Two more methods of incorrect proof, somewhat less commonly seen than those previously addressed, but also in some ways more

subtle:

**Trick: Use of Ill-Defined Expressions/Misapplication of Definitions**

We will prove that 1=2 using calculus and the definition of `x`^{2}:

(*) First note that `x`^{2} is equal to `x`*`x`, that is, `x` added to itself `x` times:

`x`^{2} = `x` + `x` + ... + `x` (done `x` times)

Take the derivative of each side, on the left using the power rule, and on the right using the sum rule:

2*`x` = 1 + 1 + ... + 1 (done `x` times)

Now 1 added to itself `x` times is simply `x`:

2*`x` = `x`

Divide each side by `x`...

2 = 1

And QED! A falsehood is proved, mathematics as we know it is inconsistent, the stars fall into themselves, and the universe ends. Uh-oh.

The main flaw in this proof is that it relies on our alternate expression of `x`^{2} above at (*). This is a prefectly adequate definition for `x`^{2}, but *only if x is a positive integer!* Our definition does not hold if `x` is any other real number; in particular using this definition to create the function `y`=`x`^{2} does not give us a continuous function. So its derivative is undefined, and the second step of our proof is not correct. Since we are trying to perform a continuous operation (differentiation) on something defined discretely, we are committing a major error.

**Trick: Poorly Constructed or Misleading Diagram**

It is difficult to node about geometry, since one's graphical abilities are limited in this medium, but I shall do my best to give a written explanation.

The canonical example of false proofs of this type are puzzles where one dissects a square into various pieces and reassembles them into a rectangle of an apparently different area than the original. See a similar puzzle at the disappearing area problem.

A neater example of this trick, given explicitly in proof form, involves "proving" that a particular triangle has more than 180 degrees in its three angles, a violation of the triangle sum rule. Although it relies on extensive use of diagrams that cannot be duplicated here, I will do my best to accurately describe the process by which this so-called proof does its work. Once again, the key here is to draw this diagram poorly:

Disproof of the Triangle Sum Rule:

Begin by drawing two circles which intersect each other at two points. Call the two points of intersection A and B. From A, draw a diameter of each circle.

Call the other endpoints of the diameters P and Q. Now our two diameters are AP and AQ. Connect P and Q to form triangle APQ. Now PQ will intersect each of the two circles, call these two points of intersection H (on the same circle as P) and K (on the same circle as Q).

Now draw in AH and AK. Consider angles AHP and AKQ. Since each of these angles subtends a diameter, each angle's measure is 90 degrees. But this means that triangle AHK has two 90 degree angles, an impossibility, since the third angle is yet unaccounted for and every triangle must have exactly 180 degrees by the Triangle Sum Rule. What has gone wrong?

The trick here relies on a sloppy diagram. If you construct your diagram carefully, you will see that PQ actually passes through point B. This means that H and K are actually the same point (point B), and that there isn't really a triangle AHK--it's just a line segment, AB.

This type of trick is particularly devious since when one is reading through a proof, one is often depending on the author's own diagrams--and the author can slightly skew the diagrams however he or she sees fit to mislead you.

A similar false proof may be found at Proof that all triangles are isosceles, once again relying on a misleading diagram (or in this case an assumption about the positioning of elements in a description of a diagram).