prove 1=2 ???

Answer 1

a = b
a^2 = a*b
a^2-b^2 = a*b-b^2
(a+b)(a-b) = b(a-b)
(a+b) = b
a+a = a
2a = a
2 = 1


I do understand the problem of 2=1, but where and is the equation
incorrect? And most important, why?

a = b
multiply both sides by a
a^2 = a*b
subtract b^2 from both sides
a^2-b^2 = a*b-b^2
apply the distributive law to both sides
(a+b)(a-b) = b(a-b)
divide both sides by (a-b)
(a+b) = b
substitute all a's for b's (remember, if a = b you can do this)/
a+a = a
regroup the two a's in the left side, and rename it 2a
2a = a
divide both sides by a
2 = 1


The point which I highlighted in red is where you would have to divide each side of
(a+b)(a-b)=b(a-b)
by (a-b) to get
(a+b)=b


But you will notice that if the first part of the proof (a=b) is true, then (a-b) would equal 0.
And, as we know, we can't divide by 0, so from that point on this "proof" that 1=2 is relegated to the non-valid, but interesting, heap of other things which people try to use to prove things which are also false.

Answer 2

I'm going to present to you a *FALSE* proof that 2 = 3.

Let a = 1 and b = 1. This means that

a = b. Square both sides, and we get
a^2 = b^2. Therefore,
a^2 - b^2 = 0.

a = b. Multiply a both sides, and we get
ab = b^2. Therefore,
ab - b^2 = 0

Since a^2 - b^2 is equal to 0, and since ab - b^2 is equal to 0, we can equate them.

a^2 - b^2 = ab - b^2.

We can factor the left hand side as a difference of squares, and the right hand side normally, to give us

(a - b)(a + b) = b(a - b)

Now, notice we have (a - b) on both sides. That means we can cancel them, to give us

a + b = b

Plugging in our original values a = 1 and b = 1, it follows that

1 + 1 = 1, or
2 = 1.

Answer 3

1 does not equal 2. Unless you're willing to accept parallel lines crossing, a big square circle, and an invisible pink unicorn in the proof.

Answer 4

It’s easy to prove:

a=b --> a² =ab --> a² - b² = ab - b² --> (a+b) (a-b) = b (a-b) --> a + b = b
now put a instead of b --> a + a = a --> 2a = a --> 2 = 1 !

Answer 5

I have seen two ways on the internet, but both of them break some math rules.

Answer 6

let 1=x, then
x=2

to check
the value of x=2
then x which is 1=2

Answer 7

One with a split personality.