Proofs that A=B, for whatever A and B (e.g. that 1=2), are classical proofs, teached into school to show children students why a division by zero cannot exist.
For example:
- first of all, if A<>B then:
- A-B=C
- now, square both sides:
- A²-2AB+B²=C²
- since (A-B)(C)=C²=AC-BC, we can rewrite:
- A²-2AB+B²=AC-BC
- rearranging all, we get:
- A²-AB-AC=AB-B²-BC
- factorize both members:
- A(A-B-C)=B(A-B-C)
- cancel both members:
- A
(A-B-C)=B(A-B-C)
- A
- finally
- A=B QED
The catch is that since A-B=C, then A-B-C=0, and we have performed an illegal division by zero.
See also: paradox