18

I believe that most middle-school math curricula have at least a brief section about irrational numbers, in which students are taught (among other things) that $\sqrt{2}$ is irrational and $\pi$ is irrational.

What I'm wondering is if students are ever told that the irrationality of $\pi$ is not at all obvious, but there is a way to prove it, though the proof requires advanced techniques. I suspect that most of the time, the irrationality of $\pi$ is asserted as a brute fact that the students are expected to accept because authority figures say so, or perhaps on the basis of very limited empirical evidence. But maybe I'm wrong.

Is there any textbook (or any kind of written curriculum—I understand that nowadays, many teachers eschew traditional textbooks in favor of free materials that they download from some website) at the middle school level which indicates that irrationality of a number such as $\pi$ is something beyond the empirical observation that its decimal expansion does not seem to repeat?

Timothy Chow
  • 493
  • 3
  • 9
  • 7
    I'm curious whether this is even addressed in high school. – Sue VanHattum Apr 20 '23 at 17:59
  • 5
    @SueVanHattum It was actually addressed when I was in high school, even though the proof was not given, but it was almost 40 years ago and we had an excellent teacher. – Massimo Ortolano Apr 20 '23 at 19:42
  • 1
    @MassimoOrtolano I'm not surprised that excellent teachers would mention this, but what I'm curious about is whether it appears in written course materials. – Timothy Chow Apr 20 '23 at 19:49
  • Unfortunately I cannot recall whether it was in the textbook or not, and I no longer have my high-school math textbooks (and I'm pretty sure that they are no longer printed: they were too streamlined and black-and-white for these modern times). – Massimo Ortolano Apr 20 '23 at 20:04
  • 14
    "[..] is asserted as a brute fact that the students are expected to accept because authority figures say so" I think there's leeway here to distinguish between a refusal to explain due to authority, and the omission of overly complex details for the target audience. By definition, teachers (especially in lower grades) have to establish some unproven truths simply because the proof of the truth is more complex than its applications. – Flater Apr 21 '23 at 02:01
  • 5
    @Flater Yes, I agree. But I'd further distinguish between omitting complex details and omitting to mention that complex details are being omitted. The latter is what I suspect is happening almost all the time. By the way, I don't know of any "applications" of the fact that $\pi$ is irrational other than as furnishing an example of an irrational number. That is, the only reason to mention it is that it is considered to be a fact that students "should" know. It's not needed for any further study in mathematics, unless you're going to be a professional number theorist. – Timothy Chow Apr 21 '23 at 04:02
  • 1
    @TimothyChow: I was referring to applications of the truth (= the approximate value of π), not applications of the proof of the truth (= π being provably irrational). In other words, using π is teachable to students significantly earlier than defining π with any shred of accuracy, let alone proving that it is an irrational number. Personally, I learned about π 6 years before I was even introduced to the concept of irrational numbers (year 3 primary vs year 3 of middle school). In those 6 years; π was equal to 3.14, and the only mention that it wasn't accurate was an offhand verbal comment. – Flater Apr 21 '23 at 04:36
  • I'm pretty sure I learned the proof for sqrt(2) in the 70's, not so sure about pi. But I was also in "honors" classes, I wouldn't be surprised if the regular classes didn't cover this. – Barmar Apr 21 '23 at 14:33
  • In the UK A level it is assumed knowledge but "rational" and "irrational" are not defined in some textbooks. Questions on the level of: "if $a$ is rational and $b$ is irrational, prove $a-b$ is irrational" are set but students are quite allergic to the proof chapter: when explaining this question to a friend (who isn't bad at maths), I learnt that they did not even know the definition of (ir)rational number. In class it's just some 'boring, weird' thing where the teacher writes $x=p/q$. Certainly, irrationality of $\pi$ or $e$ are not discussed as interesting facts. – FShrike Apr 21 '23 at 14:46
  • 3
    I remember when my maths teacher (Germany, grade 9 or 10) presented us with the irrationality proof for $\sqrt{2}$ and said this was actually the first time we got in contact with maths. Probably nobody understood what he meant by that; I only got it years later in university. – leftaroundabout Apr 21 '23 at 15:14
  • 1
    @leftaroundabout It really bothers me when people say things like that, and I think it's revealing that nobody understood his comment. You hadn't played around with the even/odd arithmetic rules, or graphed lines and parabolas, or just stumbled upon things like "to go from 5^2 to 6^2 just add 2 times 6 minus 1"? All of that is getting in contact with math, and certainly no no-name high school teacher (I was one myself at one point so I don't say this to be insulting) is the arbiter of what "real" math is. – Thierry Apr 22 '23 at 03:47
  • 3
    @Thierry I see your point, and indeed would now say it's a bit silly to draw the line of what's real maths particularly in front of something like $\sqrt2\not\in\mathbb{Q}$. However, I also agree with my teacher that most of what's done in school maths lessons is indeed not maths but only calculating etc.. The irrationality proof does contain several of the bread-and-butter things that make maths so powerful (and fun) but scarcely ever turn up in school. – leftaroundabout Apr 22 '23 at 08:33
  • "asserted as a brute fact that the students are expected to accept because authority figures say so" -- I'd expect that to be rather common in middle school, regardless of the subject. – ilkkachu Apr 22 '23 at 16:50
  • Before you want to prove that pi is irrational, you should probably start by proving that pi is defined at all, i.e., that the ratio circumference / diameter is constant for all circles. Which requires either defining the length of a curve (via integration?) or proving that the circumference of inscribed regular polygons converges as the number of sides tends to infinity. So you need either integrals or limits. – Stef Apr 23 '23 at 19:43
  • @leftaroundabout The irrationality proof is indeed beautiful (every one of the maybe 4 I've seen is). I showed it to my class, along with both of Euclid's proofs of the Pythagorean theorem. That doesn't mean that until that time they hadn't got in contact with math, whatever that means. Besides, I've been told that math is not a spectator sport, so unfortunately watching a teacher write a proof at the board doesn't count as real math. At least the students do their own calculations. – Thierry Apr 24 '23 at 00:48

7 Answers7

13

Here's a quote from the syllabus for the 9th grade in the school type "Gymnasium" in the federal state of Bavaria in Germany:

Kompetenzerwartungen und Inhalte

Die Schülerinnen und Schüler [...] verstehen das Grundprinzip eines indirekten Beweises, vollziehen damit den Beweis für die Irrationalität von Wurzel aus 2 nach und erläutern diesen; dabei erfassen sie auch, dass das Beweisen eine zentrale Bedeutung für die Mathematik und deren stringenten Aufbau hat.

In English (translation mine):

Expected competences and contents:

The students [...] understand the basic principle of an indirect proof and therefore follow and explain the proof that the square root of $2$ is irrational; thereby they also recognize that proofs play a central role for mathematics and its stringent structure.

Link to the source.

Context information:

  • 9th graders are typically about 14 to 16 years old in Germany.

  • There are different type of schools in Germany, with the details depending on the federal state. Attending a "Gymansium" takes more years than (most) other school types (in Bavaria, it lasts from grade 5 to 13) and is typically considered to be more theoretical than the other schools. This school type offers the most direct (but not the only) way to get access to tertiary education in Germany. A quick internet search suggests that currently about 40% of all 5th graders in Bavaria attend a gymnasium (but some of them switch to another type of school at some point later).

  • From my experience as a middle school and highschool student (at a gymnasium in Bavaria, about 25 to 15 years ago) I severely doubt that the majority of students in 9th grade achieve the "expected competences" cited above.

Jochen Glueck
  • 2,195
  • 6
  • 21
  • 1
    Thanks; this is a good partial answer. I guess, though, I'm most interested in whether something like this is ever said: "This requires proof but the proof is beyond the scope of this class." But I suppose the students have to have some concept of what it would even mean to prove something, before a statement like that would make sense to them. – Timothy Chow Apr 21 '23 at 00:55
  • 2
    @TimothyChow What is described here is considerably more that the mere concept of a proof. Students are supposed to understand a specific and at that level rather non-obvious proof technique (indirect proofs). The idea that facts in maths have to proven and you may or may not have the means to do so is well before that. – quarague Apr 21 '23 at 06:57
  • 2
    To me, a Brit, middle school is ages around 9-12, maybe as high as 13 in some areas. A Gymansium is definitely a high school rather than a middle school. – Jack Aidley Apr 21 '23 at 09:13
  • 2
    @JackAidley: That is very useful information, thank you! I still think my answer is relevant, though: I read the question mainly as "When school students are taught that $\sqrt{2}$ is irrational, are they also taught that this requires a proof?" In the specific situation I described, the answer is "yes" since irrational numbers first occur there in grade 9 (along with the proof that $\sqrt{2}$ is irrational). – Jochen Glueck Apr 21 '23 at 11:05
  • @TimothyChow "This requires proof but the proof is beyond the scope of this class": I would be very happy if such words were ever said in UK secondary education. But unless you have a very keen teacher, they certainly are not. For example, students are taught the limit definition of the derivative but are not taught what a limit actually is. Students are taught things like the derivative of $x\mapsto e^x$ is $e^x$, and no one questions it. There is a strong emphasis on just accepting what you're told (even when the textbooks occasionally just lie about what's going on) – FShrike Apr 21 '23 at 14:51
10

Here are two typical examples in print and digital educational content of how this is done:

From Open-Up Resources:

In your future studies, you may have opportunities to understand or write a proof that √2 is irrational, but for now, we just take it as a fact that √2 is irrational.

From the College of the Redwoods:

Archimedes (287-212 BC) was...able to prove that 223/71 < π < 22/7...Modern mathematicians have proved that π is an irrational number.

There is a set of proofs of irrationality intended for advanced students in 7th-8th grade or older at https://www.algebra.com/algebra/homework/Number-Line/Proving-irrationality-of-some-real-numbers.lesson

Marc Shelikoff
  • 424
  • 3
  • Excellent! I'll give it a little more time, but I may accept this answer since it is the closest answer so far to what I was looking for. – Timothy Chow Apr 22 '23 at 19:46
  • I have now accepted your answer. I wonder if you can make your citation a bit more specific, because the materials do not appear to be freely available online. Which grade, which book, which page? – Timothy Chow Apr 23 '23 at 12:19
  • 2
    The Open-Up Resources quote is from their Grade 8, unit 8, lesson 3 summary at https://access.openupresources.org/curricula/our6-8math/en/grade-8/unit-8/lesson-3/index.html . The College of the Redwoods quote is from p 378 of their prealgebra textbook available at https://www.redwoods.edu/Portals/121/PreAlgText/Prealgebra.pdf . It isn't specific for any grade. I've used it for remedial help for college students and adults. My suspicion is that most of the major textbook publishers in the US use similar language which is why I described the language as "typical," but that's just a suspicion. – Marc Shelikoff Apr 23 '23 at 21:16
7

Here is a page with teacher's instructions (and some homework problems) for a Russian 8th grade class in a specialized "mathematical" school (or, it is a student handout, I am not sure). Among other things, it discusses transcendental numbers and says:

Такие числа называются трансцендентными. Наиболее известное трансцендентное число — π, но доказать, что оно трансцендентно, очень сложно. Указать явно хотя бы одно трансцендентное число, тоже непросто.

My translation:

Such numbers are called transcendental. The most famous transcendental number is π, but it is very hard to prove that it is indeed transcendental. It is even not so easy to give an explicit example of a transcendental number.

Moishe Kohan
  • 171
  • 4
4

I looked through my old math textbooks that were given to me, having gone to school in Austria.

In 4th grade of Gymnasium (8th grade of school in total, i.e. for students who are normally 13 or 14 years old), I had the book "Das ist Mathematik 4", 2nd edition printed in 2007, ISBN-13 978-3-209-03636-0.

Irrational numbers are first introduced on page 16, specifically giving the example of the square root of 32. It asks the question: Could we, if we had enough digits on our calculator, display the exact value of the square root of 32? Then it points out that no, we could not.

On page 17, a proof by contradiction is written down for the example of the square root of 32. (As I am not actually a math educator and just found this question through the hot network questions, I am not going to reproduce that here. I think you can imagine it, it seems familiar to me from when the irrationality of the square root of 2 was discussed in university.)

For pi specifically, the closest I could find in that book is on page 205, which discusses the history of pi. The last two paragraphs talk about the historical problem of "squaring the circle" and how only the German mathematician F. v. Lindemann managed near the end of the 19th century to prove that that problem cannot be solved in an exact way. It points out that mathematics can also prove the impossibility of something.

That page does not use the term "irrational" at all. When a few pages earlier (on page 187) the term is mentioned, it is simply stated as a fact that pi is irrational.

I was unable to find any more discussion of this in any textbooks of later grades either.

wonderbear
  • 141
  • 1
3

A web search* of several state public school math standards and online textbooks and lesson plans showed that nowadays grade 8 seems to emphasize understanding the concept of irrational numbers and that they are out there. But not requiring proof (especially for "struggling learners", a descriptor that is a direct quote).

Of course, any time you say "are there any texts" the answer is probably yes (at least one exists). For example, when I learned the topic in middle school, we were shown the sqrt(2) proof. So, I'm sure there are some places still teaching that, even if it is deprecated a bit more now. The right question is probably not are there any, but how common is it. (If you just want a single example, try the AoPS pre-algebra text. But that book may be a bit harder than optimal, for average students...and is expensive.)

In some cases, from the search, it seems like the proof was mentioned as a concept (e.g. with the name of a Greek) but not demonstrated. It appears to be current thinking to not require the sqrt(2) demonstration. Which I don't really have a problem with, the whole point is to just introduce the concept. For what it's worth, I've probably never seen the proof that pi is irrational but just accepted it. After all, people like bragging about how many decimals they know and pi sure doesn't look rational for several decimals out so it probably is irrational and I'll accept it that someone proved it without me needing to see the details.

Note also that some numbers are STILL not proven as irrational, but we have been unable to express them as rational numbers! E.g. pi plus e. This despite, we know about where it is on the number line and it's even a rather simple expression, not some ugly continued fraction foul thingie. But if it is rational, it's got to be a very large denominator, since we haven't found it yet. Personally, I think this is kind of cool, now, knowing what irrationals are (maybe more like an engineer knows them, not like a Rudin-luvver, since I never took pure math), that there is this richness to the topic. But I wouldn't derail the initial learning about irrationals themselves with this complexity.

FWIW, with "high track" students (US G/T or German gymnasium), I would have no issue with showing the sqrt(2) proof still. They can handle it. But maybe not with the middle/struggling tracks. Just follow the modern practice to "not require proof" (an actual quote from one standard) And even for the G/T track, I wouldn't go so far as to say "proof needed" being the emphasis. The emphasis is more conceptual, even for the good students (knowing the concept of can't be a rational fraction or expressed in terminating/repeating decimal). Just the proof is a nice flourish to add and give them some more grounding.

*Source: user "guest", Google, APR2023. I can't show more than two links without getting spam blocked, but see these:

(example state standard) https://portal.ct.gov/SDE/CT-Core-Standards/Materials-for-Teachers/Mathematics/Math-Lesson-Plans/Math/Grade-8-Rational-and-Irrational-Numbers---Real-Number-Race

An interesting paper on the math ed aspect of student's first intro to irrationals:

https://www.jstor.org/stable/3482830

guest commenter
  • 39
  • 2
  • 2
    Thanks for the pointers. To clarify, I'm not asking for examples of a proof of irrationality (of $\sqrt{2}$, say) being taught. I'm asking for examples where it is asserted that there is something here to be proved, which we are not proving. I think that most students don't grasp the distinction between conjecturing that $\pi$ is irrational based on empirical evidence, and proving that $\pi$ is irrational. Conveying that distinction is something that I think we educators should try to accomplish at some point in a student's math education. But my impression is that it is often not done. – Timothy Chow Apr 21 '23 at 01:31
  • 2
    Tim, Common Core seems to deprecate showing the sqrt(2) proof. (Not trying to be anti-CC, I'm sure it was a reform issue previous to CC.) In some cases, proof is mentioned to exist, but not demonstrated. (I think this is exactly what you want.) See, for example the "warm up" section on this web page: https://flexbooks.ck12.org/cbook/prep-for-high-school-math/section/5.4/primary/lesson/introduction-to-irrational-numbers-math-8-ccss-msm8-ccss-te/ BREAK – guest commenter Apr 21 '23 at 01:46
  • RESUME In other cases, proof is not even discussed (since "proof not required" is the current standard.) And of course, some laggards are probably still showing the sqrt(2) proof, at least, as an example of proving at least one number is irrational. – guest commenter Apr 21 '23 at 01:48
  • 1
    MORE. For what it's worth, I'm a pretty reactionary anti-reform math type, but I actually think they have the pedagogy right here. Proof may be vital for MATH itself. But is not the key concern of a new learner. He just needs to learn the classication of the beasties (non-repeating, non-terminating decimal and "stuff like pi and sqrt(2)"). That's actually the key educational achievement. Not that someone proved it. – guest commenter Apr 21 '23 at 01:52
  • 2
    "For what it's worth, I've probably never seen the proof that pi is irrational but just accepted it." Here is a proof, it's only one page: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-53/issue-6/A-simple-proof-that-pi-is-irrational/bams/1183510788.full?tab=ArticleFirstPage – Mahdi Majidi-Zolbanin Apr 21 '23 at 03:25
  • 1
    @guestcommenter Ah, yes, good! Mentioning that there is a proof without giving a proof is indeed the sort of thing I was looking for. – Timothy Chow Apr 21 '23 at 04:08
  • @guestcommenter Agreed; whether someone proved that $\pi$ is irrational is not that important at this level. But among all mathematical assertions that middle-schoolers are routinely expected to "know," "$\pi$ is irrational" is an incredible outlier, being orders of magnitude harder to prove than anything else. (Niven's proof, cited by Mahdi Majidi-Zolbanin, is short, but 99% of professional mathematicians would not be able to come up with something like that purely on their own.) It's a golden opportunity to mention the amazing fact that "deep theorems exist" (not in those words, of course). – Timothy Chow Apr 21 '23 at 04:25
  • 1
    @guestcommenter Re "Proof may be vital for MATH itself. But is not the key concern of a new learner.": Not sure. I think one mistake an educator can make is to underestimate and "under-challenge" the students. It is certainly true that the majority of children will never use "scientific math" (rigorous reasoning, proofs) in their lives ever but instead merely compute things. But they should know how scientific math works in principle. Because it is one of the core achievements of human civilization, and nobody should leave school without knowing about it. – Peter - Reinstate Monica Apr 21 '23 at 08:37
  • 1
    @guestcommenter Therefore, I would argue that the concept of proof should be presented latest in what Americans call middle school (early teens), and some accessible proofs should be presented and, if possible, followed and understood. Only then can the teacher refer to that when statements are presented whose proof is beyond school math. The students will at least know what it means that a proof has been done, and what the consequences are (namely perfect reliability as a building block for further proofs). – Peter - Reinstate Monica Apr 21 '23 at 08:40
  • 2
    @guestcommenter I think my ideas follow the education in other subjects. Children may never use, let alone produce art or music or literature in their later lives but they should be familiar with the concepts and masterworks, and perhaps try to produce or copy something in order to understand and appreciate it properly. Not least, a challenge like that is an opportunity for some gifted kids to latch on to their calling (also in math). – Peter - Reinstate Monica Apr 21 '23 at 08:44
  • @Peter-ReinstateMonica (+1)x3. It's depressing that I almost never to to hear these sentiments in real life. – ryang Apr 21 '23 at 16:22
  • "FWIW, with "high track" students (US G/T or German gymnasium), I would have no issue with showing the sqrt(2) proof still. They can handle it. But maybe not with the middle/struggling tracks" FWIW, it is the 21-st century now, not the 13-th one. If somebody cannot handle an elementary argument by contradiction about odd and even numbers in the upper middle school, there is a problem with their mental development, IMHO. Inventing such proofs is a completely different story, of course, but if our kids cannot do what ChatGPT is capable of, we risk becoming an obsolete model of intelligence. – fedja Apr 21 '23 at 22:59
  • @fedja: I feel your comment indicates you may not have teaching experience at that level? I have second-year math majors at the community college where I teach (U.S.) who can't handle that argument. To say nothing of the other 99.8% of the students at the college. (The basic school algebra requirement for graduation was removed a few years back as being an intractable blocker for students.) – Daniel R. Collins Apr 22 '23 at 17:26
  • 2
    @DanielR.Collins You are absolutely right that at least 90% of the students currently have a problem here. That is the result of the long-standing policy of lowering the requirements to the level of the students instead of raising the students to the level of the requirements. We are becoming an obsolete model of intelligence and the only way to revert the process (IMHO) is to raise the plank. It should be done slowly and in the beginning it will be rather painful, but I see no other way not to go back into the caves or to become peripheral data input devices for intelligent machines. – fedja Apr 22 '23 at 17:52
  • @fedja: Well put, I agree. – Daniel R. Collins Apr 22 '23 at 18:57
3

(originally a comment, but started getting too long)

My Algebra 1 text (I think published in 1970) definitely dealt with the meaning of irrational numbers. In my school everyone took Algebra 1 in the 9th grade, the first year of high school (ages 14 to 15). Even I took Algebra 1 then (1973-1974 school year), although by this time I was 2-3 years ahead of my classmates in math from reading ahead on my own (algebra was not offered in our middle school, but public libraries existed back then). However, since Algebra 1 (in the U.S.) is now universally taught in middle school, at least for average to above-average level students, maybe this counts?

I can't find our actual textbook online (freely available), but I did find essentially the next best thing -- the teacher's edition of the version used by students 2+ years older than me. (I believe textbook editions were changed 2 years before I took the class.) The teacher's edition includes some blue comments, answers in blue, etc. that should clearly indicate what was present in the actual students' book. As you can see, this has quite a bit relevant to your question. See Chapter 11 (The Real Numbers, pp. 396-433), especially pp. 401-402 (proof of decimal eventual periodicity characterization of rational numbers) and p. 407 (proof that if $n$ is a positive integer, then $\sqrt n$ is either an integer or irrational).

I know that in class we covered all the details of the decimal characterization of rational numbers. Being in the BC (Before Calculators) era, this involves ideas that students were definitely familiar with from many years of VERY many long division calculations. On the other hand, in class I'm pretty sure that only the result of the p. 407 stuff was mentioned, and the proof was skipped.

For what it's worth, I think our Algebra 1 year-long course ended with a brief introduction to quadratic equations (last few days of class for the year), which is dealt with in Chapter 13. During those last few days I believe our teacher only introduced the quadratic formula (without proof) and we used it to solve some numerically simple quadratic equations. Completing the square was not done until our 11th grade Algebra 2 course (ages 16-17). Possibly worth mentioning is that typically 0-2 students each year in my high school doubled-up with 10th grade Geometry and 11th grade Algebra 2 during their 10th grade, which allowed them to take 12th grade precalculus in the 11th grade, thereby allowing them to take calculus at a university about 20 miles away in their last year of high school (our high school at that time did not offer calculus).

Dave L Renfro
  • 5,828
  • 1
  • 16
  • 28
0

The irrationality of the square root of two is easy to proof. Irrationality of e is actually even easier. The irrationality of π is very hard to proof. So it's not so much a matter of "believing authority", but accepting that some maths is beyond you. And beyond the teacher. BTW proving that e / π is irrational is one level harder.

It's like in engineering, a bunch of talented 18 year olds with lots of time might be able to build a hang glider. A Wright brothers' style airplane is very likely beyond them, even if they are allowed to use a motorbike engine and as many parts from a scrap yard as they like. Building a Concorde - no way.

gnasher729
  • 1,197
  • 7
  • 5
  • 5
    I draw a distinction between believing that $\pi$ is irrational because some authority says so, and believing that there is some proof that $\pi$ is irrational that someday I might come to understand if I dedicate myself sufficiently. The latter is perfectly fine; the former, IMO, fails to convey what is different about mathematics compared to other subjects. BTW I agree that proving that $e/\pi$ is harder, but it's more than "one level" harder since it remains unproven to this day! (Maybe you meant to say that proving that $e^\pi$ is irrational is one level harder.) – Timothy Chow Apr 22 '23 at 19:42
  • @gnasher729 When you say it's easier to prove $e$ is irrational than $\sqrt{2}$, are you referring to Fourier's proof? I'm not convinced that this is easier than the proof of irrationality of $\sqrt{2}$. – A. Goodier Apr 26 '23 at 20:59