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I am teaching an introduction to number theory for high schoolers right now, and there seems to be quite a bit of confusion on what the difference between the natural numbers, the integers, the rational numbers, the real numbers, the complex numbers, and Gaussian integers right now. Is there a good and quick way to explain the difference to high schoolers who might be muddled amongst these differently defined number systems, as so far as working with them in the context of elementary number theory?

Thanks!

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    $\begingroup$ What do you mean by "the difference": how they are constructed, how they are usually represented, the properties they follow, their applications, their history, or other? $\endgroup$ – Rory Daulton Jul 5 '16 at 22:30
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    $\begingroup$ I visualize these sets as a series of circles of sets of numbers, with each subsequent larger one containing all the smaller ones inside of it. $\endgroup$ – Andrew Sanfratello Jul 5 '16 at 22:32
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    $\begingroup$ Are they really confusing all of these or just a small number of them (e.g., natural numbers and integers)? Have you tried writing down a description of typical numbers in each case, say $\mathbf N =\{0,1,2,3,,\ldots\}, \mathbf Z = \{\ldots,-2,-1,0,1,2,\ldots\}, \mathbf Q = \{1/2, 2/3, -3/4,\ldots\}$? $\endgroup$ – KCd Jul 5 '16 at 23:04
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    $\begingroup$ After some thoughts, I find it difficult to try answer your question right now, as I feel some information is lacking. Could you give several examples of the kind of confusions that occur? I can imagine several (such as thinking that $2$ cannot be a real number since it is an integer, or forgetting that natural numbers are nonnegative by definition) and thay may ask for different explanations. $\endgroup$ – Benoît Kloeckner Jul 7 '16 at 11:51
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    $\begingroup$ I have three thoughts. The first is what @BenoîtKloeckner asks. The second is, although building up (incl. equiv relations etc) is a typical way to introduce these sets, I'm wondering why a high school number theory course wants to work with complex numbers at all. Are natural numbers alone sufficient? If not, then perhaps an explanation of why you want (e.g. R, C) would help. The third is, I think it's important to emphasize more than the embedding of sets into bigger sets. E.g., what do you lose going from N to Z? (PMI, for example.) What about Z to Q? (Discreteness.) Etc. $\endgroup$ – Benjamin Dickman Jul 7 '16 at 17:31
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I am not sure whether this really answers your question, but I could think of the following strategy of introducing these sets of numbers. It is not based on any kind of research and just based on personal experience with first year university students where I sometimes give something similar as a "naive idea" in addition to the proper definitions.

The natural numbers can be introduced as the numbers which appear "naturally" during counting and then the "definition" $$ \mathbb{N} := \{0,1,2,\ldots\} $$ could be given. I do not think that introducing the Peano axioms or something similar would be really appropriate for high school students but it might be of interest to specially interested or gifted students.

Then I would use the theme of "extending the set of numbers so that more equations have solutions": Working with the natural numbers there are many equations which can be solved, but unfortunately the equation $$ x+1 = 0 $$ does not have a solution. This problem can be solved by adding negative numbers which yields the integers.

Then the same steps can be made from the integers to the rational number and (slightly cheating) from the rational numbers to the real numbers and from the real numbers to the complex numbers.

Depending on whether the students already know the concept of limits you can avoid the cheating in the extension from the rational numbers to the real numbers, but talking about the completion is much more difficult. Of course I think that it should be mentioned that the step from the rational numbers to the real numbers is a bit different and that just "adding square roots" is not the right construction.

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    $\begingroup$ While I've had this built into my lecture in the past, it doesn't seem to connect with my students very well, and takes a LOT more time than is dedicated in the curriculum for the subject. More recently I've pretty much been cutting the motivation part. (Which hurts a bit because personally it's very satisfying to have that.) $\endgroup$ – Daniel R. Collins Jul 6 '16 at 21:19
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This is a good opportunity to emphasize the critical status of starting with unambiguous definitions as the first step of any proper mathematical presentation (*). Be sure to provide them with the definitions as stated in the textbook that you're using. Then, granted that students may not be accustomed to reading with such laser-intense care to each word in a definition, it's likely important to run a rather large series of exercises to emphasize that each word of the definitions is important, and to internalize the meaning of each type of number.

This needs to be an "automatic" skill for students, such that they can instantaneously recognize each type of number later on. I would recommend the timed quiz site that I developed so as to exercise exactly this topic (and others like it):

Automatic Algebra: Sets of Numbers

In addition, look for opportunities throughout the course later on to use these definitions and reflect on them again. For example: Perhaps we allow exponents $a^n$ where the base $a$ is real and the exponent $n$ is an integer. Review: What does that mean? Define radicals $\sqrt[n]{a}$ where the index $n$ is natural and the radicand $a$ is real. What does that mean? Define polynomials as a sum of terms $ax^n$ where the coefficient $a$ is real and the power $n$ is natural or zero. What does that mean? Run a series of introductory exercises in each case where students must identify whether expressions qualify for each definition or not.

Overall, this subject is among your best chances to highlight the importance of careful definitions, and how they will be used to define and illuminate later concepts throughout the course.

(*) Fine print: It should go without saying that any presentation starts with some undefined intuitive terms; in this case it's the natural numbers. But integers and rationals can then be rigorously defined, and this is a prime opportunity to practice the axiomatic thought process in a basic algebra class. Real numbers will be somewhat hand-waved at this level.

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    $\begingroup$ I am not sure that unambiguous "definitions" are the first thing, or, rather, fairly unambiguous goals. That is, why make rules that have no avowed purpose? That kind of thing... $\endgroup$ – paul garrett Jul 6 '16 at 2:33
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    $\begingroup$ I hardly see how to give high-school level unambiguous definition of the natural numbers for example, unless you assume some pretty high-level background accepted (e.g. the real numbers taken as granted). $\endgroup$ – Benoît Kloeckner Jul 6 '16 at 7:33

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