Foundations — Absolute value - modulus — definition, number line interpretation

This page builds the toolbox before the toolbox is used. The parent note throws around symbols like $x$, $|x|$, $\mathbb{R}$, $\ge$, $\{\begin{array}{}\end{array}$, and words like "non-negative" and "displacement". If any of those are fuzzy, you cannot follow the real topic. So we earn each one, from the ground up, in an order where every step leans on the previous.

0. The picture everything sits on: the number line

Before any symbol, there is a road. A straight, endless road with a marked centre.

Why the topic needs it: absolute value is defined as a distance on this line. Without the line, "distance from zero" is just words. This is the ground floor — see Number line and ordering for the full construction.

• Right of zero → positive numbers ($1,2,3,\dots$).
• Left of zero → negative numbers ($-1,-2,-3,\dots$).
• The middle mark → zero itself, belonging to neither side.

1. The symbols for numbers: signs, and what "$-$" really means

A number has two pieces of information hidden inside it: how far it is from zero, and which side it is on. The little sign in front carries the "which side" part.

The minus sign in "$-5$" is a direction flag, not part of the size. The size of $-5$ is $5$; the minus only says "on the left". Keep this split in your head — it is the whole game.

2. "Non-negative", and the symbols $\ge$, $\le$, $<$, $>$

The parent note keeps saying "non-negative". We need the exact symbols for comparing numbers, because absolute value's whole promise is that its output is never on the wrong side of zero.

Why the topic needs this: the definition of $|x|$ splits into the case $x\ge 0$ and the case $x<0$. The whole point of choosing "$\ge$" (and not "$>$") is that zero must land in the first case, so that $|0|=0$ works cleanly. If we'd written "$>$", zero would fall through the cracks. See Inequalities for how these symbols behave.

3. The letter $x$: a placeholder for "any number"

Picture: $x$ is an empty seat on the number line's road. It could be filled by $7$, by $-2.5$, by $0$ — the rule you write must hold whoever sits there.

Why needed: the definition $|x|=\dots$ is a rule for all numbers at once. Without a placeholder we'd have to write infinitely many separate statements.

4. The symbol $\mathbb{R}$: "all the real numbers"

Why needed: statements like "$|x|\ge 0$ for all $x\in \mathbb{R}$" are claims about every point on the entire line. $\mathbb{R}$ is the shorthand for "the whole line".

5. The star of the show: the bars $|\cdot |$ and distance

Now we can define the actual symbol of this chapter. First, distance on the line.

The bars $|\cdot |$ are the "read the size, drop the direction" machine. Look at the figure: the arrow's length is what $|x|$ reports; the arrow's direction is what it throws away.

6. The curly brace $\{\begin{array}{}\end{array}$: a "which rule applies?" switch

That big curly bracket in the definition is a piecewise rule — it hands you different instructions depending on which region your input is in.

Look at the figure — it is a fork in the road:

• If your number is at or right of zero ($x\ge 0$): take the top branch, output $x$ unchanged.
• If your number is left of zero ($x<0$): take the bottom branch, output $-x$ (the flipped, positive version).

Why the topic needs it: absolute value behaves differently on the two sides of zero, so it cannot be one single formula. The brace is the honest way to say "two rules, glued at zero". See Piecewise functions for the general idea, and Solving absolute value equations for how this fork splits every equation into cases.

7. Subtraction as displacement: why $|b-a|$ is a distance

The parent note leaps to "distance from $a$ to $b$ is $|b-a|$". We must earn the subtraction.

Now wrap it in the bars: $|b-a|$ keeps the size of that move and drops the direction — giving pure distance. And because flipping the move ($a-b$ instead of $b-a$) only changes the direction flag, the bars erase the difference: $|b-a|=|a-b|.$ This is why distance from $a$ to $b$ equals distance from $b$ to $a$. See Distance and metric spaces and Triangle inequality for where this leads.

Prerequisite map

Equipment checklist

Cover the right side and test yourself. Each ::: is one skill you need before the main topic.

Connections

• Number line and ordering — the road every symbol here sits on
• Inequalities — the meaning of $\ge ,\le ,<,>$ that split the cases
• Piecewise functions — the curly-brace switch generalised
• Distance and metric spaces — $|x-y|$ becomes the distance rule
• Triangle inequality — the deepest property built on these foundations
• Solving absolute value equations — where the two-branch fork earns its keep
• Complex numbers — modulus — the same "size, drop direction" idea one dimension up