The reals click for some people, and fail to click for others. What can I say?

For me it was once I really understood what a Cauchy sequence of rational numbers is, and how that is a real number. Let's see if I can explain that.

In math we have the following kind of construction in lots of places. We take some simple system, we construct some way of representing things from a more complex relationship. And then define some sort of equivalence. The new

This is a mouthful but you've seen it. Take the construction of the rational numbers from the integers. A rational number is just a pair of integers (n, m) with the second one not zero. It represents n/m. However there is an equivalence, 1/2 is the same number is 2/4. The equivalence is that (n, m) = (n', m') if and only if n * m' = n' * m.

You finish by defining operations as (n, m) + (n', m') = (n * m' + n' * m, m * m') and (n, m) * (n', m') = (n * n', m * m'). This looks like a mouthful, but it is exactly the rule that you're used to.

So we've seen this kind of construction before. (You do the same when constructing the integers from the natural numbers.)

So constructing the reals from the rationals is done as follows. Intuitively a real is a sequence of rationals that is converging. And two sequences of rationals are equivalent if they should converge to the same thing.

Where "converging" means that you have a sequence of rationals (x_1, x_2, x_3, ...) such that if we pick n, m "big enough", then x_n - x_m will be as close to 0 as we want. Or in usual Calculus notation, for every epsilon > 0 there is an N such that for every n and m both bigger than N, abs(x_n - x_m) < epsilon.

And (x_1, x_2, x_3, ...) should "converge to the same thing" as (y_1, y_2, y_3, ...) if (x_1 - y_1, x_2 - y_2, x_3 - y_3, ...) converges to 0. Or in usual Calculus notation, for every epsilon > 0 there is an N such that for every n bigger than N, abs(x_n - y_n) < epsilon.

You define operations pairwise. So (x_1, x_2, x_3, ...) + (y_1, y_2, y_3, ...) = (x_1 + y_1, x_2 + y_2, x_3 + y_3, ...).

Here is a sanity check. If you have a decimal representation, that gives us a sequence of rationals converging to that real, (3, 3.1, 3.14, 3.141, ....). Switch from base 10 to base 2, and you get a different sequence, but it is the same real. And the old chestnut, 1 = 0.99999... repeating is easy to verify.

And now work your way through the following axioms:

The algebraic axioms are easy.

1. There is a well-defined binary operation + such that x+y is always defined.

2. + is commutative, so x+y = y+x.

3. + is associative, so (x+y)+z = x+(y+z).

4. There is an additive identity 0 such that x+0 = x for all x.

5. Every x has an additive inverse called -x such that x + (-x) = 0

6. There is another binary operation called .

7. is commutative, x * y = y * x

8. * is associative, (x * y) * z = x * (y * z).

9. The distributive property holds. x * (y + z) = (x * y) + (x * z).

10. There is a multiplicative identity 1 different from 0.

11. Every x other than 0 has a multiplicative inverse 1/x such that x * (1/x) = 1.

And now the order axiom.

12. Every number is exactly one of positive, negative or 0. Or, more formally, there is a set P closed under addition and multiplication such that for all x, exactly one of three things is true: x is 0, x is in P, or -x is in P.

And then the tricky one. Completeness.

13. If X is a non-empty set of reals with an upper bound, it has a least upper bound. (For example the set of x such that x^2 - 2 < 0 is non-empty, it has an upper bound, and therefore it has a least upper bound. Which happens to be sqrt(2).)

To see that the order axiom holds, let x_1 be a rational number below the value of something in X, and y_1 be a rational number above an upper bound. And now we construct two sequences as follows.

At each step if (x_n + y_n) / 2 is an upper bound, then x_(n+1) = x_n and y_(n+1) = (x_n + y_n) / 2. Else x_(n+1) = (x_n + y_n) / 2 and y_(n+1) = y_n.

We can prove three things.

1. (y_1, y_2, y_3, ...) converges to an upper bound.

2. No upper bound can be below what (x_1, x_2, x_3, ...) converges to.

3. Both sequences are equivalent, they represent the same real.

The conclusion is that that real has to be the least upper bound.

If you can really get that, then congratulations! You understand the reals!