And so, in an effort to make proofs somewhat more enjoyable for my students, I told them all about false proofs in mathematics (I also give them brain teasers and logic puzzles). False proofs are basically just "proving" something silly or obviously wrong in mathematics, using trickery, cleverness, or just a misunderstanding of a math concept.

An example of a false proof would be: To simplify 16/64, you just cancel out the 6's and you get 1/4. It turns out that 16/64

*actually does*simplify to be 1/4, but obviously not via cancelling out digits. It's a coincidence, and can be easily shown not to work with many other fractions (e.g., 24/48 = 1/2, not 2/8 or 1/4). Simplifying fractions is all about dividing out common factors, not digits.

So anyways, there's a pretty famous false proof that 1 = 2. And usually, when I teach geometry or trigonometry (or any other course with proofs), I write out the false proof that 1 = 2, complete with seemingly valid mathematical reasons for every step, and ask the students to figure out where I went wrong. And the students generally enjoy analyzing the proof, even if they rarely figure out what's technically invalid about it (I purposely fast-talk and word the proof in such a way that my students will keep nodding in agreement until they realize I've "proved" that 1 = 2; then they get confused. This is about as devious as I get in real life.)

So this semester, I decided to try something new. I was thinking about math in the shower (because what better thing is there to think about in the shower?), and I started going off on a tangent (pun intended) and reciting the false proof. At first, I figured I'd probably just demonstrate the false proof for the students and have them analyze it as a whole, like usual. But as I said it aloud over and over again, I started to realize how easy it was to rhyme some of the key words. Then I figured: Why not turn it into some sort of song or slam poem? Surely that would make it more interesting.

So it took me about five minutes to write the poem, and then another five minutes to make a relevant powerpoint presentation so that students could see every step I was invoking in the false proof. As I got through nearly every line, I would click the next slide to express the next step in the proof. Unfortunately, I'm not able to post the powerpoint in here (and it would be weird to just show the images without changing them at the key times in the poem), so I'll just paste my poem and attempt to write out the proof, line by line, matching the lyrics. It's a pity you're unable to listen to my dreamy voice recite this lovely mathematics, but text is better than nothing, I suppose. And so, without further ado...

A false proof in math is incredibly fun,

And now I will show you that 2 equals 1.

So what is the

*Given*? Let’s choose

*a*and

*b*. . . . . . . . . . .1. a = b.....Given

Please note that these letters are arbitrary.

Let’s multiply both of the sides by

*a*, . . . . . . . . . . . . . .2. a^2 = ab.....Multiplication

Then subtract

*b*-squared without delay. . . . . . . . . . . . .3. a^2 - b^2 = ab - b^2.....Subtraction

From here we can factor and leave

*a*minus

*b*, . . . . . . .4. (a + b) (a – b) = b (a – b).....Factoring

Divide that out now and let’s see what we see.

*a*plus

*b*equals

*b*, so let’s use substitution; . . . . . . . . . . .5. a + b = b.....Division

Turn

*a*into

*b*to work towards our solution. . . . . . . . . . . . .6. b + b = b.....Substitution

*b*plus

*b*on the left, so let’s do the addition,

2

*b*and 1

*b*are a matching condition. . . . . . . . . . . . . . . . .7. 2b = 1b.....Addition

So now end with division of

*b*and you’re through,

And we clearly see how 1 equals 2. . . . . . . . . . . . . . . . . . . . . . .8. 2 = 1.....Division

+ Show Spoiler +

(Now don’t thank me, because I’m no hero;

Turns out that back in step 5 I divided by zero.)

Turns out that back in step 5 I divided by zero.)

The kids absolutely loved the poem Cheers, everyone!

~DPB