# Absolute Value

## What does it mean?

### Definitions:

The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.

For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line. Furthermore, the absolute value of the difference of two real numbers is the distance between them.

The absolute value has the following four fundamental properties:

Non-negativity $$|a| ≥ 0$$ Positive-definiteness $$|a| = 0 ⇔ a = 0$$ Multiplicativeness $$|ab| = |a||b|$$ Subadditivity $$|a + b| ≤ |a| + |b|$$

Other important properties of the absolute value include:

Idempotence (the absolute value of the absolute value is the absolute value) $$||a|| = |a|$$ Symmetry $$|-a| = |a|$$ Identity of indiscernibles (equivalent to positive-definiteness) $$|a - b| = 0 ⇔ a = b$$ Triangle inequality (equivalent to subadditivity) $$|a - b| ≤ |a - c| + |c - b|$$ Preservation of division (equivalent to multiplicativeness) $$|a / b| = |a| / |b| \space\space if \space\space b ≠ 0$$ (equivalent to subadditivity) $$|a - b| ≥ ||a| - |b||$$

Two other useful properties concerning inequalities are: $$|a| ≤ b ⇔ -b ≤ a ≤ b$$ $$|a| ≥ b ⇔ a ≤ -b \space or \space b ≤ a$$

These relations may be used to solve inequalities involving absolute values. For example: $$|x - 3| ≤ 9 ⇔ -9 < x - 3 < 9$$ $$⇔ -6 < x < 12$$

## What does it look like?

The absolute value of 5 is 5, it is the distance from 0, 5 units.

The absolute value of -5 is 5 it is the distance from 0, 5 units.

$$|x| = 2$$

$$|x| > 2$$

$$For \space |x| < 2, -2 < x < 2$$ $$For \space |x| = 4, -4 = x = 4$$

|7| = 7 means the absolute value of 7 is 7.

|-7| = 7 means the absolute value of -7 is 7.

|-2 - x| means the absolute value of -2 minus x.

-|x| means the negative of the absolute value of x.

## You'll use it...

The absolute value for real numbers occurs in a wide variety of mathematical settings, for example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. In real life, the absolute value is closely related to the notions of magnitude, distance, and norms.Like depth of an ocean, time: 500 B.C. as opposed to 500 A.D.

## Video

### Absolute Value and Number Lines

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Length: 5:39

## Video

### Absolute Value of Integers

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Length: 2:22

## Video

### Absolute Value Equations

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Length: 10:41