Real Numbers and the Number Line

The Continuous Line

The real numbers $\mathbb{R}$ fill every gap of the rationals, forming a continuous line. Their defining trait is completeness: every set bounded above has a least upper bound (supremum) in $\mathbb{R}$ . The rationals fail this, as the set of rationals with square under 2 misses $\sqrt{2}$ .

Positions and distances are measured via tools:

Tool	Notation / Form	Meaning
Closed Interval	$[a, b]$	Endpoints included
Open Interval	$(a, b)$	Endpoints excluded
Absolute Value	$\|x\|$	Distance to 0 ( $x$ if $\ge 0$ , else $-x$ )
Distance	$\|x - y\|$	Gap between $x$ and $y$

A rational search for the least upper bound of $\\{x \\in \\mathbb{Q}:

Inequalities and Bounds

A set is bounded above if a number exceeds all its elements (an upper bound). Completeness ensures the supremum exists in $\mathbb{R}$ , though it need not belong to the set itself.

The triangle inequality &|x + y| \le |x| + |y|& states the combined step length never exceeds the sum of parts. It is vital for analysis.

Common pitfall: Confusing an upper bound with the least upper bound. Any number above the set works, but the supremum is the single smallest one.

The Continuous Line

Inequalities and Bounds

Practise this lesson

Arithmetic and Number Theory