# Absolute value

In the field of **mathematics** there are many important terms, among them we can find the **absolute value** which is used in order to be able to name the **value** that a certain figure has beyond its sign. It is also known by the name of **module** .

**Function**

f (x) = | x |

## What is the absolute value?

The absolute value of a **real number,** no matter what it is, is the **same number** but with the difference that it must always have a **positive sign** . It is a numerical value that does not take into account its sign, positive or negative.

- Definition of absolute value
- Function
- Properties of absolute value
- Symbol
- How it differs from the relative value
- Importance
- Examples of absolute value

## Definition of absolute value

Basically we can say that the **absolute value** of a number refers to the **value** it has regardless of the **sign** . Although in the field of **algebra** the size, the value and the sign matter, there are some cases in mathematics and daily life in which that sign is not of importance but what really matters is the **size** , this is the absolute value of a number.

In its definition, the concept tells us in other words that the absolute value that a certain number has will always be **equal to** or **greater than 0** but that **it can never be negative** . In this case it is important to mention that due to this, the absolute value of the numbers, for example, 4 and -4 will always be | 4 |.

When found on a **number line** , the absolute value is represented as the **distance** from a certain **point of origin. **In other words, if we travel seven units from zero to the left or to the right side , we will be arriving on the line at the number 7 or -7, so the absolute value of these values will always be 7.

An important aspect of absolute value is that it is represented by two **bars** known as **absolute value bars** . It is very important that when working with it, these bars are not confused by **parentheses** or **brackets** because in mathematics, this could change all the rules and even the definitions.

## Function

The main function of the **absolute value** is to be able to represent the **distance** that exists from the **origin** or from the **zero** of a number on a number line until reaching the **destination** number or **point** . This distance will always be **positive** or **null** .

The absolute value function has its own equation which is:

**f (x) = | x |**

This function is represented by **distances** or **intervals** which are known by the name of **pieces** or **sections** and these can be calculated if the following steps are followed:

- The function will be set equal to zero, without taking the absolute value and then the roots or the values of x are calculated.
- Intervals are created with the roots of the x-values and then the sign of each of these intervals is evaluated.
- The function is then defined in intervals always keeping in mind that when x is negative, the sign of the function is changed.

## Properties of absolute value

Among the main properties that we can find in the absolute value of a number, the following are mentioned:

- The absolute value can be explored
**graphically**or**numerically**. - It will always be
**positive**or**zero**. In this regard, it is important to remember that if the original value of the number is positive or zero, then the value will always be the same. When it is negative, only the sign should be removed.

We can also find several fundamental properties that are the following:

**Multiplicative property**: | x ∙ y | = | x | ∙ | y |**Positive definition**: | x | = 0 ↔ x = 0**It has no negativity**: | x | > 0**Triangular inequality pose**: | x + y | ≤ | x | + | and |**It has symmetry**: | -x | = | x |**It is equivalent to the addition property**: | a – b | ≥ | (| a | – | b |) |**It has the identity of indiscernibles**: | a – b | = 0 ↔ a = b**Preservation of division**, which is also equivalent to the multiplicative property: | x ÷ y | = | x | ÷ | and | if b ≠ 0

## Symbol

The **symbol** that is used to represent the absolute value is a bar that is placed on each side of the number. Instead of saying “the absolute value of 3” we can represent it as follows:

**∣ ****3 ****∣**

## How it differs from the relative value

It is important to mention that all numbers have an **absolute value** and a **relative value** . The main difference is that the absolute value is one that specifically refers to **the numerical value** and does not take into account anything that can be found before or after the number, for example, the absolute value of 5 and -5 will always be 5.

On the other hand, the **relative value** can be found taking into account the **value of** the **numbers** that make up a figure, in other words, if we have for example the number 589, each of these numbers will have its own relative value also taking into account your position. The 9 will be nine because it is in the units position, the 8 will have a value of 80 because it is in the hundreds place, while the 5 will have a value of 500 because it is in the hundreds place.

## Importance

Its importance lies in the moment of carrying out different calculations that must be positive because for this the lower value must be taken first. It is also a term that is closely related to terms such as **magnitude** , **distance** and **norm** in a wide field of **mathematical** and **physical ****contexts** .

## Examples of absolute value

Some examples of absolute value are mentioned below:

- (6) = 6, because 6> O
- (-9) = (-9) because -9 <Or we take its inverse
- If (x) = 5 then x = 5 or x = -5