Subject: wanted: proofs that 0 = 1
Newsgroups: sci.math,k12.ed.math

Hi, i'm looking for short (but convincing :-) proofs that 0 = 1.
  (The most obvious ones abuse the square-root sign,
  or divide by zero in a subtle way.)
Do people have ideas/pointers to other such proofs?
For instance, perhaps induction proofs whose inductive steps are valid, but
whose base case(s) are innocuously deceptive?

As an example, a cute high school geometry proof follows.

thanks in advance,
ian

========
Theorem:  0 = 1

Proof:
(The notation "Y =~ Z" means "Y is congruent to Z".)

Construct quadrilateral ABCD as follows:
(1) angle ABC is a right angle;
(2) angle BCD measures 100 degrees;
(3) side AB =~ CD.

(Note that AD and BC aren't quite parallel, since
             D is slightly closer to the line BC than A is.)
Construct the perpendicular bisectors of BC and AD.
Since AD and BC are not parallel, their perpendicular bisectors
aren't parallel, and thus meet at a point X.
(The same "proof" holds whether X is inside or outside ABCD.)
[Sorry, too hard to draw all this in ascii.]

First we show that triangle ABX =~ DCX:

AB =~ DC, by construction.
BX =~ CX, since X is on the perpendicular bisector of BC.
AX =~ DX, since X is on the perpendicular bisector of AD.

Thus triangle ABX =~ DCX (since they have three corresponding congruent sides).
Therefore angle ABX =~ angle DCX (corresponding angles of congruent triangles).
But also, angle XBC =~ angle XCB
                 (since they are the base angles of isosceles triangle XBC).
Thus we have that right angle ABC is congruent to obtuse angle BCD--
90 degrees = 100 degrees; it follows immediately that 0=1 as claimed.