Real Numbers and the Number Line

The Continuous Line

The real numbers $\mathbb{R}$ complete the rationals by filling every gap, forming a continuous line with no holes. The property that distinguishes $\mathbb{R}$ from $\mathbb{Q}$ is completeness: every set of reals that is bounded above has a least upper bound (supremum) that is itself a real number. The rationals lack this — the set of rationals with square less than 2 has no rational least upper bound (it "wants" to be $\sqrt{2}$, which is missing). Completeness is what makes calculus possible; limits always land somewhere real.

Everyday tools describe positions and distances on this line:

• Intervals name connected stretches: $[a, b]$ is closed (endpoints included), $(a, b)$ is open (endpoints excluded), and half-open forms mix.
• Absolute value $|x|$ is the distance from $x$ to $0$: $|x| = x$ if $x \ge 0$, else $-x$. More usefully, $|x - y|$ is the distance between $x$ and $y$.
• A set is bounded above if some number exceeds all its elements (an upper bound); bounded below likewise. Bounded both ways means simply bounded.

The absolute value obeys the triangle inequality $|x + y| \le |x| + |y|$ — the length of a combined step never exceeds the sum of the individual step lengths. This small inequality is one of the most-used tools in all of analysis, underlying estimates and the very definition of limits.

Between the density of the rationals and the completeness of the reals lies the full picture: the rationals are everywhere but full of invisible gaps, and the reals seal those gaps to make the continuum on which calculus is built.

Common pitfall: confusing an upper bound with the least upper bound (supremum), or treating open and closed intervals as interchangeable. Any number above a set is an upper bound; the supremum is the smallest such number. And $[a, b]$ includes its endpoints while $(a, b)$ excludes them — a distinction that matters for whether a maximum is attained. Completeness guarantees the supremum exists in $\mathbb{R}$, but it need not be a member of the set.