Happy 3/3 everyone!

I just graded a bunch of proofs that √3 is irrational. The proofs had a lot of holes in them. This didn’t surprise me too much, in large part because the students weren’t math majors; rather, it was for a liberal arts math class taken largely as a gen ed requirement, and the whole proof by contradiction thing is really pretty scary and abstract for most people the first time around under the best of circumstances.

But actually, even when I’ve assigned this to math majors, they struggle. They can have the proof that √2 is irrational right in front of them, be instructed that instead of even numbers they want to look at multiples of 3, and despite my Find and Replace instructions, they still don’t understand what to do. The most common mistake is to replace “even” with “odd”.

In some ways this doesn’t surprise me, but in some ways it does. Why is it such a conceptual leap to go from 2 to 3? It’s a HUGE leap for many people. And so I was pondering this while grading, and Batman suggested it might be because we have a special word for “divisible by 2” but don’t for “divisible by 3”. So you get, what, 10 years of reinforcement that there is just this one special way to divide the integers, and it doesn’t generalize.

What we need is a new word for these numbers.

And fortunately we have one: threeven. So 0, ±3, ±6, ±9, … are all threeven, and the rest are…umm, not. (Maybe we need two new words). This word isn’t mine or even Batman’s; it actually was suggested by one of his students in response to this exact same problem.

As a bonus, it generalizes: there’s fourven, fiven, sixen, seven-en (sev-en? )…as far as you want. Which, admittedly, might not be very far but it still makes for a smoother sounding proof.

Happy threeven day!

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March 4, 2011 at 12:56 am |

To test whether they really understand it, after they have proved that the square root of 2 is irrational and the square root of 3 is irrational, ask them to use the same technique to prove that the square root of 4 is irrational.

What percentage of the students will realize that it’s not going to work? How many will be able to point at the specific step that works for 2 and 3 but not 4?

March 4, 2011 at 6:57 am |

We think alike! With this class I was pleased that three people understood enough to laugh when I suggested it, and I only briefly outlined the proof and where it went wrong [there’s a huge disparity in mathematical comfort in this class, probably larger than in any I’ve ever taught].

But when I’ve done that in our Intro to Proofs class for math majors, typically as HW right after the proof about 3, I’d say that most of them immediately realize that the question doesn’t make sense but only a few of them can find the spot where the proof fails.

March 4, 2011 at 10:27 am |

I’ve had the same problem when teaching this. Recently, I started teaching a slightly different proof of the irrationality of 2 than I was taught and it seems to make a difference. I start in the usual way, assuming that sqrt{2} = a/b for relatively prime positive integers a and b. Squaring gives 2 = a^2/b^2. Here is where the change comes in: multiply through by b, leaving a single b in the denominator on the right. This gives 2b = a^2/b. This implies that b divides a^2—since the left side is an integer—which is only possible if b = 1 (as a and b are relatively prime). Therefore, 2 = a^2, which is clearly impossible since 2 is not a square. I find that students generalize this proof much more easily—even to higher roots—and they can see immediately why it doesn’t work for 4.

If you work just a little bit more with it—which I don’t think you would want to do with liberal arts majors—you can prove easily that any non-integer (real) root of a monic polynomial with integer coefficients is irrational.

March 5, 2011 at 9:29 am |

It should really be sixven and sevenven, and so on. Easier to say and easier to understand. And of course, expressions divisible by x would be X-Ven: The Last Stand.

March 7, 2011 at 11:43 am |

LOL Yaacov! And yes, you’re right about the pronunciation, especially since it leads to a punchline.

Adam, I don’t think I’ve seen that proof before, but I really like how it makes it almost transparent that square roots of integers are either integers or irrational. (Hmmm; does it generalize to roots of any order? I think it would.) Despite this I wouldn’t have thought automatically that it was an easier proof for students to understand, so I’m glad for your experience on it. [I think I’m done with this proof for the non-majors, but I’ll be teaching it again next year both to majors and non-majors and might use this version instead.]

March 24, 2011 at 4:18 pm |

I always thought a number was “even” because if you have that many objects you can line them up in two rows and they line up evenly. But if you don’t have an even number, there’s an odd one out. But I might have just made that up somewhere along the way.

But then you could say “threeven” is for lining them up in three rows. Anyway. What a good word.

April 12, 2011 at 2:07 pm |

throdd.