The real numbers are much stranger than they first appear, and a lot of early pedagogy is designed so that you don't look at this too hard.
After all, almost all reals are transcendental (but interestingly we only have proved a small handful of them).
Most numbers people have used in "regular life" are algebraic, obviously, but they are a countable subset so occur almost never, in the set.
It's a standard example trotted out in elementary real analysis courses along with Cantor sets, etc. If you have a good lecturer, they might make you try and come up with an example like it as an exercise before you've seen it...
A good part of becoming competent at analysis is building up a grab bag of examples like this you can throw at new situations.
It is similarly used as an example, for a function that is continuous but not absolutely continuous.
x sin(1/x) is continuous and non-differentiable at 0, but I wouldn't say it has a corner there.
A sharp corner to me means a point where the left and right derivatives both exist but are different.
It's even worse, almost all reals aren't even computable (let alone transcendental)
This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers
Or equivalently, from the natural numbers to the real numbers.
The dart will bounce off the metal framey bit and stick into the forehead of local barfly, 'Very Angry Ron'. Then when you graph the result, violent discontinuities will appear in the function, across a wide range of scales.
These already have their own special set in category theory and are sometimes referred to, within the field, as; 'Assorted and Unreasonable Associates of Very Angry Ron'. That is, the other side of a reasonably big field, ideally with some cattle in the way.
You can learn to prove things about them rigorously, and memorize a lot of properties, but very few people will ever develop an accurate intuition for them.
Is that the same as making sense? Well to paraphrase a physicist, you don't so much understand it as get used to it.
One of the fun things about teaching this material is that usually students have been through years of calculus prior to it, and you get to watch the moment when they realize all of this stuff has been "hiding in plain sight".
Does the followong seem right to you?
Intuition often comes from relating a new thing to something known. So maths, as an abstraction of reality, initially has many sources of intuition.
Later maths never has exactly the same patterns as earlier maths (it's already abstracted; so same patterns would be the same thing, though it cam build-on). Eventually, it doesn't relate to anything known, and you have to create that familiarity from scratch.
That's hard... but if some stuff became known in the first place, why not this too? (One counter is that the other stuff was instinctively known, e.g. 3D space, or at least our minds are pre-shaped to know it, e.g. language).
Assuming math is open-ended, there'll always be new stuff that doesn't relate.
Edit: A great way to unlearn bad intuition is through the study of counterexamples. A good starting point might be chapter 1 of Counterexamples in Analysis [1].
[1]https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...
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!
However I personally found it very helpful in real analysis to be able to take any question about the reals back to how it relates to this construction. This greatly helped my intuition.
Dedekind cuts have a special case at all of the rationals. Which is weird. And the whole construct the completion using equivalence classes of sequences construction is one you'll encounter a bunch of times in topology. So it is worth learning it properly.
Plus if you're into that kind of thing, you can also construct the p-adics this way. :-)