infobahn said:
Let a = b.
Then multiply each side by b, so: ab = a^2
Now subtract b^2 from each side: ab - b^2 = a^2 - b^2
Factorise: b(a - b) = (a + b)(a - b)
Divide each side by common term: b = a + b
Substitute a (because a = b): b = 2b
Divide by b: 1 = 2
QED.
;-)
I like it!
Of course, there are some flaws in your proof (but well-hidden, which is
what makes this fun!)...
Your conclusion itself is incorrect. Given the formula b = 2b, there are
two "potential" solutions. Either 1=2, or b=0. Since no value can be
substituted for 1 which makes it equal 2 (as it is a constant, and will
accept only the value 1), the only answer available is that b equals zero.
So, does that prove that if a=b, then a=b=0?
Obviously, the real culprit is that you can't divide by zero. The operation
where you divide both sides by some value (a-b, in this case), is only valid
iff that value is non-zero. As I recall, the form that line should take
would be
b = a + b { a-b != 0 }
(Which is a fallacy, since we already said a=b, which implies a-b=0.)
But it's still great fun! I'd like to give it to an early Algebra class and
see what they have to say about it... :-0
-Howard