At this level, associating numbers with potentially infinite decimals would not be too bad. The examples of π and √2 are shown as decimals, suggesting that students are already expected to think something like this.

I think that this pair of definitions for real and irrational numbers are unacceptable - the examples of irrationals mean that they don't mislead anyone (unless you can find someone who thinks of complex numbers as numbers but hasn't encountered the term "real numbers"), but it's wrong to pretend that this is a mathematical definition. Even just removing the words "defined as" would be fine.

> At this level, associating numbers with potentially infinite decimals would not be too bad.

I find number with an infinite number of decimals quite an abstract concept. Another issue with this is that it confuses numbers with their representation.

> I find number with an infinite number of decimals quite an abstract concept.

If students are familiar with the idea that fractions can be represented as repeating decimals and are introduced to π as a non-repeating decimal, then I think the idea of the set of all possibly repeating decimals should make sense to them.

> Another issue with this is that it confuses numbers with their representation.

It's probably worth acknowledging that some numbers have multiple representations just to avoid arguments about whether 0.999.. = 1.

However, is it much worse to say that a real number is an equivalence class containing 1 or 2 infinite decimals than to say that a real number is a non-empty, downwards closed, proper subset of the rationals? I agree it's nicer to define the reals as a field with some extra properties, but I'm not convinced it's wrong to say that they are their representation.

At that level, I'd use a geometric definition. I would postulate the existence of a set R of numbers which represent the distance along a line based on a given unit. From there, you "show" that natural and rational numbers belong to R. You show that some numbers are irrational (easy to prove that sqrt(2) is).

There are some postulate there but I think it provides the right intuition and it's not a obviously circular definition.

I don’t give a definition. The students know real numbers are the “decimal numbers”. They don’t need to know a definition at this point and giving one more precise than “decimal numbers” will just confuse them. They gain familiarity by practice in the same way a Fahrenheit person like myself can only get intuition with Celsius temperatures is by using them.

From Wikipedia:

> In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line

That's the intuitive definition that makes the most sense to me.

Now define distance :)

You can make this argument on literally any definition - you have to start with something.

So dictionary.com says the defintion of distance is:

> the extent or amount of space between two things, points, lines, etc.

That works for me.

I think it makes sense to start with something physical, in the real world that people - and especially kids, can intuatively understand. Our brains by default understand what distance is. Maybe that's true for numbers as well- and that's ok too.

Don't forget "continuous" and "line". :)

Two historically trick ones!

I'm sure there are more elegant approaches, but I would start like this:

A number is an imaginary (as in made up by humans, not as in complex numbers) object that solves certain equations. It is not well defined and should not be used as basis for other definitions.

"Natural numbers" can be constructed directly and have a well known notation. Integer numbers are used to make the equation "a + x = b" solvable by x for every natural number a and b.

Rational numbers (Q) make the equation "ax = b" solvable by x for any non-zero integer a and b.

The algebraic closure of Q (=: Q*) makes the equation P(x)=0 solvable by x for any polynomial P.

The definition of R is more complex. I guess I would simply say it contains numbers to solve equations like sin(x)=0 by x.

This makes it very clear that -3 or 0 apples don't exist. These numbers just have the purpose to solve equations which makes them handy if you want to compute stuff.

> A number is an imaginary object that solves certain equations.

TBH this is the worst explanation of anything I have ever seen.

>A number is an imaginary object

and now the time has come for Intro to Complex Analysis, good luck not leaving everyone completely dumbfounded.

Also, a natural number is not “imaginary” in any sense. (E.g. you do not imagine the number of fingers on your hand - you discover it, by counting.)

Personally I find these definitions a bit circular.

Equations like "a + x = b", "ax = b" and "sin(x) = 0" have no meaning unless you have an understanding of what the terms mean, or could mean.

You can easily define + on natural numbers and * on integer numbers without running into problems! Thus there is no cirularity here.

I understand very well that the peano axioms for example are logical and not circular.

My point was that I think your explanation relies on the experience on what a number is, having used those equations before with the terms actually representing real numbers (such as 3 + 2 = 5 instead of a + x = b and sin(180) = 0 instead of sin(x) = 0).

So as a definition of what a number is it's circular as you presume experience with numbers and what they mean.

Usually it begins with "consider the number line..."

A number is an informal term within mathematics.

Not exactly true. You can formally construct the set of real numbers with set theory and then say a number is an element of that set Wikipedia page on different approaches: https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...

If we're talking about "number" in general, yes, it is true that this is an informal term among mathematicians.

What mathematicians are you talking about? Set theorists and number theoriests have very precise definitions of numbers.

Maybe by "number" in general do you mean something that encapsulates both "real numbers" as in lengths, and "integers" as in the discrete counting numbers? In which you can quite easily do so by defining real numbers, either formally or saying that something like the Wikipedia definition that they are a distance along a line, and then saying integers are a subset of those numbers.

Mathematicians have precise definitions of "real numbers", "complex numbers", etc. but not of "number." For example, are the hyperreals numbers? Nonstandard integers? Quaternions?

Mathematicians have definitions of all those things. And if you are working in something like the quaternions so do all the previous you mentioned (real numbers, complex numbers) because they are subsets of the well defined quaternions.

I agree in a sense there is no agreed-upon simple non-formal definition which encapsulates all of those together but I think that's unfair to try to do since hyperreals, quaternions, etc are extreme extensions which aren't used in most ordinary mathematics.

That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc

You can go down the same rabbit hole I guess and say "yeah! true! distance has no definition either!" but I don't find that very helpful. At this point you are just saying nothing can be defined, at which point the phrase "we have no precise definition of X" has no meaning as it is true for all X.

Some things are formally defined terms of art and some things aren't. There are some people here who don't merely want to learn a model for the naturals, they want some kind of spiritual essence of a number. It's useful to say that number is an informal term, and that the fuzzy boundaries you think with are adequate for communication.

Similarly, "crazy" is not a term of art for psychiatry. That doesn't mean a lack of definition, and lacking formalism does not mean inadequate for communication. By using a fuzzy term for communication, you imply a taste in curation and abstraction. It means you don't want to get into the details of what crazy means.

> That is like saying that physicists have no precise definition of "distance" because there is Euclidean distance, Geodesic distance, Hyperbolic distance, Hamming distance, Levenshtein distance etc etc etc

A metric is a term of art, and if that's the level of specificity you wanted, then it's adequate.