2 March 2024

In mathematics there are various terms that are used in many areas of study and in solving numerical operations. However, we do not recognize many of them by their technical names or are rarely used in other areas.

One of these terms is absolute value of a number, which tends to be used in branches such as science, technology and engineering. If you want to know more about absolute values, then we will explain in more depth the existing types, their properties and uses.

## Meaning and concept of the absolute value of a number

When speaking of the absolute value of a number, it refers to the measure of distance or magnitude that has that number to zero inside the number line. This means that the absolute value of a number indicates the number of units that moves that number away from zero, either positive or negative.

An example of this would be the absolute value of -5, which would be 5, since it is located at a distance of five (5) units from zero. This would happen whether it is in negative numbers, as is the case with -5, or in positive numbers, since positive 5 is also five (5) units away from zero, within the number line.

To represent that an absolute number two vertical bars are used to surround the numberfor example -7, whose absolute value would be |-7|, which is equal to 7. Note that the absolute value of a number will always result in a value between any positive number and zero, since the distance between them will never it can be negative.

In general, absolute values ​​are considered a vital mathematical tool and that it is used in diverse sciences where measures of magnitude and distance are used. In addition, it also has different applications in areas of study such as economics, physics, statistics, among other fields.

### What is the absolute value of an integer?

Integers are those that do not have decimals or fractions and that form the numerical set of integers. These include any positive, negative and zero numbersuch as the numbers -2, -1, 0, 1, 2…, etc.

For its part, the absolute value of an integer is understood as the distance between that integer and zero, regardless of the direction on the number line. In this way it is understood that the absolute value of an integer is the magnitude of that number without taking the sign that accompanies it, be it positive or negative.

In general, to differentiate a normal integer from the absolute value of an integer, two vertical bars are used surrounding the number in question. These bars are the ones that symbolize that the number is an absolute value and are used in the following way: |Z|, where Z is an integer.

For example, |-2| represents the absolute value of -2, and |2| would be the absolute value of 2. In both cases, the result would equally be 2, since they are both the same two (2) units away from zero (0).

### What is the absolute value of a real number?

The real numbers are any number that is located in a specific position in the number line. These numbers include whole numbers, rational numbers (expressed as fractions or decimals), irrational numbers (cannot be expressed as fractions), and complex numbers (expressed in real, imaginary, or polar form).

Now when talking about absolute value of a real number, is understood as that measure of distance marked between a number and zero on the number line. In general, when talking about the absolute value of a real number, it refers to the value of that same number without taking into account whether it is positive or negative.

Absolute values ​​are expressed by enclosing the number in question within two vertical bars. For example, |3/5| would represent the absolute value of the real number 3/5.

While |-3/5| symbolizes the absolute value of the real number -3/5. Similarly, in the case of both examples the result would be 3/5. This is because both are the same distance (3/5) from zero on the number line.

### What is the absolute value of a decimal number?

Decimal numbers are those that express their value using a comma or period to separate its integer part from its fractional part. In general, decimal numbers are used to represent quantities that are not integers, but that need greater precision, such as measurements of mass, volume, length, etc.

In a decimal number you can find the integer part, which is a whole number located to the left of the decimal point. While, the fractional part is the one that is on the right side of the semicolon.

An example would be the number 1,235, where, the 1 is the integer part and the fractional part would be 0.235.

For its part, the absolute value of a decimal number is the exact distance between the point, in where is the number on the number line, and zero. Absolute values ​​are calculated by measuring the total distance between the decimal number and zero, whether the value is positive or negative.

These values ​​are represented using two vertical bars on the sides of the numbers. For example, if we have the number 3.1416, the absolute value would be represented by |3.1416|and the result would be 3.1416.

In the same way it happens if the number is negative, like him -5.31, which would be represented |-5.31|, and its absolute value would be 5.31. In summary, the absolute value of a real number is the same number that, when applying the absolute value, it would become positive.

## What are the properties of an absolute value?

As we have said before, the absolute value is known as a mathematical function with which we always a positive or zero result is obtained. Likewise, it has some important properties that define it, among which are the following:

1. Non-negativity: when we obtain the absolute value of any real number, it is always equal to or greater than zero. This means that all the results are obtained positive and never negative.
2. Symmetry: It is always obtained that the absolute value of a number will be equal to the absolute value of its opposite. Also, it can be expressed as follows: |a|=|-a|and it applies to any real number a.
3. Triangular inequality: it applies to any pair of real numbers a and b, where it is fulfilled that |a+b| ≤ |a|+|b|. This means that the sum of the absolute values ​​of two numbers will always be greater than or equal to the absolute value of the sum.
4. Products and quotients: this is a rule that for any pair of real numbers a and b, which satisfies that |ab|=|a| |b| and |a/b|=|a|/|b|. This will be so as long as b is not equal to zero.
5. Inverted triangular identity: this explains that in every pair of real numbers a and b it is true that |ab|=|ba|.

### What is the symbol for an absolute value?

The symbol used to denote an absolute value of a number is a vertical double bar, where a bar will be placed first, followed by the real number and then the other bar. For example, if we want to denote the absolute value of the number -3, it would be written as follows: |-3|.

If we want to represent the absolute value of quotients or products, the symbol can be used on each individual real number or with the function within it. This means that these functions can be represented in the following ways: |ab|=|a| |b| or |a/b|=|a|/|b|.

## Examples of an absolute value in mathematics

An absolute value is understood as a relative space or distance between a real number and zero, regardless of whether the value is negative or positive. Therefore, it has different applications in various areas, but mainly in mathematics, some examples of this are:

• vector calculation: if we have a vector V=(-3, 5) inside the Cartesian plane, to calculate its absolute value we use the formula |v|=√(x²+y²). Where we have that, ‘x’ and ‘y’ are the components of the vector.

Therefore, we would have the following operation: |v|=√((-3)²+4²), which would give us as a result: |v|=5. This means that the absolute value of vector V is 5which means that the length of the vector to zero is 5 units.

• Calculation of the distance between two points: if we want to obtain the distance between two points A and B on the number line. Where A=-3 and B=4, we can use the distance formula that involves absolute values, which is represented as follows: distance AB=|BA|

With it we obtain that: distance AB=|4-(-3)| distance AB=|7| distance AB=7. This means that the distance between points A and B is 7 units on the number line.