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Here is the problem that I asked undergraduate students of an introductory number theory course to prove:

Prove that if $N$ is a nonsquare natural number, then $\sqrt N $ is irrational.

Many of them proceeded as follows:

Suppose we can write it as $\frac{p}{q}$ where $p$ and $q$ have no common factor. Thus, $N=\frac{p^2}{q^2}$. This can only happen if $q^2=1$; that means $N=p^2$, contradicting the fact that $N$ is a nonsquare natural number.

Any idea of how I could possibly explain what is going wrong in a constructive way.

I also love to have some similar examples of such "proofs", though I don't know what a similar example might look like. I am just hoping your answer help me to see a general phenomenon for which the "proof" above is just an example!!

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    $\begingroup$ "This can only happen if $q^2=1$." Why is that?? $\qquad$ $\endgroup$ – Michael Hardy Nov 27 '16 at 0:09
  • $\begingroup$ @MichaelHardy Could you please see my answer below. $\endgroup$ – Amir Asghari Nov 27 '16 at 9:35
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This is indeed tricky, and it seems to me the most effective way (in far more general, similar situations) is to show them the problem would be to have them apply their method to another, close problem where the answer is actually opposite. Either they will explain why it does not apply there, and you can argue that the difference is subtle enough to warrant more details in their original proof, details which must stress the difference between the examples, or they first believe they can prove the same result in the second case and your point will be made in an even more compelling way.

In the present case, I suggest to ask whether there are two integers $p,q$ such that $$18 = \frac{p^2}{q^3}$$ If they think there are not, and apply the previous reasoning, then ask them about $p=12$ and $q=2$.

In the more general case, your extra knowledge as a teacher often comes in handy: you would take the most general positive result, remove the most harmless-looking hypothesis, look in the proof where it is needed, and cook up an example where the theorem is not true from there (but beware that in some cases, the hypothesis might not be needed after all if one settles for a more complicated proof, so wide culture is key). This is one of the reasons, by the way, teachers should know way much than a perfect mastering of the content they teach.

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  • $\begingroup$ Using two different powers in the same equation was a clever idea. Thank you. By the way, it is for years I try to "know way much than a perfect mastering of the content". But, you know it is too vast out there :) $\endgroup$ – Amir Asghari Nov 24 '16 at 11:48
  • $\begingroup$ @AmirAsghari: that comment about teachers' knowledge was not meant to target anyone, it is something general that I remind at every occasion because I too often see people (parents, students wanting to become teachers, ministers reforming the teachers' hiring, etc.) who don't understand why it would be difficult to be a teacher with just the high school baggage slightly consolidated. $\endgroup$ – Benoît Kloeckner Nov 24 '16 at 12:23
  • $\begingroup$ I know, that is why I finished with a smiley face. Also, mentioning that "it is too vast out there" was my attempt to say it is easier said than done :) Best regards $\endgroup$ – Amir Asghari Nov 24 '16 at 12:58
  • $\begingroup$ $p$ and $q$ in this counterexample here have common factors. $\endgroup$ – Thomas Rasberry Mar 11 '17 at 16:19
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    $\begingroup$ @ThomasRasberry: sure, but I don't know what your point is. To clarify the construction of the example: when the powers of $p$ and $q$ match, one can reduce to a coprime pair, but this should be said and is actually the main (but not the only) missing part in the student's answer. By giving an example where one cannot do this reduction (as here), one stresses that equality of powers is a notable property of the previous equation. $\endgroup$ – Benoît Kloeckner Mar 12 '17 at 9:59
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What's wrong is that most of the justification is missing. Why can $N = p^2/q^2$ only happen if $q^2 = 1$? This can be justified using the fundamental theorem of arithmetic (which states that integers have unique prime factorization), but that justification needs to be made explicit, and they need to make sure that the fundamental theorem of arithmetic has already been proven.

The first sentence also assumes that rational numbers can be written as a ratio of coprime integers, which also should be justified. The usual way to do this is to appeal to the Euclidean algorithm.

The key principle here is that every claim must be justified with precise and logically valid reasoning. To construct other examples of incomplete proofs like this, just take any valid proof and remove parts of the reasoning, leaving bare assertions in their place.

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    $\begingroup$ I understand your point. But to be honest, I found it hard to explain to the students who are just learning to give proofs, what is obviously, and in fact turn out to be essentially correct, need further justification. You know the very statement of "if $N=\frac{p^2}{q^2}$, then $q^2=1$", in a way, is equivalent to what we want to prove, but it is much more "obvious". That is exactly what makes it hard to explain. $\endgroup$ – Amir Asghari Nov 23 '16 at 23:00
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After writing for the students where they have gone wrong with their given proofs, I realized that it was just me thinking they are wrong, just because I was accustomed to seeing the fundamental theorem of arithmetic explicitly used and stated. To see the point, let me quote the proofs given by J. L. Lagrange in his Lectures on Elementary Mathematics (1898) and More than half a century earlier (1831), Augustus De Morgan in his book ON THE STUDY AND DIFFICULTIES OF MATHEMATICS (I've seen these in cut-the-knot)

Lagrange: it's impossible to find a whole number which multiplied by itself will give 2. It cannot be found in fractions, for if you take a fraction reduced to its lowest terms, the square of this fraction will again be a fraction reduced to its lowest terms, and consequently cannot be equal to the whole number 2.

Augustus De Morgan : 7 is not made by the multiplication either of any whole number or fraction by itself. The first is evident; the second cannot be readily proved to the beginner, but he may, by taking a number of instances, satisfy himself of this, that no fraction which is really such, that is whose numerator is not measured by its denominator, will give a whole number when multiplied by itself, thus 4/3×4/3=16/9 is not a whole number, and so on.

True, both of them appeal to the fundamental theorem of Arithmetic indirectly. But, both of them are based on what the authors have assumed to be correct previously. This reminds me of the role of the context of a proof as described by Ian Stewart and David Tall in The Foundations of Mathematics

One situation that is of great concern to students is what constitutes a proof acceptable in an examination. To a certain extent, the answer depends on the examiner, but part of the anxiety is due to uncertainty about the context...in an examination it may not be clear at which level a proof is required. Do all the steps have to be included? What can safely be missed out?

Thus, it seems that I should change all the comments I have written for the students, and instead, ask them to specify where they have used the fundamental theorem of arithmetic! And in the future, If the use of a certain tool/concept/approach matters to me, I should specify it at the outset!!

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  • $\begingroup$ Both of the old "proofs" that you quote have the same serious gap.. We know nowadays that it is essential to be explicit about hypotheses that were implicit in older days, for failure to do so quickly leads one astray when one passes to more general number systems (e.g. algebraic numbers). This is the reason that some eminent mathematicians gave false "proofs" of FLT. One should not repeat mistakes of the past. $\endgroup$ – Bill Dubuque May 16 '17 at 18:33
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Perhaps start by asking if $\frac{p}{q}=M\in Z$ implies $q=1$. This might focus the student on the hypothesis that $p$ and $q$ are relatively prime. Then direct the conversation to relating the factors of $p$ and $q$ to $p^2$ and $q^2$.

What you raise is an interesting problem in teaching proof. The original statement involves irrational numbers, rational numbers, and integers. Maybe it is better to start by not even bringing up the main statement about $\sqrt{N}$, and not bringing in rational numbers. Just a cold start, saying "Let's look at the (Diophantine) equation $p^2=Nq^2$." Only after analyzing this in the integers do you connect it to the notion of $\sqrt{N}$ being irrational.

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  • $\begingroup$ You know, learning proofs is very tricky business. We discussed problems like $7m^2=n^2$ or $24m^3=n^3$ in the class. Yet, when asked to prove the irrationality of $\sqrt12$ as a homework, many of them copied the proof of the irrationality of $\sqrt2$ (seen somewhere else) or something like the one I described in the body of the post. In the former case, it was somehow easy to explain where there are wrong, but I found the latter tricky to explain. Please also have a look at my previous post to see the difficulties we had with such a seemingly innocent change of structure as you suggested! $\endgroup$ – Amir Asghari Nov 24 '16 at 8:52

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