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 .
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
- Properties of absolute value
- How it differs from the relative value
- 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.
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
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.
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