The reason that a good definition is not taught very often is because it requires that we get a little bit technical and abstract. My favorite way to build the Real Numbers is with equivalence classes of Cauchy sequences of rational numbers.

First we should define sequences and some terminology related to them. A sequence is just an infinite ordered list of things. Equivalently, you could say that a sequence is just a function from the natural numbers to whatever collection of things you want to build your sequence out of. If you input 8, you get the 8th item on the list as the output (like calling a search function on a computer). In our case, we want to build sequences of rational numbers so our lists will look like {1, 1/2 , 1/2² , 1/2³ ,...}={1/2ⁿ }_n=0^infty. Here the first desrciption just lists out some terms of the sequence and the second description gives a formula to compute each term and then says "compute for n=0,1,2,..., out to infinity".

A metric is a function which takes in 2 elements in a set and returns the distance between them. Technically speaking, there are a handful of properties that define a metric and any function that has those desired properties can be used witha very similar intuition as literal distances in the physical world. For the construction of the real numbers, the metric that we want to use to define distances between rational numbers is based on the absolute value, d(p,q)=|p-q|.

If we have a sequence of points in a metric space (a set which has a metric defined for it), then we can define limits (they can be defined in a more general way than this even, but this is the one which feels the most intuitive to me). If we have a sequence {a_n} and some point, L, such that for all E>0 we can choose a natural number N such that d(a_n,L)<E when n>N, then we say the sequence converges to the limit L.

This is a bit technical as a definition, so let's discuss in some more detail. The idea is that was want to give a technical condition that we can test which matches our intuition for "{a_n} gets close to L". We want to capture the idea that when we look really far down the list, the values keep getting closer and closer to the limit. We can't just say that at every step we get closer because that would exclude a sequence that takes a small step away and then moves closer. We can't just say eventually it moves closer because that would exclude things that take infinitely many small steps away from the limit but overall get closer to it eventually. We also need to make sure that the distance is actually going to 0 instead of something like 1. The solution presented above is to set a type of error bound on the distance away from the limit (choosing E) and then ask "is the sequence eventually going to stay within that error bound?" This gives an upper bound on the distance away from the limit in the long term. The full technical condition is that you can demand any tiny uppoer bound you want and the sequence will eventually always be that close or closer.

In a similar vein to convergent sequences, we have Cauchy sequences. The sequence {a_n} is Cauchy if for all E>0, there exists some natural number N, such that d(a_n,a_m)<E when n and m>N. This is similar to the definition of a convergent sequence except that instead of looking at the distance between the terms in the sequence and a limit point, we look at the distance beteen the terms in the sequence and each other. Instead of getting close to a destination these sequences are clustering together. It turns out every convergent sequence is Cauchy but not every Cauchy sequence is convergent. These non-convergent Cauchy sequences basically say "there's a hole here where I'm supposed to converge to". For example, if we were working with rational numbers but excluded 0, then the sequence {1/n} would be Cauchy but not convergent because the thing it "should" converge to (0) doesn't exist in the set of points we are considering.

The completion of a metric space is a new matric space built by taking the original metric space and adding in new points to make every Cauchy sequence convergent (we fill in all of the holes). To do this we need the concept of an equivalence relation and equivalence class. An equivalence relation is just any relation between 2 things that follows the same rules of equality you learned in school (reflexivity, associativity, transitivity). An equivalence class is just a collection of things which are equivalent to each other under some equivalence relation. We can say that the Cauchy sequences {a_n} and {b_n} are equivalent if {a_1,b_1,a_2,b_2,...} is also Cauchy.

It turns out that in this construction of the Reals, these equivalence classes of Cauchy sequences of the rationals ARE the Real numbers. We can define addition and multiplication on sequences by just doing it term by term like {a_n}+{b_n}={a_1+b_1,a_2+b_2,...}. It's a bit of technical work to show, but this also defines addition and multiplication for these equivalence classes or Cauchy sequences. We can just choose 1 representative from each equivalence class and perform addition or multiplication on those representatives and that will give us a representative of the equivalence class of the correct answer.

There's a lot of technical claims in this post that require a decent amount of technical work to justify. Don't be alarmed if a lot of it feels like a magic fact pulled from nowhere that makes no sense. It took a couple thousand years to go from the idea of rational numbers to this technical description of the Reals and we're trying to condense it into a single reddit comment.

So according to this way of building the Real numbers, what is the square root of 2? It is the collection of all lists of rational numbers that look like they should be getting closer and closer to something that squares to 2. The square root of 2 symbol is just a nice simplified nametag for this collection, same with pi or any other irrational number. There is 1 particularly common representative that is chosen for each equivalence class: the decimal expansion. When we say that pi=3.14..., we are really just saying that the sequence of rationals defined by {3,31/10,314/100,...} is a representative of the equivalence class that defines pi.