Real number

I am not yet sure how to define real numbers. Many irrational numbers (e.g. ${\sqrt {2}}$ and $\pi$) are in fact valid concepts, but the standard definition of the reals involves infinite nonsense.

The concept of real numbers is a concept: it identifies things out in reality. In particular, it does not need to be "constructed" via some set-theoretic method like Dedekind cuts, nor do such constructions even make sense. That any real number can be approximated by fractions is obvious: a fraction is the outcome of directly measuring any quantity with a standard ruler.

Examples

Any fraction.

Any algebraic number, like ${\sqrt {2}}$.

Pi

e

The number $0.x_{1}x_{2}x_{3}x_{4}\cdots$, where $x_{i}:\{0,1\}$ is its $i$th digit in base 2, and where $x_{i}=1$ if $i$ is prime, and 0 otherwise.

The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational)