CLASSIFICATION OF REAL NUMBERS - SCIENCE - 2025

The main classification of real numbers is divided into natural numbers, whole numbers, rational numbers, and irrational numbers. Real numbers are represented by the letter R.

There are many ways in which the different real numbers can be constructed or described, ranging from simpler forms to more complex ones, depending on the mathematical work to be done.

How are real numbers classified?

- Natural numbers

Natural numbers are represented by the letter (n) and are those used to count (0,1,2,3,4…). For example "there are fifteen roses in the garden", "The population of Mexico is 126 million people" or "The sum of two and two is four ". It should be noted that some classifications include 0 as a natural number and others do not.

Two children doing a sum of two natural numbers.

Natural numbers do not include those that have a decimal part. Therefore, "The population of Mexico is 126.2 million people" or "The temperature is 24.5 degrees Celsius" could not be considered natural numbers.

In common parlance, as for example in elementary schools, natural numbers can be called counting numbers to exclude negative integers and zero.

Natural numbers are the bases with which many other sets of numbers can be constructed by extension: whole numbers, rational numbers, real numbers, and complex numbers, among others.

The properties of natural numbers, such as the divisibility and distribution of primary numbers, are studied in number theory. Problems related to counting and ordering, such as enumerations and partitioning, are studied in combinatorics.

They have several properties, such as: addition, multiplication, subtraction, division, etc.

Ordinal and cardinal numbers

Natural numbers can be ordinal or cardinal.

The cardinal numbers would be those that are used as natural numbers, as we mentioned earlier in the examples. "I have two cookies", "I am the father of three children", "The box includes two free creams".

Ordinals are those that express order or indicate a position. For example, in a race the order of arrival of the runners is listed starting with the winner and finishing with the last one who reached the finish line.

In this way, it will be said that the winner is the "first", the next the "second", the next the "third" and so on until the last. These numbers can be represented by a letter in the upper right part to simplify writing (1st, 2nd, 3rd, 4th, etc.).

- Integer numbers

The whole numbers are made up of those natural numbers and their opposites, that is, the negative numbers (0, 1, -1, 2, -2, 50, -50…). Like natural numbers, these also do not include those that have a decimal part.

An example of whole numbers would be "30º ago on average in Germany", "I stayed at 0 at the end of the month", "To go down to the basement you must press the -1 elevator button".

In turn, whole numbers cannot be written with a fractional component. For example, numbers like 8.58 or √2 are not whole numbers.

Whole numbers are represented by the letter (Z). Z is a subset of the group of rational numbers Q, which in turn form the group of real numbers R. Like natural numbers, Z is an infinite countable group.

The whole numbers make up the smallest group and the smallest set of the natural numbers. In algebraic number theory, integers are sometimes called irrational integers to distinguish them from algebraic integers.

- Rational numbers

The set of rational numbers is represented by the letter (Q) and includes all those numbers that can be written as a fraction of whole numbers.

That is, this set includes natural numbers (4/1), whole numbers (-4/1), and exact decimal numbers (15.50 = 1550/100).

The distribution of 1/6 of the cheese is a rational number.

The decimal expansion of a rational number always ends after a finite number of digits (ex: 15.50) or when the same finite sequence of digits begins to repeat over and over again (ex: 0.3456666666666666…). Therefore, within the set of rational numbers numbers are included. pure newspapers or mixed newspapers.

Additionally, any repeating or terminal decimal represents a rational number. These statements are true not only for base 10, but also for any other integer base.

A real number that is not rational is called irrational. Irrational numbers include √2, π, and e, for example. Since the entire set of rational numbers is countable, and the group of real numbers is not countable, it can be said that almost all real numbers are irrational.

Rational numbers can be formally defined as classes of equivalence of pairs of integers (p, q) such that q ≠ 0 or the equivalent relation defined by (p1, q1) (p2, q2) only if p1, q2 = p2q1.

Rational numbers, along with addition and multiplication, form fields that make up whole numbers and are contained by any branch that contains integers.

- Irrational numbers

Irrational numbers are all real numbers that are not rational numbers; irrational numbers cannot be expressed as fractions. Rational numbers are numbers made up of fractions of whole numbers.

As a consequence of Cantor's test that says that all real numbers are uncountable and that rational numbers are countable, it can be concluded that almost all real numbers are irrational.

When the radius of length of two line segments is an irrational number, it can be said that these line segments are incommensurable; meaning that there is not a sufficient length so that each of them could be "measured" with a particular integer multiple of it.

Among the irrational numbers are the radius π of a circle circumference to its diameter, the Euler number (e), the golden number (φ) and the square root of two; furthermore, all square roots of natural numbers are irrational. The only exception to this rule are perfect squares.

It can be seen that when irrational numbers are expressed in a positional way in a numeral system, (as for example in decimal numbers) they do not end or repeat.

This means that they do not contain a sequence of digits, the repetition by which one line of the representation is made.

Simplification of the irrational number pi.

For example: the decimal representation of the number π begins with 3.14159265358979, but there is no finite number of digits that can represent π exactly, nor can they be repeated.

The proof that the decimal expansion of a rational number must end or repeat is different than the proof that a decimal extension must be a rational number; Although basic and somewhat lengthy, these tests take some work.

Usually mathematicians do not generally take the notion of "ending or repeating" to define the concept of a rational number.

Irrational numbers can also be treated via non-continuous fractions.

References

1. Classifyng real numbers. Recovered from chilimath.com.
2. Natural number. Recovered from wikipedia.org.
3. Classification of numbers. Recovered from ditutor.com.
4. Recovered from wikipedia.org.
5. Irrational number. Recovered from wikipedia.org.