Real number

Informally, we say that real numbers $\mathbb{R}$ essentially extend the set of rational numbers $\mathbb{Q}$ by defining what $\mathbb{Q}$ does not, but excluding imaginary numbers.

The rigorous construction of the real numbers is premised from the idea that we can approximate a real number with a rational number, i.e., we essentially extend a decimal expansion.

The analysis of the real number line is real analysis.

Real numbers have a few properties that rational numbers do not:

• Completeness