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A negative real number is a real number $x<0$, lying to the left of zero on the number line. Equivalently, $x=-a$ for a positive real number $a$. Examples include $-3$, $-1/2$ and $-\sqrt{2}$; zero is excluded.

The sum of two negative real numbers is negative, while their product is positive. Multiplication by a negative real number reverses an inequality: if $x<y$ and $c<0$, then $cx>cy$. These are consequences of compatibility between the total order and the arithmetic of an ordered field, rather than independent rules about written minus signs.

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 $\mathbb{R}$.