# Why is it possible to teach real numbers before even rigorously defining them?

In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined. For example, without the definition the fundamental group, it is almost impossible to teach anything serious about it.

But as far as real numbers were concerned, they were taught very early without any clear definition was even given. Definitions of real numbers using say Dedekind cut were never introduced until in say Mathematical Analysis.

Pedagogically speaking, how is it even possible to teach about a mathematical concept (real number) before clearly defining it?

• Counterpoint: How can you define the real numbers properly unless you have some idea of what kind of properties they are supposed to have? – Adam Nov 2 '19 at 1:41
• I would question your hypothesis. I don't think the issue is being able to teach things without a definition, but rather what you teach about them. You could teach people to find fundamental groups of concrete objects without them knowing what a group is, let alone quotients or isotopies. – Jessica B Nov 2 '19 at 10:15
• The same way we teach natural numbers (1,2,3,4,...) before we rigorously define them (Peano axioms, or ZFC set theory,...) – Gerald Edgar Nov 2 '19 at 11:07
• "In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined." -- Really? The ancient Greeks seemed to do okay. Euclid never really had definitions that were all that rigorous. – John Coleman Nov 2 '19 at 20:58
• "In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined." -- I think it takes mathematicians many years of training before they reach the point where they can't understand anything without a formal definition. – Nathaniel Nov 3 '19 at 0:30

Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of calculus predated formal definitions of the real numbers (Dedekind cuts, Cauchy sequences) and a formal definition of continuity, individuals must first make use of real numbers for years, before they are capable of even understanding the need for a formal definition.

Thoreau wrote, “If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them.” This is how mathematical knowledge expands. First we build castles in the air, and foundations afterwards.

• This is not a disagreement with the sentiment of the answer, rather a qualification - It is unfair to Bourbaki to attribute to Bourbaki overly formal teaching practices. Bourbaki's books are far from the stale, formalist screeds some contend (their books on Lie groups/algebras are my among my favorites on these topics). One might even say they are little read and much criticized. – Dan Fox Nov 6 '19 at 8:33

It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. 376) calls the Fundamental Assumption of School Mathematics.

In terms of the nitty-gritty of classroom instruction, real numbers are handled in K–12 by what is called the Fundamental Assumption of School Mathematics (FASM; see p. 101 of [Wu2002] and p. 62 of [Wu2008b]). It states that any formula or weak inequality that is valid for all rational numbers is also valid for all real numbers. For example, in the seventh grade, let us say, the formula for the addition of fractions, $$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},$$ where $$a$$, $$b$$, $$c$$, $$d$$ are whole numbers, can be (and should be) proved to be valid when $$a$$, $$b$$, $$c$$, $$d$$ are rational numbers. By FASM, the formula is also valid for all real numbers $$a$$, $$b$$, $$c$$, $$d$$. Thus high school students can write, without blinking an eye, that $$\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{3}}=\frac{\sqrt{3}+2\sqrt{2}}{\sqrt{2}\sqrt{3}},$$ even if they know nothing about what $$1/\sqrt{2}$$ or $$\sqrt{2}\sqrt{3}$$ means. If this seems a little cut-and-dried and irrelevant, consider the useful identity $$\frac{1}{1−x}+\frac{1}{1+x}=\frac{2}{1−x^2}$$ for all real numbers $$x$$. If $$x$$ is rational, this identity is easily veriﬁed (see preceding addition formula). But the identity implies also $$\frac{1}{1−\pi}+\frac{1}{1+\pi}=\frac{2}{1−\pi^2}.$$ Without FASM, there is no way to conﬁrm this equality in K–12, so its validity is entirely an article of faith in school mathematics.

• You might want to pick an example that doesn't need to consider zero denominators. – chepner Nov 4 '19 at 15:33
• Or, just tack on $x \neq \pm 1$, easy fix. – James S. Cook Nov 9 '19 at 15:09

But as far as real numbers were concerned, they were taught very early without any clear definition was even given.

Were they?

I would say that usually no teacher really speaks of real numbers to students until the first calculus lesson. In almost all pre-calculus exercises, students only meet rational numbers, in the form of fractions or decimal expansions. The only exceptions are some roots of integers, like $$\sqrt{2}$$, and the number $$\pi$$, when doing geometry. Therefore, algebraic speaking, before calculus students are not working with real numbers, but they are working in a field extension like $$\mathbb{Q}(\pi, \sqrt{2}, \sqrt{3}, \dots)$$ or something a little bigger.

The fundamental property of real numbers, that distinguish them from said field extensions of the rationals, is their completeness, that is, $$\sup A$$ exists for every bounded $$A \subseteq \mathbb{R}$$. I doubt students ever uses or see this before calculus.

• What is $\pi$? There is a huge difference conceptually between adjoining to $\mathbb{Q}$ a root of a polynomial with integer coefficients and adjoining to $\mathbb{Q}$ a number to be called $\pi$ that has not been defined, even operationally ... How do I compute with such a thing? By calculating areas of regular polygons I can give a succession of shrinking intervals with rational endpoints that contain it ... The definition of $\pi$ is another one of these things that is problematic pedagogically in the same way as the definition of the real numbers. – Dan Fox Nov 6 '19 at 8:44
• How does one deal with exponentials and logarithms from this perspective? How would one consider the equality $e^{\log2}=2$ by working in a field extension of $\mathbb Q$? – YiFan Nov 17 '19 at 10:45

At German universities, one of the first lectures in mathematics is "Analysis 1" which is a kind of "rigorous calculus" and there one always proceeds more or less like this:

We start with an axiomatic approach to the real numbers. In short: The real numbers are a complete, ordered, Archimedean field. In practice we first introduce the axioms of a field, then the axioms of an order, and then, with the help of the Archimedean axiom, we introduce limits by epsilon/delta. Then you observe that there are rational sequences which should be converging, but there is no limit and some axiom of completeness is introduced (often "every bounded set has a least upper bound", sometimes "every Cauchy sequence converges", sometimes "the intersection of nested closed intervals is non-empty") and from then on we work with the real numbers.

Sometimes, but not always, some construction of the real number is given, but in most cases, this is skipped. It has been like this when I studied in the 1990s and it still is like this.

Of course, this is difficult for some students, but the majority of the students goes through this quite well.

Personally, I like this approach. You introduce some objects not by exactly saying what they are, but by stating a complete list of what you can do with them. (Tim Gowers wrote a great chapter on this in "A very short introduction to mathematics*.)

Finally, let me ask: If you want everything defined, where do you start? Define the natural number using set theory? But how to define a set? At some point you will be back the defining objects through their properties somehow.

• This is exactly the same approach that was used in the engineering curriculum at my university in Italy, at the very first lesson, some 30-odd years ago. We actually did a bit of the Zermelo-Fraenkel axioms of set theory, but we didn't construct any model of the real numbers (they probably later hinted at the construction through Cauchy sequences). A very nice analysis book that approaches the real numbers in this way is Mathematical Analysis I by Zorich. – Massimo Ortolano Nov 3 '19 at 9:47
• Totally agree with your emphasis. I think it is a pedagogical failure to leave out an explanation of axiomatizations (interfaces) and structures satisfying them (implementations). See also this post. – user21820 Nov 3 '19 at 12:35

Introductory mathematics is not done with formal definitions. This is because most people at the corresponding age (kids) don't know what the words "formal" or "definition" actually means. This only come to play much later.

You start teaching kids mathematics by showing them how to count things up to ten using their fingers. Then, you teach addition with little simple problems like "John had five apples. Mary gave him two more. How many apples does John have now?" - You teach by making associations of numbers and math with phenomena that are easily recognizable and verifiable in their world and day-to-day activities.

Only after they are already somewhat used to counting numbers and doing addition, subtraction, multiplication and division and they already got a few lessons on sets of oranges, sets of cats, sets of some small numbers and union and intersection of those is that the concept of the set of natural numbers ($$\mathbb{N}$$) are introduced to them.

Some time further, you introduce the concept of fractional numbers and negative numbers. And again, this is associated with things easily recognizable in the mundane world like "two and a half pizzas" or "a half orange" or "John have five dollars and promised to give seven to Jane, so he is lacking two". So far, no one gave them a formal definition of those things, these are only show up later on when the kids are (or should be) firmly familiar with the concept.

The (in)formal definition of $$\mathbb{Z}$$ only comes in after the kids did played a bit with negative numbers. Only after the kids are already familiar with fractions is that you explain that "every fraction is representable with a ratio between two integers numbers" and you "call all those numbers that can be represented in this way as the set of rational numbers denoted as $$\mathbb{Q}$$". Gosh, that was an informal definition, but it is sufficient and more than enough to be understood by kids. No kid would easily and quickly understand something like $$\mathbb{Q} = \{\frac{p}{q} | p \in \mathbb{Z} \land q \in \mathbb{Z}^*\}$$ because that notation and the rules governing them are alien-like for most of them even if they were already presented and used to all of the involved elements.

Some time later, when teaching square roots and geometry, things like $$\sqrt{2}$$ and $$\sqrt{3}$$ show up. You quickly tell them that these aren't rational numbers and that there is no way to represent them as a ratio of two integers. Again, that is an informal definition, but is enough. You might even give a further step by telling them that any square root of a prime number is irrational because that if that was not the case, by finding a ratio of two integer numbers that squared give the supposed-to-be prime number would mean that it would be composite (i.e. a proof by contradiction, but still informal). Then, you present the symbol $$\mathbb{I}$$ to represent them. Also, soon $$\pi$$ will also show up to join the irrational numbers group.

Finally, you present the real numbers just as "the union of rational and irrational numbers". This is a very simple and informal, although very precise and sufficient definition.

So, the answer is that people can learn about the real numbers without formal definitions by simply working to them with informal definitions and correlations with real-world mundane concepts and then by building up higher-level concepts on top of lower-level concepts. Formal rigorous definitions are then usable only for people that are already skilled enough in math to be able to make sense of them.

• I pretty much agree with everything except saying something like $\ll$ Finally, you present the real numbers just as "the union of rational and irrational numbers". $\gg$ Knowing what the irrational numbers are is essentially the same as knowing what the real numbers are, since "irrational" is defined as "real and not rational", so this doesn't really tell the student anything, and some young students will surely realize this. Probably better to say something like real numbers are all possible decimal expansions or all possible lengths, and some can be irrational. – Dave L Renfro Nov 3 '19 at 12:37
• @DaveLRenfro Some students who only met rational numbers so far, if not explicitly presented to irrational numbers, might not realize that things like $\sqrt{2}$ are not rational. So, presenting the real numbers before the irrational ones would make no sense for them because they could be unable to imagine a real number that is not rational. On the other hand, by first showing them that there are irrational numbers out there, makes presenting the real numbers very straightforward later. – Victor Stafusa Nov 3 '19 at 14:34
• presenting the real numbers before the irrational ones would make no sense for them because they could be unable to imagine a real number that is not rational --- I'm not saying that they have to know the existence of irrational numbers in advance, but rather simply indicate what it takes to be a (real) number, such as all possible lengths or all possible decimal expansions. The issue of whether this notion corresponds to all possible fractions comes later, when something like $\sqrt 2$ comes up in a geometrical setting or in solving equations. – Dave L Renfro Nov 3 '19 at 15:27

I believe that, unlike many other mathematical concepts (especially more abstract ones), even fairly young people have a basic understanding of distance, including fractions to any degree of units of measurement (e..g., inches, feet, miles, etc., in the imperial measurement system and/or centimetres, metres, kilometres, etc., in the metric measurement system).

By using things like a number line, you can explain & demonstrate various properties related to real numbers without needing to first give a clear & "rigorous" definition of them. For example, you can at least "show", although not necessarily "rigorously" prove (with this depending to a certain extent on what you consider to be "rigorous"), concepts like that addition is commutative, adding negative numbers is the same as subtracting the positive value of them, etc.

There are various online resources to help use number lines to teach real number concepts. One quite good basic one being Algrebra1Coach's Real Numbers and the Number Line, and a somewhat more rigorous & advanced one being LibreText's Real numbers and the Number Line. Also, a more specific resource is teachoo's Locating irrational number on number line.

• Thanks! Can you prove any theorems about real numbers without clearly defining what a real number is? – Zuriel Nov 2 '19 at 1:27
• @Zuriel You're welcome. I've updated my answer to state about at least showing commutativity of addition, adding negative is the same as subtracting positive, etc. – John Omielan Nov 2 '19 at 1:30
• And yet distances in our world are most certainly discrete – Francisco José Letterio Nov 2 '19 at 4:25

Since real numbers include whole numbers, rational numbers, and irrational numbers (almost the entire landscape) I’m uncertain why you think teaching does not occur.

Does it not begin very early with counting and learning numbers? It isn’t until later that these numbers are labeled as real, when needing to distinguish from imaginary as introduced in pre-algebra(?)

The question seems to extend into psychology and philosophy; but one could also make the point that intuitive concepts need little-to-no teaching. We don’t need to know how to spell before we say words (or even write words) and we don’t need to know their definitions before they are used - especially not their exact definitions.

Note: I’m uncertain if your question is something nuanced to mathematics that applies beyond my level of understanding of it. If that is the case I will delete this answer.

• The question is talking about the construction of real numbers that are not rational, that is irrational numbers, rather than the distinction between real and complex numbers. The issue is that while the definitions of integers and rational numbers are intuitive and accessible, the formal construction of real numbers beyond the rational numbers is neither intuitive nor accessible at an elementary level (it was not achieved by mathematicians until 1872). So at an elementary level one operates with numbers that one has not defined, and this is potentially problematic. – Dan Fox Nov 6 '19 at 8:37

You are confusing definitions and models.

A definition of real numbers is a set of axioms they obey. Different such sets exist, but they can be shown to be equivalent.

A model of real numbers is a construction like Dedekinds, which is a set of sets of sets etc. that is carefully constructed so that they obey one such set of axioms.

To work with real numbers you really only need to know what axioms apply. You use them to construct theorems.

This does not mean that models are useless, but they are approaching the subject from a different angle.

"In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined."

There is no reason, this has to be true. It is just asserted. Two examples have been given where significant content can be learned without strict axiomatic buildup of the subject: arithmetic and calculus. We could come up with others (set theory, topology, etc.).

Note: I am not disputing that the axiomatic buildup can also be beneficial at times, very much so. But we should not just cite the original statement as some sort of doctrinal statement. After all, we are discussing practical pedagogy in that statement, not the axiomatic structures of math themselves. I think we would want evidence, statistics, explication before just going with the statement as true.

This answer may seem off-topic, since this is "Math Ed" SE. But a similar question can be asked about many scientific concepts.

Based on observations of the world around us animals (including humans) have built up heuristic models of physics, biology and psychology. In order to be a useful creature, one must have some predictive models, even if they are not formal (in the sense of formal mathematics). Someone that is good at sports, e.g. basketball, has some intrinsic sense of Newtonian physics. The math that describes how a ball flies through the air uses real numbers, but knowledge of throwing a ball doesn't require any knowledge of real numbers.

Thus, to (maybe) answer the OP, when we teach math (or physics or biology), we are often telling students: "You know how when you throw a basketball you have to control the angle and the force? Well, this is how to write a formula that tells you for any basketball shot, the ideal angle and force to use!" We are connecting a formalism to a concept they already know, not building up a formalism from the bottom up.

Read Dirk's answer - we take the axioms of rational numbers (which are just fractions of integer numbers, so easy to understand) and get quite a few nice theorems. We even have Cauchy sequences (which are kind of sequences with limits but not quite, but the difference was never mentioned).

In my Analysis I, the professor then went to prove the Intermediate Value Theorem. (If f is continuous, a < b, f(a) < 0 and f(b) > 0, then there is an x, a < x < b, with f(x) = 0). So he defines the set of all y with f(y) <= 0, says this set is non-empty and has an upper bound, therefore it has a least upper bound x, and because f is continuous, f(x) can't be either less than or greater than 0, so f(x) = 0.

Muttering starts among the students and gets louder. At last someone gets up and says "That proof is not right. How do you know that this set has a least upper bound?" And you hopefully noticed that what the professor boldly claimed was exactly the missing axiom that makes the difference between rational numbers and real numbers, and that's what the professor says. "Axiom #10 of real numbers: Every non-empty set with an upper bound has a least upper bound". And immediately afterwards he proved what is now obvious: Every Cauchy sequence has a limit.

Think of the naturals instead of the reals. We learn to count around age four. We learn a lot of properties of arithmetic, like addition, subtraction, multiplication, division, prime numbers, unique factorization, etc. without reference to any axioms. This is just how numbers work. Similarly we learn how reals work before we see a careful construction of the reals. I saw and understood the proof that $$\sqrt 2$$ was irrational before I saw a construction of the reals. In a sense the careful construction of the reals is wasted effort. We wind up proving they behave as we already know they behave. Having done that, we don't reference the construction any more, we get on with life proving things based on the properties (that used to be asserted, now are theorems) we know and love.

You seem to think that calculations are impossible without definitions of terms. That is not true.

Historically, mathematical definitions were deduced from mathematical practice. That is, the practice came first, the definitions later.

… because students learning more about real numbers at university bring their understanding of real numbers that they developed through high school. Which is, real numbers are the set of rational and irrational numbers: all possible numbers on a single continuous number line. And that’s a correct working schema to begin learning more about real numbers and develop formal definitions.

In both learning and history, informal understandings typically preceded rigorous formal definitions.