51

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 ...


40

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 ...


24

As far as I know, there are four basic possibilities for defining the real numbers: The real numbers can be defined using axioms. The real numbers can be defined using decimal expansions. The real numbers can be defined using Dedekind cuts The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers. Approach #1 is ...


24

First of all, $\sqrt{2}+\sqrt{3}$ most certainly is a number. It is a real number, approximately equal to $3.14626$ Perhaps what you're asking is why the sum of two simple radicals isn't also a simple radical, when the sum of two integers is an integer, and the sum of two fractions is a fraction. Roots of integers are examples of algebraic numbers - numbers ...


22

Short Answer You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term should be introduced as such—emphasize that "real" in mathematics does not mean the same thing that "real" means in everyday vernacular English. Long Answer ...


20

I think G Tony Jacobs's answer is an excellent one, and I admit I don't quite understand what the OP's objection to it is all about. But I am going to take a stab at trying to explicate what I think might be underlying the objection that "$\sqrt{2}+\sqrt{3}$ is not a number". Looking over the examples that precede the question, we can notice that they all ...


18

When it comes to convincing younger students of something, I find that analogy can be quite useful, even if you have to squint a bit to make it technically rigourous. I imagine a younger student reasoning something like this: "Hey, I know 5 is different from 50, and 50 is different from 500. When you add zeros at the end of number, it changes the number. So ...


16

I'll focus on question 2 from a perspective of "maybe the right thing to think about is: what happens in the students' minds while they read this question?" When you say "Suppose $A⊆R$ is nonempty and bounded above," the following thing happens in my mind before I continue reading: This picture takes up very little working memory for me because I am ...


16

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 ...


13

Check out Berkeley mathematician H.H. Wu's homepage. In particular, see his textbook drafts for Pre-Algebra (pdf) and Introduction to School Algebra (pdf). For example, see p. 20 and the discussion of decimals as "a class of fractions" in the former text. Note that the book is intended for teachers of 6th-8th graders and not for students. The exposition ...


12

I think the level of the student is very important to this question. If the student has never had an abstract math course (like my students), then the lack of a definition of "number" is a great way to introduce the idea of abstract algebra. They are very happy to initially believe a definition like A number is anything that you can add and multiply, such ...


12

Turn the problem around. Say to the student: If "5.00" is not the same number as "5", then one of them is larger than the other. Which, and by how much? You're the one who claims there is a difference. Then tell me, literally "what is the difference"? Does a movie ticket priced at "five dollars and no cents" cost more or less than one priced at "five ...


12

I've approached this a couple different ways in the past, and I haven't decided what I think is best. But I'd like to point out some arguments in favor of not including any construction of the real numbers (like Dedekind cuts or limits of equivalence classes of Cauchy sequences), and simply stating the axioms which are needed for the rest of the course. (...


11

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, ...


7

I think that the set of reals isn't the one you would like to start with. The problem with it is that they have their quirks and for many it is counter-intuitive that any open interval is of uncountable size. On the other hand, there are many other objects with which the Cantor's diagonal argument works as well (sometimes it works even better). To name a few:...


7

Note: The question is broad, I agree with a lot of what Mark Meckes said. Below my take on the Cauchy sequences approach (mainly in favor to complement the other answer). The approach of constructing the real numbers as equivalence classes of Cauchy sequences of rational numbers modulo sequences with limit zero seems certainly not like the most intuitive ...


7

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 ...


6

Whether you construct the reals or just axiomatize them, at some point you need to talk about the completeness property or completeness axiom. The students should ultimately learn several versions of this (Cauchy completeness, nested intervals, least upper bounds). I've found the following version, not listed in previous answers (unless I overlooked ...


6

Let me play devil's advocate. They are not identical. In situations where complex calculations result in a number such as 6.3856, and the science teacher is asking for 4 significant digits after decimal, one wouldn't round at all. If the answer happened to be 5.0000, the student should keep the zeros to indicate the level of accuracy, that no rounding was ...


6

This comment from MSE by Kimchi Lover is fantastic. Examples every time. Can you think of a target end example result and work the course up to understanding it? If I were teaching it, I'd make the goal something like "Brownian motion paths are continuous and nowhere differentiable" or "most numbers are normal numbers" and construct measure theory and ...


6

For 95% of high school students, this sort of thing is of no interest. But: The 5% do need to be served well and helped to achieve their potential. The 95% may find such things confusing if they are never explained, so it makes sense to offer them at least some brief explanation. Even the 5% are in no position at this point to understand fully what is meant ...


5

If the student struggles with this task, what do you think he (or she) thinks of a decimal number? What is .37 for him? Let him try to explain the meaning of .37 in terms of examples or calculations where this number is used. A number must have a deeper meaning than just being a sequence of digits in order to understand how we use them. One of your examples ...


5

This would appear to be an essential problem with understanding the place value system. If the student hasn't already understood place value, showing and explaining is unlikely to be effective. First off, you may be unaware of what the student's conception of the value of these numbers is. An activity might help you get to the heart of that, if not help the ...


5

Whatever teachers may think about the nature of numbers, the foundations of "arithmetic" and the nature and concept of number in particular are very subtle. For a recent and sophisticated look at the issues, see: John Horton Conway, On Numbers and Games, second edition, A.K. Peters, 2001. Conway gives his approach to the surreal numbers, relates these to ...


5

In my experience (remedial and college algebra), these topics are always resisted and show poor performance -- very analogously to the topic of fractions (irreducible divisions). One thing I do is show the proof that $\sqrt{2}$ is irrational and tell the story of Hippasus; the lesson, as Lowell put it, "Thoughts that great hearts once broke for, we/ Breathe ...


5

Chapter 2 of Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Edwards, H. M.) is dedicated to surds and their application to Euler's treatment of the problem of whether there are integers $x,y,z$ such that $x^3+y^3=z^3.$ I would look into whether this book, or the chapter in question, can be read or captured in an accessible form in ...


5

I always emphasize comparing decimals with the same number of places to my students. Therefore before comparing 0.12 and 0.125, I would first teach them that 0.12 = 0.120. Students can than grasp that 0.120>0.012. This is further brought home when students read it as 12 hundredths is greater than 12 thousandths. I suggest you work on converting ...


5

Can all of your students correctly explain what the supremum and infimum of a set are? If not, they will be unable to reason about these concepts. It would be like expecting a student to do something with the order of an element in an abstract (arbitrary) group if they can't define what the order if an element is (e.g., thinking "order of $g$ is $n$" means ...


5

I recall having been taught different classes of numbers (in maths at school) way before we were introduced to complex numbers. Main reason was to distinguish natural numbers integers rational numbers finally ... real numbers It's reasonable to teach students things like the coverage of numbers on the number line: Why are real numbers continuous while ...


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