real number

Real numbers

$$\mathbb{R}:=\{g+r:g\in \mathbb{Z},r=\underset{n\to \infty }{\lim }\frac{a_n}{b_n},a_n,b_n\in \mathbb{N}\}$$

Here, $g$ is the integer (whole number) part and $r$ is the fractional part, $n$ is the number of digits after the decimal point and it can be finite or infinite.
The rational numbers $\mathbb{Q}\subset \mathbb{R}$, while $\mathbb{R}\setminus \mathbb{Q}$ are the irrational numbers.
The real numbers are a field under addition and multiplication.

Completeness axiom of the real numbers $\mathbb{R}$

Every non-empty subset $A\subset \mathbb{R}$ that is bounded from above has a supremum in $\mathbb{R}$, i.e. $\exists T\in \mathbb{R}$ such that $T=\sup A$ (similarily for infimum).

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Circular transclusion detected: general/bounded

Circular transclusion detected: general/bounded