7 junio, 2024

Classification of Real Numbers

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

The real numbers refer to the combination of the groups of rational and irrational numbers. To form these groups, natural numbers and integers are needed.

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

How are real numbers classified?

– Natural numbers

The natural numbers are represented by the letter (n) and are those that are used to count (0,1,2,3,4…). For example «there is fifteen roses in the garden”, “The population of Mexico is 126 millions of 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.

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

In common parlance, such as in elementary schools, the natural numbers may be called counting numbers to exclude negative integers and zero.

The natural numbers are the bases with which many other sets of numbers can be built by extension: the integers, the rational numbers, the real numbers and the complex numbers, among others.

The properties of natural numbers, such as divisibility and the 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. «Have two cookies”, “I am the father of three children”, “The box includes two gift creams.

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

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 at the top right to simplify writing (1st, 2nd, 3rd, 4th, etc.).

– Integer numbers

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

Example of integers would be «It is 30º 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 button of the elevator».

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 the natural numbers, Z is a countably infinite group.

The integers form 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 decimal expansion of a rational number always ends after a finite number of digits (eg: 15.50) or when it begins to repeat the same finite sequence of digits over and over again (eg: 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 set of real numbers is uncountable, it can be said that almost all real numbers are irrational.

The rational numbers can be formally defined as equivalence classes 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.

The rational numbers, along with addition and multiplication, form fields that make up the integers 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 proof 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, these line segments can be said to be incommensurable; meaning that there is not a sufficient length such that each of them could be «measured» with a particular integer multiple thereof.

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

It can be seen that when irrational numbers are expressed positionally in a number 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 a line of representation is made.

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 terminate or repeat is different from 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 «terminating or repeating» to define the concept of a rational number.

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

References

Classifying real numbers. Recovered from chilimath.com.
Natural number. Retrieved from wikipedia.org.
Classification of numbers. Recovered from ditutor.com.
Retrieved from wikipedia.org.
Irrational number. Retrieved from wikipedia.org.

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