Real number ($\mathbb{R}$)

A real number is an element of the complete ordered field $\mathbb{R}$. This includes integers, rational numbers such as $1/3$, and irrational numbers such as $\sqrt{2}$. The number line represents each real number by a point, and the total order distinguishes which of two different points is larger.

The field operations permit addition, subtraction, multiplication and division by nonzero real numbers. Completeness adds the least-upper-bound property: every nonempty set of real numbers that is bounded above has a supremum in $\mathbb{R}$. For example, the positive real numbers whose squares are less than two have supremum $\sqrt{2}$. That supremum is not a rational number, illustrating why the rational numbers alone leave a gap.

The density of the rational numbers means every real number can be approximated arbitrarily closely by rational numbers. Constructions through Cauchy sequences or Dedekind cuts make those approximations into a precise description of $\mathbb{R}$.

A real number is any value that can represent a quantity along a continuous line. This includes all the rational numbers (which can be expressed as the quotient of two integers, such as 1/2 or -3) and all the irrational numbers (which cannot be expressed as a simple fraction, such as the square root of 2 or pi). Real numbers can be classified into several categories: - **Rational Numbers**: Numbers that can be expressed as a fraction (e.g.