27 November 2023

Working with numbers implies knowing their nature, their type, classification and the use that can be given to them. Thanks to great scientists in history, today we have different types of numeric sets. In linear algebrathis is a great advantage for those who are beginning to study the sciences of mathematics.

At the time of solve a math problem, it is vital to know if we are in the presence of a decimal number, an integer, rational or irrational. In this opportunity, we will focus mainly on what a decimal number is and the different types that exist. In addition, you will know how we can identify this set of numbers and what is the difference between the integers.

## Decimal number definition

We could define decimal numbers as the counterpart of every whole number. That is, they are numbers that belong to the numerical decimal system and that they are composed of two parts: the integer part and the decimal part. Also, it is a real number, so they are numbers that can be both positive and negative. At the same time, these belong to the group of rationals, for the simple fact that they can be written as a fraction. The latter is a quality that only the rational have.

The most important feature of these numbers is that they have a comma that is distinctive. In this way, when we see a number, such as “20.42”, we know that we are dealing with a positive decimal number. Otherwise, it would be negative if there is a minus sign before the same number. Making a summary of the most fundamental aspects of these numbers, they would be the following:

• They have an integer part and a decimal part.

• They can be positive and negative numbers.

• They are real numbers.

• They can be written as a fraction.

• There is only one way to write decimal numbers, which is using the comma.

It is worth mentioning that these, unlike irrational numbers, have a repeating pattern. This means that a decimal number can be periodic, since it is repeated in its decimal part, something that does not happen with irrational numbers such as the number pi or other universal constants. Before moving on to the next point, these numbers are usually the result of solving a mathematical equation. For this reason, you will never see a decimal in a binomial or polynomial regardless of the number of terms it has.

### What is an exact decimal?

Not all numbers of this type have the same number of decimal places. Those that have a finite number of certain number of decimal places are known as exact decimal number. This greatly helps scientists and students to be able to use the decimal system with more precision. In the latter case, something known as numerical rounding is used, which eases and reduces the number of decimal places to use for some mathematical expression or application.

Furthermore, being exact decimal numbers, have the particularity that they can be written as a fraction more easily. Some basic examples of this type of numbers can be the following:

Each of these numbers, which have an integer part and a decimal part, can be written as a fraction thanks to its nature. Of course, for this there are simple mathematical procedures that can be applied. But this is not always the case, since sometimes it is enough to know which numerator and denominator to use to make write that decimal number as a fractional number.

### What is a repeating decimal?

They are another type of decimal numbers that differ from the previous ones for the simple fact of having a decimal part that repeats infinitely. To give you an example, if you have a number like: 4.65656565656565…n, this number will keep repeating as many times as necessary. This is known as a periodic number. It must be borne in mind that not all rational decimal numbers They have this particularity, the number pi and Euler’s, as well as other mathematical constants, have an enormous number of decimals, but they never repeat no matter how much you count.

Keep in mind that numbers of this type do not always have the same structure, that is, they will not always be in the same shape as the previous example. You may come across repeating decimals as follows:

• 0.33333…

• 0.1428571428571…

• 0.128333….

As you will notice, there are completely different decimal periodic numbers in their mathematical form. Keep in mind that all the examples we have given you are with positive numbers. By belonging to the set of real numbers, also there are mathematical expressions that can be negative. One way to identify how large or small a decimal number is is by seeing if it is greater than, less than, or equal to zero. It could be said that this is the starting point and the universal reference.

## What are the parts of a decimal number?

Every decimal number is made up of two fundamental parts: the integer part and the decimal part. The fusion of both is what allows us to work as such with this numerical set. Now, this simple concept or theoretical basis applies to any real number, either in the order of positive or negative. In addition, these are characterized by having a single comma that separates both parts, so if we are in the presence of this symbol, we know that it is a decimal.

### whole part

We could say that the integer part of these numbers can express how big a number or quantity is if we are using some unit of measurement. This applies perfectly to any scientific or academic field. As for the integer part, this will always be located on the left side of the comma. It is thanks to this simple, but powerful detail, that we can locate all integers (whether positive or negative) in a table of values.

Do you remember that class where you had to identify what the units, tens and hundreds were? Well, this only applies to the positive part, like the units, tens and hundreds of thousands, millions and so on.

### decimal part

As for the decimal part, it is located on the right side of the comma. No sign should ever be placed after the comma symbol, since it is located at the beginning of the numerical expression. Unlike the integer part, the decimal is related to the tenths, hundredths and thousandths or millionths.

An important aspect to consider is that the larger the decimal numbers, the lower their value. For example:

• The number 0.50 is considered as half of something.

• If we have 0.25, this would be half of half of that something.

• As we go down the number, the amount is getting smaller.

• The closer the decimal places are to zero, we can round off.

Keeping this last point in mind and making rounding possible, we can write a decimal number as an integer. This applies perfectly to any unit of measurement or metric system. Now, for practical purposes, doing this practice is not entirely good, since the accuracy of the number would be lost. For this reason, the greater the number of decimal places that are used, for example, in a division calculation, the more precise the result will be.

## What are the types of decimals that exist?

Generally speaking, there are only three types of decimal numbers in all of universal mathematics. Each one of them They have somewhat different characteristics.but which, after all, are still rational numbers and decimals.

### exact decimal number

A mathematical function can be made up solely of decimal numbers or something mixed, in which there are real rational and irrational numbers. As for the set of exact decimal numbers, these are those that when written or divided, the result is one that, when examining its decimal part, has a specific and reduced number. For example, if you divide 31 by 2, the result would be 15.5. This, in its simplest and most minimalist form, is what is known as an exact decimal number.

One aspect to take into account is that these numbers, in their decimal part, do not tend to reach infinity. Unlike the number pi, which, if it does and to date, no one knows for sure what the exact number that defines it is.

### repeating decimal number

These, unlike the previous ones, do have a greater number of decimal places. In fact, they share an interesting characteristic with irrational numbers such as pi, since their decimal part is infinite. However, they differ by having an infinitely repeating pattern. For example, we can have the expression: 0.5956595659565956…n, this will repeat infinitely. Therefore, you can simply use the first 2 or 4 decimal places to get an easier decimal number to manipulate.

### Non-exact and non-periodic number

The simplest example we can give you, which falls perfectly into this category, is the number pi. It has two main qualities.:

In fact, we could say that they have one more quality, which is that there are no two numbers that, when divided, give us this result. Pi is a universal constant that is unique, because it is non-periodic and infinitebesides that there is no division or fraction that can represent it.