Posted by: Dave Richeson | December 18, 2008

## Fallacies, Flaws, and Flimflam it’s been good to know ya

At the end of 2008 the College Mathematics Journal will stop running its 20-year column “Fallacies, Flaws, and Flimflam.” The column was devoted to “mistakes, fallacies, howlers, anomalies, and the like”â€”usually found on student work. In honor of the departure of this entertaining column I submit the following FFF which was submitted as a solution to a homework problem in my Discrete Mathematics class.

Prove or disprove: the quotient of any two rational numbers is rational.

False. We will present a counterexample. The numbers 22 and 7 are rational. However $\frac{22}{7}=\pi$ and $\pi$ is irrational. Thus the conjecture is false.

## Responses

1. I can beat that with a submission from my geometry class. I forget what the theorem was the student was proving, but the first line was: “We will begin by assuming the conclusion.”

2. Excellent! At least (s)he was being clear about her/his assumptions!

3. 22/7 and pi are extremely close, but they are not equal.

• Exactly. That’s what makes this funny. :-)

4. However, perhaps the student was right —> 1/0 is not a rational number. So there is a counterexample that shows the statement is false.

• Exactly—it is false, and that was the counterexample I was looking for.