Table of Contents

## Definition of Absolute Value

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

- |x| = x for a positive x,
- |x| = −x for a negative x (in which case −x is positive)
- |0| = 0.

For example, the absolute value of 5 is 5, and the absolute value of −5 is 5 too. The absolute value of a number may be thought of as its distance from zero.

For any real number x, the absolute value of x is denoted by |x| and is defined as:

### More Examples

The absolute value of −7 is 7

The absolute value of 4 is 4

The absolute value of 1 is 1

The absolute value of −25 is 25

## No Negatives Numbers

Absolute value means to remove any negative sign in front of a number, and to think of all numbers as positive (or zero).

The absolute value of x is thus always either zero or positive (never negative). If x < 0 (negative), then its absolute value is necessarily positive (|x| = −x > 0).

Absolute Value means how far a number is from zero.

“5” is 5 away from zero, and “−5” is also 5 away from zero.

So the absolute value of 5 is 5, and the absolute value of −5 is also 5

## Symbol of Absolute Value

To show that we want the absolute value of something, we put “|” marks either side (they are called “bars” and are found on the right side of a keyboard).

**Examples**

|−7| = 7

|4| = 4

Sometimes absolute value is also written as “**abs()**“, so abs(−2) = 2 is the same as |−2| = 2

## Subtract Either Way Around

And it doesn’t matter which way around we do a subtraction, the absolute value will always be the same:

|4−7| = 3, because 4−7 = −3, and |−3| = 3

|7−4| = 3, because 7−4 = 3

### More Examples

Here are more examples of how to handle absolute values:

|−2×5| = 10 | Because −2×5 = −10, and |−10| = 10 |

−|7−3| = −4 | Because 7−4 = 3 and then the first minus gets us −3 |

−|6−8| = −2 | Because 6−8 = −2 , |−2| = 2, and then the first minus gets us −2 |

−|−11| = −11 | Because |−11| = 11 and then the first minus gets us −11 |

## Absolute Value Properties

The absolute value has 4 fundamental properties (a, b are real numbers), that are used for generalization of this notion to other domains:

Properties | Rules |

Non-negativity | |a| ≥ 0 |

Positive-definiteness | |a| = 0 ⇔a = 0 |

Multiplicativity | |ab| = |a||b| |

Subadditivity (specifically triangle inequality) | |a+b|≤ |a|+|b| |

There are additional properties which are either consequences of the definition or implied by the 4 fundamental properties above.

Properties | Rules | Details |

Idempotence | ||a||= |a| | the absolute value of the absolute value is the absolute value |

Evenness | |-a| = |a| | reflection symmetry of the graph |

Preservation of division | equivalent to multiplicativity | |

Identity of indiscernibles | |a – b| = 0 ⇔a = b | equivalent to positive-definiteness |

Triangle inequality | |a – b| ≤ |a – c| + |c – b| | equivalent to subadditivity |

Reverse triangle inequality | |a – b| ≥ ||a| – |b|| | equivalent to subadditivity |

|a|≤ b ⇔ -b ≤ a ≤ b | other property concerning inequalities | |

|a|≥ b ⇔ a ≤ -b or a ≥ b | other property concerning inequalities |