Here’s the definition of 2 using the standard construction with the Peano axioms. It’s the set containing 0 and 1. The number 1 is the set containing 0 and 0 exists by one of the axioms. It’s not something a person in intermediate algebra can understand. For one, the natural question then is, “what is a set?”. Whatever one does there has to be some brain washing in order to get started. This is unavoidable unless one thinks Principia Mathematica should be the starting point.

Well, the peano arithmetic can be described directly as first order logic without set theory ;)

I'm fine with having an intuition for sets, but I think reals really should be defined properly. At least, R should not be confused with the algebraic closure of Q.

The algebraic closure of the rationals is not the reals. The reals are the completion of the rationals using the standard metric.

The intention of my post was to point out the complexity of not brain washing students at a low level. Your comments have enhanced my point by bring up considerations I didn’t want to get into!

This volume does not confuse R as the algebraic completion of Q. It is completely reasonable for an Algebra 1/2 teacher to wait for a Calculus or Analysis teacher to discuss the metric completion of Q. Describing R as rational + irrational numbers is a completely solid description.

It is only solid as long as you don't define irrational numbers literally as everything that is not rational.

What would be an example of a real number that's neither rational nor irrational? (I'm not a math guy, in case it's not obvious)

I think by the definition given at the beginning of the thread irrational numbers are numbers that can’t be expressed as a ratio of two natural numbers. i is irrational by that definition, but not a real.

But I must admit I haven’t read the whole post.

Looks like the set of reals is simply taken as the number universe (i.e. all numbers are already assumed to be real).

Just to clarify: there are lots of numbers that aren't real numbers (for example, imaginary numbers). Intuitively the real numbers are all the points along the number line, including rationals and irrationals (such as root 2, pi, or e). There are lots of other comments in this thread that give a good explanation of how that works formally.

If the students haven't yet encountered complex numbers, infinitesimals, infinities etc. then it's perfectly reasonable to say that all numbers are assumed to be real (as follows strictly from the definition in the book).

There is no such example. All real numbers are either rational or irrational.

∞

i or some hyper real numbers are neither irrational nor rational. Basically all crazy extensions of R used to solve even more crazy problems. They don't really exist naturally (and you can make up your own field extension of R as you wish), but they are handy if you want to compute stuff.

But by definition there are no real numbers that are neither rational or irrational.

You "win at the game" by learning what is taught, getting the "A", and then doing you own in-depth research about what interests you on your own time.

K-12 was, of course, invented by the Germans in order to create good little factory workers that would get up early and work all day and not complain too much. The fact we still use the word Kindergarten is a nod to this origin story.

They weren't at all interested in the students gaining any "understanding" and most certainly not in them "winning" in any sense of the word.

> Everything feels as if it was randomly defined by the teacher.

I suppose you prefer things randomly defined by Euclid? Just kidding... kinda. Seriously though, randomly defining things and then working through the consequences of that definition is a totally valid way to do math. Those random definitions are called postulates.

A student may be confused about why just these axions and not some others? It can then be explained that there in fact be alternate set of axions.

But still most teachers give us the same standard set of axions. Why? What would happen if they dropped some of them or replaced them with others?

Euclid's postulates would now be called axioms, not definitions.

One of the standard construction of the real numbers is the set of equivalence classes of rational Cauchy sequences. This definition is equivalent of the handwavey definition above (irrationals are the Cauchy sequences that don’t converge to a rational number and the reals are the rationals plus the irrationals.)

However almost any construction of the real numbers is challenging to give a simple explanation for. Even leading 19th century mathematicians didn’t truly understand the real numbers until Cantor.

A problem is that lots of lower-education math instructors don't understand these concepts deeply themselves. I think it could be okay if these things were clearly framed as "true for the problems we're looking at, but not universal", but they were typically presented as universal by teachers who themselves don't know any better, and that really caught me up too.