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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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, subtracting a negative number is the same as adding the corresponding positive value, etc.

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

Answer 7

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.

Answer 8

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.

Answer 9

"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.

Answer 10

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.

Answer 11

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.

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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.

Answer 12

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.

Answer 13

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.

Answer 14

… 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.

Answer 15

Most people except mathematicians never need to use the special properties of real numbers (ex. unique Dedekind-complete ordered field). You can teach them that the ratio of circumference to diameter, $\pi$, is about 3.14 and can't be represented exactly as a fraction, but can be approximated well and 5 or 6 decimals is enough for practical purposes. The same for common quantities in engineering like $e$ and $\sqrt 2$.

Therefore the concept of real number is never really taught or defined early on, just as an informal notion of number. The properties of reals are not as important early on in education as understanding numbers as abstractions of measurements and counting that can be manipulated symbolically using equations and operations like addition and multiplication. We know how these operations work for rational numbers and, we implicitly apply the same operations to irrational numbers without thinking about it because it works well enough. This simplification, while technically not "the whole truth", is good enough to be useful and build understanding for future math concepts where our intuitions can later be ground in formal definitions.

Answer 16

As others have said, it's because historically we did not need to have a rigorous definition of the real numbers in order to "prove" theorems about (functions of) them. Therefore, depending on how serious the student is, they may not need it either. I could see it being useful for very serious students who are on the track to becoming career mathematicians, though.