Lecture · 12 min · basic

The number line

Integers as positions, absolute value as distance

Euler, Elements of Algebra · §§1–12

A number is not a pile of marks. It is a place. Once you treat integers as positions on a line, subtraction, negatives, and absolute value stop being three separate tricks and become one geometric fact.

On the board today

—Place integers on a directed line with a chosen origin and unit.
—Read $a < b$ as “ $a$ lies to the left of $b$ ."
—Treat $∣ x ∣$ as distance from the origin, not as “drop the minus.”

Distance on the line

∣ a - b ∣ = distance between a and b

The formula does not care which point is larger.

Fix an origin $0$ and a unit length. Every integer $n$ is the point you reach by walking $n$ units to the right, or $- n$ units to the left. Order is direction: $a < b$ means the walk from $a$ to $b$ points the same way as the positive unit. Absolute value forgets direction and keeps length: $∣ x ∣$ is how far $x$ sits from $0$ .

Triangle inequality

For all real $x, y$ , $∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ .

∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣

A detour through $0$ cannot be shorter than the direct path. Walking $x$ then $y$ covers at most the two separate lengths.

Worked example · Two points

Find the distance between $- 7$ and $4$ , and decide which is larger.

1.Larger means farther right: $4$ is to the right of $- 7$ , so $- 7 < 4$ .
2.Distance is absolute difference.
$∣4 - (- 7) ∣ = ∣11∣ = 11$

The points are $11$ units apart, and $4$ is larger.

Pause the lecture

Which numbers $x$ satisfy $∣ x - 3∣ = 5$ ? Think geometrically before you algebraically.

Common mistake. Writing $∣ - x ∣ = - x$ is false whenever $x$ is positive. $∣ - x ∣ = ∣ x ∣$ always. Absolute value never returns a negative number.

Practice this lecture