Number Line
A number line is a straight line on which numbers are represented as points, with positive numbers to the right of zero and negative numbers to the left.
Definition
A number line is a line where every point represents a number: numbers increase as you move right and decrease as you move left, with zero in the middle. More formally, it is a visual representation of the real number system with a one-to-one correspondence between points and real numbers, where the distance between two points $a$ and $b$ is $|a - b|$, modeling addition (right), subtraction (left), order ($<, >$), and absolute value (distance from the origin). Topologically, the number line is the metric space $(\mathbb{R}, d)$ with $d(x,y) = |x-y|$, the unique (up to isometry) complete, separable, connected $1$-manifold without boundary, underlying the definitions of limits, continuity, and the derivative.
Example
On a number line, $-3, -2, -1, 0, 1, 2, 3$ are evenly spaced, with $2$ to the right of $0$ and $-2$ to the left. Solving $|-4 - x| = 3$ geometrically means finding all points $x$ whose distance from $-4$ is $3$: they are $x = -7$ and $x = -1$. Cantor's construction removes the middle third of $[0, 1]$, then the middle thirds of the remaining intervals, repeatedly; the resulting Cantor set is a subset of the number line with measure $0$ but uncountably many points.
Key Insight
A number line turns addition and subtraction into movement: adding means moving right, subtracting means moving left. The same idea in $2$D gives the coordinate plane, in $3$D gives $3$-space, and in any dimension gives $\mathbb{R}^n$, with all of calculus and analysis built on this model. The completeness of the real line, every Cauchy sequence converges, is what makes the intermediate value theorem, mean value theorem, and fundamental theorem of calculus provable.