1=2

When I was in high school, I was told that you can't divide by zero, but never shown why. "It's undefined," they tell you, but that's just a restatement of the rule. I recently came across (on some blogs called "mathematics learning" and "spirit of mathematics") a fun and effective way to justify this rule to your students. You simply prove that 1 equals 2, as in the following:

1. A = B let A equal B
2. A^2 = AB multiply both sides by A
3. A^2 - B^2 = AB - B^2 subtract B squared from both sides
4. (A+B)(A-B) = B (A-B) factor both sides
5. B = A + B divide both sides by (A-B)
6. B = 2B substitute B for A, since A=B
7. 1 = 2 divide both sides by B

Dividing by zero (A-B) in step 5 is what leads to the result that 1=2. Since every other operation is valid, this is a valid and effective (albeit informal) reductio ad absurdum. I think teenagers could get it, since it is so simple and powerful.

Here is another version with simpler operations (no factoring):

1. A = B
2. A^2 = AB
3. A^2 + A^2 = A^2 + AB
4. 2(A^2) = A^2 + AB
5. 2(A^2) - 2AB = A^2 + AB -2AB
6. 2(A^2) - 2AB = A^2 - AB
7. 2(A^2 - AB) = A^2 - AB
8. 2 = 1

How do you explain to your algebra students why they can't divide by zero?