🏵 Absolute Value

⚜️ The absolute value represents the distance of a number from zero on the number line and is always positive or zero.

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Measuring Distance in One Dimension

It is natural to ask: what is the distance between two numbers? How far is a number from zero?

The absolute value of a real number $x$, denoted $|x|$, is defined as:

$$ |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases} $$

It gives the non-negative value of $x$, regardless of sign. Intuitively, $|x|$ is the distance from $x$ to the origin $0$ on the number line.

Examples:

• $|5| = 5$
• $|−3| = 3$
• $|0| = 0$

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Distance Between Two Numbers

The distance between two real numbers $a$ and $b$ is:

$|a - b|$

This expression is always non-negative and symmetric:

$$ |a - b| = |b - a| $$

Example: The distance between 2 and 7 is $|2 - 7| = |-5| = 5$.

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Properties of Absolute Value

These properties follow directly from the definition and are useful throughout algebra and analysis:

1. Positivity:

   • $|x| \geq 0, \quad \text{and } |x| = 0 \Leftrightarrow x = 0$

2. Multiplicativity:

   • $|ab| = |a| \cdot |b|$

3. Division (if $b \neq 0$):

   • $\left| \frac{a}{b} \right| = \frac{|a|}{|b|}$

4. Symmetry:

   • $|-x| = |x|$
   • $|a + b| = |b + a|$
   • $|a - b| = |b - a|$

5. Triangle Inequality:

   • $|a + b| \leq |a| + |b|$ Equality holds if and only if $a$ and $b$ have the same sign or one is a multiple of the other.

6. Containment:

   • $-|a| \leq a \leq |a|$
     This may be useful dealing with expressions.
   • $|a|-|b| \leq |a + b| \leq |a| + |b|$
   • $|a|-|b| \leq |a - b| \leq |a| + |b|$

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Applications

1. Defining Precision and Approximation

When we say $x$ is approximately $a$, within a margin $\varepsilon > 0$, we write:

$|x - a| < \varepsilon$

This describes all $x$ within a radius $\varepsilon$ of $a$. It defines an open interval:

$$ |x - a| < \varepsilon \quad \Leftrightarrow \quad x \in (a - \varepsilon, a + \varepsilon) $$

This becomes the foundation of limits, continuity, and neighborhoods in topology.

2. Solving Inequalities

Inequalities with absolute value can be converted into compound inequalities:

• $|x - 3| < 2$ means $1 < x < 5$
• $|x + 1| > 4$ means $x < -5$ or $x > 3$

These are essential in analysis and calculus.

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Graphical Interpretation

The graph of $y = |x|$ is a V-shaped curve with its vertex at the origin. It’s linear for positive and negative values, reflecting at $x = 0$:

$y = \begin{cases} x & x \ge 0 \ -x & x < 0 \end{cases}$

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Summary

Absolute value is a simple but powerful tool:

• It allows us to measure distances in $\mathbb{R}$
• It forms the foundation for defining limits and continuity
• It lets us handle precision, approximations, and inequalities systematically

From here, we can move on to intervals, topology, and limits—all built on this intuitive notion of distance.