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 fini

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 ⸻ 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.