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Since the question might be unclear, let me provide some details about how I came to asking it:

Providing rigorous definitions to non-mathematics undergrads is a non-trivial task -- yet important. However, the equilibrium between rigour and sloppiness is unstable at best:

  • Be sloppy and leave out so such much that the ''definition'' gets stripped of any use.

  • Be rigorous and risk losing 99,9% of the class because students just can't follow.

One example that recently came back to my mind is the definition of the real numbers. If one goes for the currently accepted definition, then you can bet your bottom dollar that only a few chosen ones will understand it. Then, I recalled having seen the below definition of the real numbers. It goes as follows (and I thank @Blue for the great edits to my first post):

$\mathbb{R} = \{e \mid e^2 \ge 0\}$

Source. MIT OpenCourseWare. Lecture by Herbert Gross: "Part I: Complex Variables, Lec 1: The Complex Numbers". (03:00) Transitioning from a discussion of how solutions to $2x=3$ could not be integers, Prof. Gross sets up a similar discussion about how solutions to $x^2=-1$ can not be real numbers, remarking (emphasis mine):

By definition, a real number is simply any number whose square is non-negative. Assuming that to be the definition of real numbers, we come to the equation $x^2=-1$, [...]

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Now, would that be a good way of doing for non-mathematics undergraduates? Why would it and why would it not? If not, then how would you proceed? What would your best ''definition'' be for the real numbers?

The purpose of the question is less about ripping apart a simplified definition than about communicating the right message to the right audience -- without getting stuck with futile definitions.

DeeCeeDelux
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  • I don't like that as it is not even evident that this class is a set. In fact, this seems to contain the proper class of ordinals. -- Moreover, we need to define squaring and comparing numbers before we know which numbers we are talking about – Hagen von Eitzen Sep 23 '18 at 15:50
  • I don't think so, you would have to state from where to choose your elements ($e$ here), which would essentially be a loop, since you are assuming that $e$ already is a real number. – Pink Panther Sep 23 '18 at 15:52
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    What is the meaning of $e^2$ if real number $e$ is yet to be defined?... – drhab Sep 23 '18 at 15:53
  • Those are exactly my points... But how would one proceed except for saying that it is the 'set of all numbers'? Or should I accept that there is no other intuitive way except the latter? – DeeCeeDelux Sep 23 '18 at 15:56
  • What kind of definition is that? Nothing about complete ordered fields? – Angina Seng Sep 23 '18 at 15:57
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    I don't see how that defines anything. – lulu Sep 23 '18 at 16:03
  • In what context did you see that "definition"? – Blue Sep 23 '18 at 16:41
  • MIT OpenCourseWare https://www.youtube.com/watch?v=BOx8LRyr8mU&list=PLD971E94905A70448 03:00 – DeeCeeDelux Sep 23 '18 at 17:05
  • @DeeCeeDelux: Thanks for the link. I took the liberty of adding some context from the video to your question. I think the instructor was being a bit loose with the term "definition", because what he gave is not a formal definition of real numbers. (His informality is excusable, being just 3 minutes into the first lecture on a pretty sophisticated topic.) In context, he seems to mean, "If we know anything about numbers that we have encountered (so far), it's that their squares are non-negative. Which raises the question ... What about numbers whose squares are negative? [etc, etc, etc]" – Blue Sep 23 '18 at 17:54

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Off hand, the only possible use of that characterization of $\mathbb{R}$ that I can imagine is as a solution the very specific problem:

Identify which complex numbers are real.

And even then I don't think it really achieves that purpose, since using the relation $\geq$ in such a context pretty much already assumes you've identified the real numbers.