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”—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.