Unreal numbers

I was thinking about the ability of representing different kinds of numbers. Imagine that we had a certain CPU that could process algorithms, and the final output of the algorithm is a number. The CPU has a certain number of operations (At least https://en.wikipedia.org/wiki/One-instruction_set_computer). Then, if the algorithm can be described with an integer (since the algorithm can be described with binary), then... can integers describe Real numbers?

Algebraic numbers can be very handy.

I was making myself toy, tapered kaleidoscopes using one piece cardboard plans. One needed to ensure that the dihedral angles between the mirrors be precise.

This is not easy to do with the usual middle-school geometry-box rulers and protractors. Graduations not fine enough, precise enough, extending long straight lines using a small ruler not straight enough ...

However the dimensions being all algebraic numbers one could use entirely straight edge and compass constructions. Had much better luck this way, with a pointy enough pencil.

A proper drafting board and a T-square or a drafter would have made things easier for parallel and perpendicular translations. But one can do those with compass too.

> Of course, we don’t teach about computable numbers in school. Instead, the most common “upgrade” from ℚ are reals:

While "computable" numbers are a recent concept, already for a few centuries, since the early 18th century, mathematics has taught about another set of numbers intermediate between rational numbers and "real" numbers: the algebraic numbers, which are a subset of the computable numbers.

Like the "real" numbers, the "complex" numbers have also been partitioned since that time into "complex" integer numbers, "complex" rational numbers, "complex" algebraic numbers, "complex" transcendental numbers.

Everything that is now discussed in terms of "computable" and "non-computable" numbers, was previously discussed in terms of algebraic numbers and transcendental numbers.

While "computable" numbers is a more general concept that more precisely defines the limit between what is countable and what is not, the practical importance of this concept is reduced, because few of the computable numbers that are not algebraic are interesting, the main exceptions being the numbers that are algebraic expressions containing "2*Pi" and/or "ln 2".

I think this article should’ve used the Cauchy sequence method to construct the reals instead of Dedekind cuts. It would’ve built on the earlier mention of equivalence classes.

> But what would be an example of an uncomputable number? That’s a good question. Most obviously, we could be talking about numbers that encode the solution to the halting problem. It would lead to a paradox to have a computer program that allows us to decide, in the general case, whether a given computer program halts. So, if a procedure to approximate a particular real requires solving the halting problem, we can’t have that.

This doesn’t make sense to me. Given that there’s no generic way to compute halting, how would we make the leap to saying that there’s a specific number which represents the solution to that problem?

Previously: https://news.ycombinator.com/item?id=45424648

A number that can predict the future?

This is the first time I've seen this way to show that Q does not have a higher cardinality than N, is it a common method?

I don't remember exactly how I learned about it in high school, infinity cardinalities have rarely come up since then, but it was some other method or at least another form of presentation, i.e. symbols and prose.