Are all numbers equal? What is mathematically wrong with this proof?

Question

Theorem: All numbers are equal (We are trying to prove that all numbers are equal)
Proof: Choose arbitrary a and b, and let t = a + b. Then

a + b = t
(a + b)(a - b) = t(a - b) ........multiply both sides by (a - b)
a^2 - b^2 = ta - tb  ..............Remove brackets
a^2 - ta = b^2 - tb  ..............Put terms with b on one side, and those with a.
a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4   ..............Add (t^2)/4 on both sides.
(a - t/2)^2 = (b - t/2)^2  ....................Factorize both sides
a - t/2 = b - t/2   .............................Find sqrt on both sides
a = b   ......................conclusion

So all numbers are the same

Dan A · Accepted Answer

"
(a - t/2)^2 = (b - t/2)^2 ....................Factorize both sides
a - t/2 = b - t/2 .............................Find sqrt on both sides
"

(a - t/2)^2 = (b - t/2)^2 does not necessarily mean that a - t/2 = b - t/2. The other alternative is that a - t/2 = -(b - t/2)

eg. if a=7, b=3, t=10, then:

(a - t/2)^2 = (b - t/2)^2
(7-5)^2 = (3-5)^2
(2)^2 = (-2)^2
4 = 4

i.e. that step is ok, however:

a - t/2 = b - t/2
7 - 5 = 3 - 5
2 = -2

That step is not ok! This is because you assumed that a - t/2 and b - t/2 were both non-negative, which only occurs when a=b i.e. your "proof" is only valid when a=b to begin with, which is why it leads to that conclusion.

EDIT: x^2 = y^2 does not mean that either x=y or -x=-y, it means that x=y or x=-y. This is the problem with your justification. I do agree with you though that a-b=0 is not a problem. If this is the case, then every step is valid. However, if a-b=0, then a=b, which is what you end up proving anyway.

mark p · Answer

Your proof is fine until the next-to-last step, where you take the square root.
sqrt(a*a) = +/-a, not just a.

While it is true that (-3)^2 = (3)^2 this does not imply that 3 = -3

Notice that t/2 is basically the average of a and b.
So, (a-t/2) and (b-t/2) have the same magnitude, but opposite signs. The squares are equal, but that doesn't mean the numbers are equal.