Absolute value

1. Geometric definition of absolute value and the non-negativity property

Geometric Definition of Absolute Value

The absolute value of a real number $x$, denoted $|x|$, is the distance from $x$ to zero on the number line. Distance is always measured as a non-negative quantity, so $|x| \geq 0$ for all real numbers $x$.

Intuitively, absolute value strips away the sign of a number, leaving only its magnitude.

Core Rules:

• $|x| = x$ if $x \geq 0$
• $|x| = -x$ if $x < 0$
• $|x| \geq 0$ for all $x \in \mathbb{R}$ (non-negativity)
• $|x| = 0$ if and only if $x = 0$

This non-negativity property ensures that absolute value always produces a result on the non-negative portion of the number line.

Example: $|5| = 5$ because 5 is 5 units from zero; $|-5| = 5$ because -5 is also 5 units from zero.

2. Evaluating absolute value of negative numbers and the distance between two numbers

Distance Between Two Numbers

For any negative number $x < 0$, the absolute value $|x|$ equals $-x$, which is positive. The expression $|a - b|$ represents the distance between $a$ and $b$ on the number line, regardless of their order.

This distance interpretation is symmetric: $|a - b| = |b - a|$ because distance does not depend on direction.

Core Rules:

• If $x < 0$, then $|x| = -x > 0$
• $|a - b|$ measures the gap between $a$ and $b$
• $|a - b| = |b - a|$ (symmetry)
• Distance is always non-negative

This formulation unifies the concept of separation between any two points.

Example: $|-7| = 7$; the distance between 3 and 8 is $|3 - 8| = |-5| = 5$, which equals $|8 - 3| = 5$.

3. Understanding the piecewise nature of the absolute value function

Piecewise Nature of Absolute Value

The absolute value function $f(x) = |x|$ is defined piecewise because its formula changes depending on whether the input is non-negative or negative. This creates a V-shaped graph with a corner at the origin.

The function is continuous everywhere but not differentiable at $x = 0$ due to the sharp turn.

Core Rules:

• $|x| = x$ when $x \geq 0$ (right branch, slope = 1)
• $|x| = -x$ when $x < 0$ (left branch, slope = -1)
• The graph has a vertex at $(0, 0)$
• Not differentiable at $x = 0$ (sharp corner)

This piecewise structure is fundamental to solving equations and inequalities involving absolute value.

Example: For $x = -3$, use the second piece: $|-3| = -(-3) = 3$. For $x = 4$, use the first piece: $|4| = 4$.

4. Solving basic absolute value equations and simple inequalities

Solving Absolute Value Equations and Inequalities

An equation $|x| = k$ where $k > 0$ has exactly two solutions: $x = k$ or $x = -k$, because both values are distance $k$ from zero. If $k < 0$, there is no solution since absolute value cannot be negative.

For inequalities, $|x| < k$ means $-k < x < k$ (values within distance $k$ of zero), while $|x| > k$ means $x < -k$ or $x > k$ (values beyond distance $k$).

Core Rules:

• $|x| = k$ (where $k > 0$) gives $x = k$ or $x = -k$
• $|x| < k$ gives $-k < x < k$
• $|x| > k$ gives $x < -k$ or $x > k$
• If $k \leq 0$, then $|x| = k$ has no solution

Example: $|x| = 7$ yields $x = 7$ or $x = -7$. The inequality $|x| < 3$ gives $-3 < x < 3$.

5. Applications: Calculating total magnitude of forces or net change volatility in economics

Applications of Absolute Value

Absolute value quantifies magnitude without regard to direction, making it essential in physics for computing total force magnitudes and in economics for measuring volatility. In finance, $|\Delta P|$ captures price change size regardless of whether the market rose or fell.

This abstraction allows analysts to assess variability, risk, or total effect independent of sign.

Core Rules:

• Use $|F|$ to find force magnitude (ignoring direction)
• Sum $\sum |\Delta x_i|$ to compute total volatility or cumulative change
• Absolute deviations $|x - \mu|$ measure spread from a mean
• Always non-negative, suitable for distance-based metrics

Absolute value transforms signed quantities into comparable magnitudes.

Example: If stock prices change by -5 dollars, +3 dollars, -2 dollars over three days, total volatility is $|-5| + |3| + |-2| = 5 + 3 + 2 = 10$ dollars.