Real number

I am not yet sure how to define real numbers. Many irrational numbers (e.g. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} and $\pi$) are in fact valid concepts, but the standard definition of the reals involves infinite nonsense.

I don't need to give them a set-theoretic definition. Real numbers are basically just the concept of numbers.

Examples

Any fraction.

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

Pi

e

The number $0.x_{1}x_{2}x_{3}x_{4}\cdots$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i : \{0,1\}} is its Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} th digit in base 2, and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i = 1} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} is prime, and 0 otherwise.

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