What if I told you that I can prove, mathematically, that one plus one equals...one, not two? That, in fact, one equals two? Not philosophically, but
literally—mathematically. Wouldn't that shake things up a bit in every field that employs algebra?
Let's consider:
a = b
Then,
a2 = b2. (That's squared; I don't know how to use the proper "raised to" sign here on the computer...)
Then we can subtract b2 from both sides,
a2 - b2 = b2 - b2
but we know that a2 - b2 = (a+b) x (a-b)
So, we have...
(a+b)(a-b) = b2 - b2.
Now, b2 can be written as ab, as a = b. So...
(a+b)(a-b) = ab - b2.
Take b out common from the RHS...
(a+b)(a-b) = b(a-b)
The term (a-b) is common to both sides, and can be eliminated.
Then we have...
a + b = b.
a = b. So, substitute a by b...
b + b = b; (That is...1 + 1 = 2.)
2b = b.
Or...2 = 1.
Note: I don't claim to "invent" this or anything. I remember reading about this somewhere many years ago, and I'm merely presenting it here.
P.S.: The catch...there is always one....is that the term (a-b) cannot be eliminated from both sides. The reason? Well, a-b is a zero-value term
because a equals b, and division by zero is not acknowledged. So, alas, technically, this was a fail, but for someone who doesn't quite notice this
little thing...it's a very interesting math prank.