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Comment by ysofunny | Hacker News Reader

> Surely they mean algebraic numbers not real numbers.

that's an extremely subtle distinction! I don't really understand this distinction to my own satisfaction.

in my mind the concept I need to understand better is the generalized notion of 'decimal point'

this presents me 2 avenues of further investigation:

'decimal' which just means ten which is already known as the general concept of the base of a positional numeral system (so this links to exponents and modulo arithmetic and the modern algebraic framework)

and the other avenue is the 'point'. so here's my question: up to which extent should I attempt to realize the notion of "fixed point" (like in Tarski's theorem) and the specific application of a "fractional"-point as an instance (where 'decimal' is the specific applied case n=10=2*5) of a "fixpoint" used as part of a positional numeral value determination system?

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At the risk of telling you something you don’t already know: algebraic numbers are an extension of integers¹ which adds in those numbers which are zeros of finite polynomials with integer coefficients.² So, to pick a trivial example, ±√2 are algebraic number because they are solutions to x²-2=0.

But there are many real numbers which are not algebraic. π and e are the best known non-algebraic numbers (we call these numbers transcendental. In common parlance, unless otherwise specified, the term transcendental number is generally used to refer to the set ℝ\ where refers to the set of algebraic numbers in this instance³). In fact, statistically speaking, the probability that any given real is algebraic is 0.⁴

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1. It’s worth noting that while it’s easy to assume that algebraic numbers are a subset of the reals, the fact that it includes complex numbers (i is algebraic since it’s a solution to x²+1=0) means it’s not. n.b., just as not all reals are not algebraic, not all complex numbers are algebraic.

2. Rational coefficients of a polynomial can be trivially eliminated by simply multiplying all terms of the polynomial by the lcm of the denominators of any non-integer polynomial, so the set of zeros of polynomials with integer coefficients and the set of zeros of polynomials with rational coefficients will be identical.

3. I’m aware is also often used to label the adele ring, but I’m happy to borrow it for the purposes of a Hacker News comment.

4. The proof of this statement is left as an exercise to the reader.

Weirdly, almost all real numbers are non algebraic, but only a handful have ever been named.

FYI HN ate the unicode glyphs you used for adele and the superscript 3.

Also, yes I could have said "real algebraic" to be more clear. But no one says "real rational" or "real integer" when that's what they mean :-)

And in the 2D plane, you can usually cast a subspace of 2D reals as a subspace of 1-cooridinate complex numbers.

A decimal "point" is called that because it's written as a dot: .

"Fixed point" arithmetic has nothing to do with "fixed point theorems", it's a name collision.

Base ten has nothing to do with the distinction between real and algebraic numbers.

If you Google "algebraic vs transcendental number" you'll find many good resources. Pi is an example of a transcendental number.

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