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Among the mathematically ignorant one often finds a mistaken proposed definition of "irrational number", which says that it is a number whose decimal expansion does not terminate or repeat.

The definition found in all textbooks I've seen and used by mathematicians is that an irrational number is one that cannot be expressed as a ratio of two integers.

Of course, one can prove that a number is irrational if and only if its decimal expansion neither terminates nor repeats. But that's lousy as definition, because it doesn't say motivate the concept as well and it makes it appear that we're singling out base $10$ rather than some other base, and that takes the whole account of the matter down a path that doesn't bear a lot of fruit.

The wrong definition has also in some cases led students to wonder if a number can be irrational in base $10$ and rational in some other base. That is confusion.

So my question is: Why does this mistaken idea persist for some many decades without ever being taught? (Or might I be mistaken in thinking that it is never taught? (And if so, is there solid evidence that such teaching is widespread?))

quid
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Michael Hardy
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    [Just a guess...] I've seen lessons done on converting between fractions and repeating decimals (you could say repeating or terminating, though I think of the latter as having repeating zeros). This is a frequent topic covered in elementary classrooms and written about (example). Anyway: Once one knows that repeating decimals can be written as fractions, you can say that the other decimal representations are the irrational numbers, and together they make the real numbers. (Yes, this elides over the 0.999... caveat...) – Benjamin Dickman Jul 08 '16 at 20:50
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    Still, infinite decimals are a fast way to introduce the real numbers as a set, as opposed to the other three (?) main ways of developing them: axiomatically (e.g., axiom of completeness), via Dedekind cuts, or via (Cauchy) convergent sequences of rationals. The decimal definition is okay for the set of reals, but somewhat inconvenient in terms of defining e.g. how to compute the quotient of two reals; I don't recall ever seeing a full treatment of the reals and the four operations using this definition. – Benjamin Dickman Jul 08 '16 at 20:51
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    You seem to be working with the assumption that the words of a definition decide whether it is 'right' or 'wrong'. I disagree. What makes a definition 'wrong' is that it is not equivalent to the canonical definition (or doesn't express the desired concept, in the case of a newly developed definition). What you are describing is not a wrong definition, only an alternative definition that you find less natural. – Jessica B Jul 08 '16 at 21:05
  • @JessicaB : If one were to judge ONLY by the standards of deductive logic, you would be right. Likewise, we cannot prove by logic that a certain city in France must be called "Paris", but error as to that point can cause you to flunk a geography test. $\qquad$ – Michael Hardy Jul 08 '16 at 21:09
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    A recognizable real number system was originally developed by Simon Stevin using infinite decimals. See http://arxiv.org/abs/1202.4153 . There may be a useful question here, but it would be easier to see what it was if the OP could cut the erroneous material claiming that the definition is erroneous. –  Jul 09 '16 at 00:57
  • When we talk about mathematics and use the word 'erroneous', isn't deductive logic kind of the gold standard? The geography of France isn't analogous, unless it's been formalized - wouldn't put it past the French to do that... – Spike0xff Jul 09 '16 at 01:48
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    @Spike0xff : There are many conventions in mathematics. For example $A+BC$ means $A+(BC)$ and not $(A+B)C$. Nobody hesitates to tell a student that construing it to mean $(A+B)C$ is incorrect. Also, deductive logic alone shouldn't be the gold standard. Definitions should get motivated far more than they typically are. $\qquad$ – Michael Hardy Jul 09 '16 at 01:51
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    I think (as is often the case...) it would be helpful for those who are down-voting the question to articulate precisely what they have in mind, so that the post can, potentially, be improved... – Benjamin Dickman Jul 09 '16 at 02:09
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    @MichaelHardy No argument, but I don't understand the point. To me the irrational numbers are first and foremost a set of numbers. They have the interesting property that they can't be written as ratios of integers. They also have this interesting property of endless non-repeating base-N representation. I imagine on some planet, some intelligent fish refer to them with a whistle that means 'numbers-with-endlessly-surprising-binary-representation'. It's the same set of numbers either way, and by whatever name we choose to label them. – Spike0xff Jul 09 '16 at 02:12
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    Well then, my suggestion would be to replace the word erroneous, which to me strongly implies deductive logic (when used about mathematics) and ask "Why are irrational numbers introduced as non-repeating decimals instead of as, well, numbers that aren't integer ratios?" I notice the answers already tend to be to that question, or it seems so to me. – Spike0xff Jul 09 '16 at 02:16
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    A word used in the body to describe the definition is 'lousy' and it is explained why OP thinks it is a not a good way to define it, even though it is a way. I think the 'mistaken', 'erroneous' and 'wrong' should be read against that background, like: 'it is an error of judgment to use/consider this as the definition (as opposed to a characterization).' Now, this is an opinion. Yet, the question leaves open room for answers explaining that and why this definition is or should be taught. I'd recommend that those that think so write up their opinion and experience as answers. – quid Jul 09 '16 at 10:17
  • What is your source for the claim that this definition is not taught? Students first encounter the notion of irrationality in elementary school, when they meet $\pi$ for the first time, and in that context pretty much the only thing they learn about what it means to be an "irrational number" is that its decimal expansion goes on forever without repeating. – mweiss Jul 11 '16 at 23:16
  • @mweiss : I haven't seen any book that says that. Have you? $\qquad$ – Michael Hardy Jul 12 '16 at 00:02
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    Oh, I have seen that and much worse. For example: https://drive.google.com/file/d/0B4wpC387vwzeTF94dExDTEVrV0U (You may want to avert your gaze, as the wrongness in that image is blinding.) – mweiss Jul 12 '16 at 04:42
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    I can't answer as to why (you feel) that this mistaken idea persists, but I might draw a similar analogy. Students very often learn the basics of Calculus without the proper rigor that would be required of Analysts. Calculus is focused more on the intuition, whereas Analysis hopes to rectify the vagaries by formalizing it. (To me, this falls under the concept that "mathematics is not taught and learned linearly" [citation needed].) It would seem to me that if the less technical definition might only be superseded by the more accurate one after more students have stopped learning math. – Andrew Sanfratello Jul 12 '16 at 05:19
  • @AndrewSanfratello I do not see this analogy. Everything to give the 'right' definition ought to be present in the/a given context. But I will better expand my answer. – quid Jul 12 '16 at 19:48
  • @mweiss : What book is that from? $\qquad$ – Michael Hardy Jul 12 '16 at 20:08
  • @MichaelHardy It's from Cool Math by Christy Maganzini (https://books.google.com/books/about/Cool_Math.html?id=3UMbH4KRZO0C). Not strictly speaking a textbook, but I think fairly typical of how confused most elementary-level presentations of this material are. – mweiss Jul 13 '16 at 03:29
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    "The wrong definition has also in some cases led students to wonder if a number can be irrational in base 10 and rational in some other base. That is confusion" Seems like a natural question (given the definition) rather than a confusion. A question which is a dumb question when using one definition isn't necessarily a dumb question when using a different but equivalent definition, especially when the equivalence of the definitions is a nontrivial theorem. Even with the standard definition, it isn't obvious that a nonrepeating decimal in base 10 is nonrepeating in other bases as well. – John Coleman Jul 14 '16 at 14:22
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    It is certainly a natural question if, but there is confusion in the terminology, and the question would be clearer without that. One could ask whether a number that is terminating or repeating in one base must always be terminating or repeating in another base. THAT is a natural question. And the natural answer is: observe that either way it's an integer over an integer, so it doesn't matter which base you pick. $\qquad$ – Michael Hardy Jul 15 '16 at 20:06
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    I've been asking this question myself ever since I saw the definition of the real numbers as "the union of the rational numbers and the irrational numbers" when they taught us number sets in middle school. It's a completely unsatisfactory definition of the real numbers. – Joe Z. Jul 21 '16 at 21:00

6 Answers6

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I can think of two related reasons:

  1. The characterization via the decimal expansion might be perceived more strongly like a property of the number: "This number is irrational, because this number's decimal expansion does not terminate." The other one is rather a non-property and thus not perceived as a definition of some thing.

  2. The characterization via the decimal expansion seems more readily useable. The idea "I look at the decimal expansion and then I 'see' if it is periodic or not." seems natural and appealing. Instead the other one might lead to a: "How in the world should I check there are no integers that give this number as a fraction?"

Let me add that I agree with OP that the definition of irrational should be a (real) number that is not rational. The above answers the question why this might not stick.

How rigorous a notion of real number one has in a given context seems orthogonal to me. In any case, there ought to be one available if one can talk about non-periodic decimal expansion.

quid
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    I agree especially with hypothesis #2. Even if initially defined correctly, students will be mostly exercising the decimal expansion analysis in problem sets. Granted that they're fuzzy about definitions versus theorems, that more-exercised concepts winds up taking precedence in their minds. – Daniel R. Collins Jul 09 '16 at 00:00
  • Your second point makes sense only when the number is seen in decimal form. Consider the ratio of the long to short sides of the triangle that is one of the "points" of the "star" that is a regular pentagram. Suppose one asks whether that is rational. One doesn't have it in decimal form, but in the form of a picture of the five-pointed star. $\qquad$ – Michael Hardy Jul 09 '16 at 01:03
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    @MichaelHardy that's true, but for the most part students are confronted with them in that form and not with problems to decide the commensurability of lengths via geometric arguments. – quid Jul 09 '16 at 09:59
  • There is still a problem with the second characterisation - how in the world should we check that the infinite decimal expansion never starts repeating itself? But it's not as bad as the case with ratios of integers, because there are some numbers (e.g. 0.101001000100001...) that can be easily shown to have non-repeating decimal expansions by definition. And, of course, the negative property is a bit better hidden so students are less likely to notice it. – kaya3 Jan 28 '23 at 09:00
  • @kaya3 indeed maybe I described it a bit backwards. One "sees" the repetition and might check rigorous then. – quid Jan 29 '23 at 12:49
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Note that this definition of rational and irrational numbers is most commonly presented to high school students, who tend to have a strong and natural intuition of numbers in base $10$. At this stage, it's much easier to understand what it means for a number to be irrational in terms of its decimal expansion rather than the statement "a number which cannot be expressed as $\frac mn$ where $m,n \in \mathbb{Z}$ ". This more 'classical' definition raises the deeper question of what precisely it means for a number to be irrational, and, indeed, whether or not such numbers exist isn't apparent. On the other hand, it's easy to "imagine" numbers which have non-terminating non-repeating decimal expansions.

A deeper analysis of the real number system, in which one considers the notion of a number in a more general and abstract sense than a mere decimal expression, is generally beyond the scope of high school study, and consequently the more "intuitive" decimal explanation is preferred.

MathematicsStudent1122
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    While I upvoted this answer, I still stink that the definition of a (real) irrational number should be it is a (real) number that is not rational. Not only but also since usually a discussion of irrational numbers will come after knowing the fact that every rational number, in the sense of quotient of two integers a concept known earlier, has a periodic (or terminating) decimal expansion. (As opposed to decimal expansions being introduced, then being split up into periodic and non-periodic ones and then observing the periodic ones can be obtained as a quotient of integers). – quid Jul 09 '16 at 13:25
  • So you're saying I am in fact mistaken in thinking this is never taught. You say it is "presented to high school students". Do you mean teachers or textbooks present it? Can you cite such textbooks? What evidence do you have that high-school teachers are doing that? $\qquad$ – Michael Hardy Jul 15 '16 at 14:37
  • @MichaelHardy Here is one study from 2013: http://www.tandfonline.com/doi/abs/10.1080/14794802.2013.803778 Though the full study is not freely available, the abstract speaks for itself. – MathematicsStudent1122 Jul 15 '16 at 16:38
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    ok, That explains part of what happens in Brazil. As for "the [alleged] mathematical need for the construction of the field of real numbers remains unclear in the textbooks" (in secondary schools?? a need???) I say time that the author may want to spend on that in secondary schools is better spent on things that are worthwhile instead. $\qquad$ – Michael Hardy Jul 15 '16 at 19:29
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The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to probably does not know anything about Cauchy sequences or Dedekind cuts. So the complement of "rational" is at best only intuitively defined for them.

user52817
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  • I do not think that OP meant to define irrational number literally just as number which is not rational. But rather as real number (or maybe complex number) which is not rational. Where the notion real number was introduced in some (possibly informal) way, such as via decimal representation or 'points on the number line' or still something else. – quid Jul 12 '16 at 18:23
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    @quid I don't think that affects this answer, particularly. If real numbers are only defined by decimal expansion then defining rationals by decimal expansion seems pretty reasonable. If you've used a number line, decimal expansion again makes more sense, because those appear in order along the number line, unlike integer ratios. I'd be interested to know of an intuitive understanding of the reals where division is easy to understand. – Jessica B Jul 14 '16 at 05:57
  • @JessicaB the answer says that the complement of "rational" is at best intuitively defined, and this pattern of argument came up elsewhere, too. I do not agree this objection is quite to the point, since as soon as one has infinite decimal expansions, usually the complement of rationals relative to this collection is just as good defined as is the notion of infinite decimal expansion present in any given context (which is somehow the baseline of rigor for this discussion). – quid Jul 14 '16 at 10:37
  • You said "then defining rationals by decimal expansion seems pretty reasonable." I would agree with this. If somebody really also defines rationals in terms of decimal expansion, then this is a consistent approach, like we split up all expansion into eventually periodic ones and non-eventually-periodic ones. And then one continues from there. ( @JessicaB ) – quid Jul 14 '16 at 10:37
  • I find this answer very silly. You say this is problematic BECAUSE they don't know about --- of all things! --- Cauchy sequences and Dedekind cuts?? I'd say their lack of suspicions about those make it unproblematic, and we should recall that the notion of the ratio of the diagonal of a square to the side is something that arose in geometry long before anyone had any concept of what today we call a real number, and was at that early date (500 BC or so?) proved to be irrational. $\qquad$ – Michael Hardy Jul 15 '16 at 19:22
  • I've been posting to stackexchange daily for years and I've rarely down-voted anything, but I down-voted this answer. $\qquad$ – Michael Hardy Jul 15 '16 at 19:25
  • @MichaelHardy--Your original question sounds like a quest for a mathematical definition of "irrational numbers." It is of course straightforward to give examples of irrational numbers such as $\sqrt(2)$, as you mention. But this is not the same thing as providing a mathematical definition of the set of irrational numbers. My point is that a mathematical definition of this set is essentially equivalent to a mathematical definition of the set of real numbers, which is non-trivial. – user52817 Jul 15 '16 at 21:20
  • The definition of irrational is very straight-forward: all those numbers which aren't rational. The "ir" of irrational, just means, "not". We know that rational numbers exist. The problem is to show that there is such a things as irrational. Once we've shown that $\sqrt{2}$ is not rational, the job is done. Now, unless you're a constructivist, you shouldn't have a problem with the Law of the Excluded Middle. Nor should your students. Dedekind cuts are not necessary; they're just a very cleaver (and fascinating) 19th approach to building a mathematical foundation. – A.Ellett Sep 04 '16 at 16:21
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    @A.Ellett it is not this straight-forward. By your definition $\sqrt{-1}$ irrational, because after all it is a number that is not rational. For your approach to defining irrational numbers to work, you first need to somehow define the set of real numbers, so that you can take the complement of the set of rational numbers in that set. – user52817 Sep 05 '16 at 15:43
  • @user52817 It is worth emphasis that many esteemed number theorists and algebraists do use the most general definition, i.e. irrational means "not rational", so $i,$ is irrational. See here for further discussion. – Bill Dubuque May 09 '18 at 19:43
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    @Number Still there must be a universal set in the background. Perhaps for the algebraists you mention, it is the complex numbers. If no universal set is assumed, then the irrationals would be too big to be a set. – user52817 May 09 '18 at 22:48
  • @user52817 Usually the "universe" is clear from the ambient context. There are many similar definitions, e.g. transcendental := not algebraic, e.g. the "indeterminate" $x$ is transcendental over $,\Bbb C$ in $\Bbb C(x) = $ rational functions in $x$ (fractions of polynomials in $x$) – Bill Dubuque May 09 '18 at 23:32
  • There is a good reason why they would not know it. These are topological constructs and someone learning for the first time (or recalling due to a prolonged lack of use) what the are irrational numbers is not the same person who has an understanding of either Cauchy sequences or Dedekind cuts. In fact, by the time someone reaches the stage of learning what is a Cauchy sequence one already knows intuitively what the rational numbers are (likely both definitions). Math is rarely studied through formalisms as the 1st principles. – wrod Feb 05 '24 at 12:39
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I Googled "high school algebra textbook PDF" and immediately found this.

enter image description here

Here they confusingly define "rational" as the ratio of two integers and "irrational" as the decimal expansion.

Also there is this:

enter image description here

Somehow defines "irrational" as both definitions, with no mention that they're equivalent or which one is the "real" definition.

There is another example on the first page of Google":

enter image description here

Here they with no proof or explanation say "doesn't repeat or terminate thus is irrational". This one is a little better as they formally define the "correct" definition. But there is no explanation of the two.

So I think the issue isn't that some books teach the "erroneous" definition it's that they don't teach any single definition at all! They never even mention that the two definitions are different, or why one might expect the two to mean the same thing.

So the question isn't "why are they presenting definition A instead of definition B", it is "why aren't they presenting ANY coherent definition at all??".

I would be very interested to find ANY high school algebra textbook that presents a single coherent definition of irrational number.

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    A reason why a coherent definition of irrational numbers is not presented at this level is because it requires a coherent definition of the reals. This is typically done in a undergraduate analysis course. Notice it is easy to show the square root of 2 is not a ratio of integers, but it is nontrivial to show it has a decimal expansion. In an undergraduate analysis course, the first theorem that appears after the construction of the reals is that the square root of 2 exists, using the completeness property. Once existence is established, irrationality is immediate. – user52817 Jul 16 '16 at 18:21
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    @user52817 a coherent definition is not the same thing as a rigorous definition, is not the same as a rigorous development of a body of theory. To wit, the usual proof of the irrationality of $\sqrt{2}$ given in an undergraduate analysis course takes for granted that the integers are a UFD, often without this every having been proven, let alone a rigorous definition of the integers having been given. – quid Jul 17 '16 at 10:26
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    Richard Askey found a real howler in a 1996 teacher's edition of a U.S. school textbook that wound up in Edward Barbeau's Sept. 1997 column Fallacies, Flaws, and Flimflam in College Mathematics Journal. It's also on p. 10 of Barbeau's 2000 book. Problem: Explain why $4^{1/2}$ is rational while $5^{1/2}$ is irrational. Solution: $4^{1/2}=2$ which is rational. $5^{1/2},$ in its decimal form, does not terminate or repeat and therefore cannot be written as an integer over an integer. (CONTINUED) – Dave L Renfro Jul 19 '16 at 21:33
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    (CONTINUATION) Askey wonders how one could show this about the decimal expansion of $5^{1/2}$ without first proving that $5^{1/2}$ is irrational. – Dave L Renfro Jul 19 '16 at 21:33
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    The two grey boxes in your last picture form a tautology. The set of irrational numbers is defined as the set of numbers that are not rational, and the set of rational numbers plus the set of not-rational numbers forms the real numbers. Hooray, we just proved that all numbers are real numbers! – Joe Z. Jul 21 '16 at 21:06
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Many good arguments have been presented. I would like to add that working with the decimal expansion does require much less understanding of what a real number is. The decimal expansion gives you rather touchable things to work with so that even without being able to see a "sequence of digits" as a number (which would allow arithmetic operations), pupils can participate in evaluating whether this sequence should represent an irrational number. In Zachary's post, you see that shortly after introducing irrationality, you can discuss the irrationality of 0.424224222422224... Proofs of irrationality based on the a/b-definition are much more complex, I think. Since being operational is always a good criterion for a definition, this definition may not be so bad in the context of school.

So, although from a more conceptually point of view, the a/b-definition is much more appropriate, for teaching in schools with a mix of gifted and not-so-gifted students, the alternative definition might better fit the purpose of enabling the whole class to participate.

I guess, some might find it pointless to present a concept that allows students to participate withouth thorugh understanding of the foundations. This, however, seems to be standard in schools around the globe and is a discussion in its own right...

Anschewski
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    "Proofs of irrationality based on the a/b-definition are much more complex, I think." But, usually one will show or at least explain that every rational has a periodic decimal expansion (and the converse). This characterization (yet not definition) of rationals then can be used immediately to characterize irrationals as those with non-periodic expansion. Then, one can apply this characterization instead of the original definition. – quid Jul 19 '16 at 09:26
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    You are absolutely right, quid. Stating the a/b-definition and then deriving the periodic/finite-decimals-characterization, however, takes more time than giving the periodic/finite-decimals-definition directly. And time is always restricted in school teaching.

    I think if you want to encourage teachers to use the a/b-definition, they should at least be given some tasks which use this definition. Interestingly, I have never seen such tasks in school although spontaneously, I could imagine some (e.g., show that the double of an irrational is always irrational again).

     – Anschewski Jul 19 '16 at 11:00
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    "Proofs of irrationality based on the a/b-definition are much more complex, I think." NOT TRUE. Let's try a few: THEOREM: $\log_2 3$ is irrational. PROOF: Suppose $\log_2 3 = a/b$ for some positive integers $a,b$. Then $2^{a/b} =3$, so $2^a = 3^b$. But $2^a$ is even and $3^b$ is odd, so that is impossible. $\blacksquare\quad$ How is that "much more complex" than a proof not based on the a/b definition? Indeed, how can you prove it without the a/b definition? THEOREM: $\sqrt 2$ is irrational. PROOF: Suppose $\sqrt 2 = a/b$ for some positive integers $a,b$, and$,\ldots\ldots\qquad$ – Michael Hardy Jul 19 '16 at 16:58
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    $\ldots,a/b$ is in lowest terms. Then $(2b-a)/(a-b)$ is $\sqrt 2$ in lower-than-lowest terms. Contradiction. $\qquad$ – Michael Hardy Jul 19 '16 at 17:00
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    $\ldots,$Now can you please tell us HOW to prove irrationality of either $\log_2 3$ or $\sqrt 2$ WITHOUT using the "a/b-definition"?? $\qquad$ – Michael Hardy Jul 19 '16 at 17:01
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    And how can you prove $\pi$ is irrational without using the $a/b$ definition? Or that $e$ is irrational? $\qquad$ – Michael Hardy Jul 19 '16 at 17:05
  • @MichaelHardy: Consider looking for the existence of any irrationals (can we identify any example which is clearly irrational?), not proofs for all irrationals in general. Would you agree that determining the irrationality of 0.424224... is easier than any of your examples? – Daniel R. Collins Jul 19 '16 at 18:07
  • @Anschewski I admit I hardly have any experiencing teaching at the relevant level. It may well be that the slight indirection the approach I propose entails already loses some students there; while there are only few that would gain something from it. – quid Jul 19 '16 at 19:21
  • @DanielR.Collins : No. How would you prove it more simply than by the proofs above? You can go through the long-division argument and say that only a certain number of possible remainders exist and therefore the decimal expansion must terminate or repeat. But is that simpler than the argument given above for the irrationality of $\log_2 3\text{ ?}\qquad$ – Michael Hardy Jul 19 '16 at 19:24
  • @MichaelHary 1st: The examples of $e$, $\pi$ and $\log_2 3$ are good examples to use the a/b-definition but do not show up in the curriculum soon after introducing irrational numbers. The irrationality of $\sqrt2$ is in fact best proven in the style you presented. Still, as Daniel R. Collins said, I think the irrationality of 0.424224... is more easily seen than the irrationality of $\sqrt2$. 2nd: I rather take an educational or phsychological point of view than looking at mathematical elegance. Thus, indirect proofs, the use of variables, etc. are seen as additional cognitive load. – Anschewski Jul 20 '16 at 16:13
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    3rd: It seems to me we got into a discussion on which definition is the best. I did not intend this discussion since the question we started with asks for reasons for the common practice in schools. I give possible reasons, not more. I do not even think one should start with the digit-definition in school but my opinion on this point does not matter. I want to help people understanding possible reasons. @quid: My experience does not build on teaching in school but teacher education only. However, I have the very strong feeling that in school you always need to include everyone in class. – Anschewski Jul 20 '16 at 16:24
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It always makes me cringe when I hear someone present the definition of irrational as a number whose decimal expansion does not repeat. It does a huge disservice to the history of math, its beauty, and the notion of rational.

A definition of irrational as a decimal number which doesn't repeat lacks beauty in that the word irrational bears no relation to what you're describing. The traditional, the Greek definition, of rational as a ratio makes very clear then what irrational is talking about.

The definition of rational and irrational as inherited from the Greeks is simple, elegant, and beautiful. It presents a particular view of the world and its nature. Teaching this definition is an opportunity (a window from a modern classroom looking out onto an ancient world) to understand something about the human desire to comprehend and make sense of the world. It's also an opportunity to teach the students the degree to which this ancient world view is still present with us. We expect, to a certain degree, that the world itself is rational. This expectation has played a huge role in the development of mathematics and physics.

Mathematics, as taught in high school, frequently comes across as dry with little or no connection to the world of human drama. Why not tell of the Pythagoreans and their punishment for anyone who dared reveal that the square root of two cannot be expressed as the ratio of two whole unit lengths. In various guises this plays out again and again through history. Move forward in time to Copernicus, Kepler, and Gallileo's discoveries and observations challenging accepted conventional wisdom.

The history of the square root of $-1$ itself is fascinating. Resistance to the idea being revealed in the names imaginary and real: such as your imaginary friends and your real friends. But then as non-Euclidean geometry is developed and theories of electro-magnetism worked out, imaginary numbers gain greater credibility as their complexity is discovered.

When it comes to how we teach what the irrational numbers, there are a variety of reasons for resorting to a definition which hinges on the nature of the decimal expansion. One reason is perhaps laziness. To understand why $\sqrt{2}$ cannot be expressed as the ratio of two whole numbers requires a degree of logical reasoning that students may be resistant to (or, unfamiliar with). The teacher may not want to deal with that factor and goes down the path of least resistance. Of course, the laziness may be impossed upon the teacher by time factors. To appreciate the Greek's definition requires time to carefully consider and construct a logically coherent argument. You're not going to accomplish this in one class period--not if you want to get all the students to be on-board with you. (Other reasons could be the teacher's own lack of understanding, but here let's just assume for the sake of argument that we're talking about someone who is highly qualified to teach mathematics.)

I think it is worth pointing out that there really is nothing simple about defining irrational in terms of a nonrepeating decimal expansion. It presents an opportunity for a great challenge to give the students: prove to me that such a number exists. How do you show that no matter how far out you go in the decimal expansion of a number that it doesn't eventual start repeating itself? Perhaps the students come up with something clever, but then how do they show that their clever approach really does produce what they claim it produces? If you can show that such numbers exist, what does that tell you about the nature of numbers? (Oh, there are so many wonderfully juicy questions to explore. The tie-ins with philosophy and how we look at the outside world are potentially enormous.)

I've never heard of anyone taking the following approach. However, I can't imagine that I'm the only person to have tried this in a high school classroom.

Here's the approach:

Define rational numbers as numbers that can be expressed as the ratio of two whole numbers. Next, show that you can enumerate the rational numbers: you don't have to use the word enumerate, just tell the kids, "Hey, look at this! We can write out a list of them." Then look at the decimal expansion of those numbers; fill in trailing zeros where necessary; don't worry about whether the decimals repeat or not. Now, use the diagonalization approach to create a new number that's not in that list. Voila! A number that is not rational. [This can easily take two or three days of class time if you do it right. And, if you do it right, there are lots of places to emphasize the curriculum you are expected to cover.]

Why approach it this way? Well, it's a clever argument. But it's also one that is easier for the students to follow and make sense of. It's very tangible. You actually have the try to write out a list of all the rational numbers. Let them figure out their own approach to doing this. They're bright enough; they'll figure something out. And, if they don't come up with what you already have in mind, that's fine. [This is part of the reason this takes several days to accomplish.]

This appoach shows a way to indesputably create an irrational number. (Students buy into it more because they like decimal better than fractions.) If the students came up with different ways of listing the rational numbers, then have them compare their answers to see that they all have come up with different irrational numbes.

Importantly, this approach has actually taught them something else: while you can make a list of all the rational numbers, you cannot do the same with all real numbers (since by forming any such list you can create a number not in the list). And, this conclusion can be part of a wrap up exercise. [It also creates a great opportunity to talk about other matter such as to why $0.99999$, repeating, is equivalent to $1$.]

Next you can then have the students show why rational numbers must have repeating decimals. Conversely, given a repeating decimal, you can teach them to show how to find two whole numbers whose ration will reproduce the desired expansion. Now you've shown that not only do rational numbers have a repeating decimal, irrational numbers must not have such an expansion. Into this you can build a whole series of lessons about primes and the natures of remainders upon division. And there are various cool tangents to go down: such as looking at cyclic primes and the decimal expansions of fractions whose denominators are such primes.

And then, you can ask the students, "Are there any lengths which we can construct with a compass and ruler that cannot be expressed rationally? In comes the usual proof of the irrationality of $\sqrt{2}$.

All of this is quite feasible in a geometry class. And it would be a lesson suitably taught toward the end of the year after the students have gained plenty of experience building logical arguments. It could be taught over the course of several weeks with other topics peppers in (or rather the other way around, "irrationality" peppered in among other topics). But, however you try to go about it, it takes a lot of time.

Given my experience teaching geometry in a high school classroom, the curriculum usually doesn't allow you the luxury to spend that much time building toward a particular topic in a manner that will create a substantial buy-in and appreciation from the students. It's not necessarily that you can't squeeze in all the various topics the curriculum demands. If you have the luxury of writing your own lesson plans and structuring out how you will teach what over the year, it's very feasible. However, that's not the reality for most classroom teachers. Frequently, the teacher is shackled by a curriculum impossed from outside the four walls of their classroom, or the teacher may be forced to follow a scripted lesson plan, and required to follow lock-step what the other teachers in his/her department are doing. If the teacher must work with other teachers, either in their own department or across a school district, things get even more difficult. And, so the teacher sighs, throws up their hands, teaches a definition that lacks elegance, and loses an opportunity to show students some of the inherent of mathematics.

Parting thought. I love the word "expansion". It suggests an unpacking of the ratio into decimal form. What exactly do we mean then when we say "the decimal expansion of an irrational number"? I think it's a great question to have your students ponder and philosophize over.

A.Ellett
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  • I don't agree that using decimal expansions is the best way to show irrational numbers exist. In particular the diagonal argument is NOT how Cantor showed the uncountability of the reals. Look at this: https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article – Michael Hardy Sep 04 '16 at 18:43
  • Moreover, the ratio of diagonal to side of a square is how the idea emerges naturally from geometry. – Michael Hardy Sep 04 '16 at 18:43
  • @MichaelHardy Well put regarding the diagonal of the square. That's part of what I was trying to drive at. – A.Ellett Sep 04 '16 at 18:44
  • @MichaelHardy Regarding decimal expansions and diagonalization, it's an approach that has worked for me in the classroom. Diagonalization is relatively easy for the students to grasp, but I would never just leave it there. Hence, I always wrap up the sequence of lessons with the usual proof for the irrationality of $\sqrt{2}$. I'm not sure why you bring up Cantor or Cantor's proof. – A.Ellett Sep 04 '16 at 18:55
  • Where do you teach? – Michael Hardy Sep 04 '16 at 19:11
  • @MichaelHardy I've taught in a variety of venues including at a small liberal arts college in the midwest, and most recently in an inner-city high school in a large metropolitan area. – A.Ellett Sep 04 '16 at 19:15
  • The fact that the square root of 2 exists as a real number should eventually be established analytically, not geometrically. The standard Euclidean proof you cited using the diagonal of a square depends on the parallel postulate. So here you are working in a to establish existence. The analytic approach uses Dedekind cuts or Cauchy sequences. This is not an antiquated 19th century approach! – user52817 Sep 04 '16 at 22:54
  • Working in a model to establish existence. – user52817 Sep 04 '16 at 23:39
  • @user52817 i never said dedekind cuts etc were antiquated. ZF set theory (and is kin) are not necessary here. and i could not imagine trying to get high school students to understand them (there's a lot of very high level math involved with d cuts). but i would be interested to know why you believe a geometric proof is inadequate. – A.Ellett Sep 04 '16 at 23:47
  • The geometrical proof that sqrt(2) exists is indeed the best you can do for high school students. But since it relies on the parallel postulate, by modern standards of rigor, it's weak. Since the premise of the question concerns defining the real numbers as analytic objects, I just want to point out this shortcoming. You did not say that d-cuts are antiquated, but you did say in reply to a earlier answer that they are not necessary. I am trying to point out that such an approach is necessary for a modern definition of the reals. – user52817 Sep 05 '16 at 01:01
  • @user52817 Dedekind cuts are built on ZF which has its own set of axioms. If the problem of a geometric proof is its reliance upon a particular postulate, then similarly for proofs using Dedekind cuts. Just as there are geometries (such as projective geometry) that don't rely on the parallel postulate, so too there are set theories that work with alternate axioms. The beauty of ZF is it points out a very nice way to build models for mathematics. But ZF isn't necessary. (You could approach math formalistically.) ZF is a model for much of mathematics; mathematics isn't necessarily ZF. – A.Ellett Sep 05 '16 at 01:16