**Absolute value in R** : If we have the real numbers on a number line, by *Dedekind’s Postulate* , we know that each point on the line corresponds to a single real number and vice versa. Since there are positive numbers and there are negative numbers, we place the positives to the right of the origin O of the coordinate system of the line and the negatives to the left. Two things interest us the distance between two points on the number line and the distance from one point to the origin. Precisely this distance gives us the concept of absolute value of a real number, this understood as the coordinate of a point on the line.

## Summary

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- 1 Definition
- 2 Equivalent definitions
- 3 Properties
- 1 equations
- 2 Basic inequalities

- 4 Generalization
- 1 Norm of a complex
- 2 Modulus of a vector

- 5 Applications
- 6 References
- 7 Sources

## Definition

It is agreed that the distance from point A (a) to the origin O (0), denoted by d (A, O), represents the absolute value of the number a, which locates point A.

- d (A, 0) equal to the absolute value of the real number a, denoted by | a |

| a | = a when a is non-negative

| a | = -a as long as a is negative

In case of a positive number, arithmetically, the absolute value is the same number; the value of 0 is 0, and the value of a negative number is its opposite.

## Equivalent definitions

- | to | = (a
^{2})^{5}, in other words, the absolute value of a is equal to the arithmetic square root of its square. - | a | = max {a; -a}, when a ≠ 0, a ≠ -ay have a different sign, but one of them is positive, just the one that is positive is the largest and is its absolute value.
- | a | = a × sgn (a), where sgn (a) represents the sign of a, which is 1 if a> 0; is 0, when a = 0 and -1, when a is negative.

## Properties

- | a | ≥ 0; the absolute value is non-negative and is zero only when a = 0.
- | a | = | -a |, both the number a and its opposite -a have the same absolute value.
- | a |
^{2n}= a^{2n}for n positive integer. The even power of the absolute value of every number is equal to the even power of it. - | ab | = | a || b |, the absolute value of a product is equal to the product of the absolute values of the factors.
- | a ÷ b | = | a | ÷ | b |, for the quotient, a particular case of the product, that is, by the multiplicative inverse of the divisor, which must not be null.
- The number a is is one of the extremes of the interval [- | a | , | a |]
- | a ± b | ≤ | a | + | b |, the absolute value of a sum or difference of any two numbers does not exceed the sum of their respective absolute values. As an example: | -15 – (- 35) | ≤ | -15 | + | -35 |.
^{[1]}. - || a | – | b || ≤ | ab | the distance between the absolute values of two numbers does not exceed the distance between those numbers.

### Equations

- | x | + | and | = 1 its graph is a square whose vertices are at (1,0), (0,1), (-1,0) and (0; -1)
- y = | x-1 | its graph is a pair of perpendicular rays in the positive half plane with a common point (1.0)
- y = 1 – | x |, its graph is a pair of perpendicular rays, in the direction of the negative half plane, with its common point, and vertex of the angle formed by the rays, the point (0.1)

### Basic inequalities

- | x | <c, sss -c <x <c if the absolute value of a number is less than a positive, the first is in the open end interval -c and c.
- | x | > c> 0 if i x> c or x <-c, x is outside the open interval (-c; 0)

## Generalization

### Norm of a complex

### Module of a vector

## Applications

- The absolute value is used to define a bounded and unbounded sequence of real numbers.
- It is also used to find the limit of a limited sequence
^{[2]} - It is used when defining the limit of a real function of real variable.
- It is used to find the distance between two points M and N of a number line, d (M, N) = | mn | where m and n are the coordinates of the points cited.
- It can be used to define a continuous function of R in R, without resorting to the concept of limit.