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 
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.”
By: Robert Talbert on December 18, 2008
at 10:54 am
Excellent! At least (s)he was being clear about her/his assumptions!
By: Dave Richeson on December 18, 2008
at 11:14 am
22/7 and pi are extremely close, but they are not equal.
By: Stephen on February 22, 2012
at 7:12 pm
Exactly. That’s what makes this funny. :-)
By: Dave Richeson on February 22, 2012
at 8:33 pm
However, perhaps the student was right —> 1/0 is not a rational number. So there is a counterexample that shows the statement is false.
By: Steven on January 30, 2013
at 1:32 am
Exactly—it is false, and that was the counterexample I was looking for.
By: Dave Richeson on January 30, 2013
at 10:36 am