, by subtracting the greatest integer not exceeding it. Thus, pi is equivalent to 3.14159... - 3 = .14159....
and -3.2 is equivalent to -3.2 - (-4) = .8
In this system, which has many practical applications, 1 and 2 are equal, for both are equivalent to 0.

II. Define an equivalence relation on the set of real numbers by classifying all real numbers according to their sign. In one form of this, we write all positive numbers as equivalent to +1, all negative numbers as equivalent to -1, and 0 as equivalent to 0.

Answer 3

Let a=b,
Then a^2=ab(multiplying both sides by a)
a^2-b^2=ab-b^2(subtracting b^2 from both sides)
(a-b)(a+b)=b(a-b)(factoring)
a+b=b{dividing by (a-b)}(This step is incorrect.)
2b=b( replacing a by b)
2=1
Well there are many such mathematical pardoxes, like how to prove 1Re=1ps, how all triangles can be proven to be isosceles, how sqrt(1)=+-1, how convergence of infinite series gives S=0 at one time and S=1 at the another, how (cosx)^2+(sinx)^2=1 gives 0=4. You can refer 'Math Charmers-Alfred Posamentier' to know more about such paradoxes.

Answer 4

1=1
then 1^2=1^2
=> 1^2-1^2=1^2-1^2 [subtracting 1^2 from both side]
1(1-1)=(1-1)(1+1)
=> 1=1+1
=> 1=2*1
=> 1=2

hence proved

Answer 5

multiply both sides by 0

1=2
1*0=2*0
0=0

Answer 6

a=a then a^2=a^2 => a^2-a^2=a^2-a^2
a(a-a)=(a-a)(a+a) => a=a+a => a=2a => 1=2

Answer 7

No. The "proof" is a logical absurdity.