How can I disprove that 1 = 2??

Question

My physics teacher told me that 1 = 2 because:
0 = 0
a^2 - a^2 = a^2 - a^2
a(a-a) = (a+a)(a-a)
cancel out (a-a)
a = a+a
divide by a
1=2????
how is this possible and how can I disprove it??

Answer 1

I was going to answer this before anyone else got in but got called away before I could enter my explanation (honestly).

Just about everyone here is on the right tracks right you cannot divide or multiply by zero and therefore cannot "cancel out" (a-a) from each side of the equation.

You cannot disprove his theory as what he is effectively saying is that 1 x 0 = 2 x 0 or for that matter = 1million x 0 = 0 which taken at face value is true, they all = 0. Ergo 1 = 2 = 1 million etc.

You can however, tell him his theory is flawed because it relies on being able to divide or multiply each side of the equation by zero which is not possible. Zero is zero . Add if you want to p*ss him off, that as your educator he should be educating you not playing stupid games! Ha ha.

But I suppose he will say he's done his job coz your thinking about it.

Answer 2

There was wrong in your interpretation of the whole formula from the start.

0=0
say a=1
1^2-1^2=1^2-1^2
so
1(1-1)=(1+1)(1-1)
1(0)=(2)(0)
going back to math

1/0=2/0 these mean that infinite=infinite
note anything by 0 is infinite or undefined

now if you do not want to cancel it out but multiply
this will happen
1*0=0
2*0=0

therefore 0=0which is your first assumption

where is the confusion? none
hope that this help. ok

Answer 3

When you cancel ( a - a ), what you really do is divide both sides by 0 which is an absurd ( incorrect ) mathematical operation. One can never divide any number by zero.

What you have done is 2 * 0 = 1 * 0
Cancel 0 on both sides to get 2 = 1.

Answer 4

You're not disproving that 1=2; you're simply showing the flaw in this one derivation.

Almost all such derivations depend on dividing by 0, which is not legal in most number systems. Look through the "proof", and you'll likely spot it quickly enough. The rest is left as an exercise for the student (Aaaaaarrrghh!!).

There are other proofs that depend on equating multiple roots (such as assuming that +1 = -1 because they're both sqrt(1) ... but being very subtle about that assumption).

Answer 5

Just go back to the basic rules of mathematics,
You have to solve calculations in brackets first,
so:
0=0
a^2 - a^2 = a^2 - a^2
a(a-a) = (a+a)(a-a)
a(0) = (2a)(0) -----> equivalent to a x 0 = 2a x 0
0=0

Therefore, it is zero from the start!!!
Obey the rules or otherwise you will get unlogical answer.
Am i correct enough?

Answer 6

2 = 1+1, by definition.
Cancelling out (a-a) is the same as dividing by zero and is meaningless.
2 x 0 = 1 x 0, but that doesn't make 2 = 1

Answer 7

1+1=2
1-1=0
2-1=1
1=2-1
1=1

Answer 8

old know it all has it correct!

Canceling out (a-a) means that you are dividing both sides of the equation by (a-a). By definition (a-a)=0 so you are diving both sides of the equation by zero. Dividing by zero is not a valid mathematical operation, therefore the result is not true.

Answer 9

cancelling out a-a is dividing by 0, which is not allowed, because, obviously, if you could divide by 0, any number would equal any other.

In math and logic, a system is consistent if following its rules never leads to a contradiction. 1 = 2 is an obvious contradiction, so no rule that would make it provable is allowed.

Answer 10

this is the wrong step:
cancel out (a-a)

when manipulating an equation you can multiply (or divide) both sides by any number BUT zero. And (a-a) = 0.