The Notion of Real Numbers
The real numbers are a mathematical object that appeared from the measurement requirements of geometrical and physical quantities of the world, as well as the carrying out of such computational operations as root extraction, the computation of logarithms, solution of algebraic equations, and studying the behavior of the functions.
While natural numbers arose in the process of counting, rational numbers appeared due to the need to operate with parts of the whole, the real digits were invented for measurement of continuous quantities. Thus, the expansion of the stock of numbers under consideration led to the set of real numbers, which in addition to rational numbers also include other elements, called irrational numbers.
Visually the concept of real numbers can be imagined with the help of the number line. If one selects the direction on the line, the start point, and the unit of length for measuring lengths, so that every number can be associated with a particular point on the line, and vice versa, while each point will be only one number. As a result, the term of the number line is usually used as a synonym for the set of real digits.
History of Real Numbers
The concept of a real number has passed a long way of formation. Even in Ancient Greece, at the school of Pythagoras, which considered all integers and their relationships as the basis, was discovered the existence of incommensurable magnitudes (incommensurability of the side and diagonal of a square), that is, in modern terminology – numbers that are not rational. Following this, Eudoxus of Cnidus made an attempt to build a general theory of numbers, which included the disparate values. After that, for over two thousand years, no one felt the need for a precise definition of the concept of real numbers, in spite of the gradual expansion of this concept. Only in the second half of the 19th century, when the development of mathematical analysis required the reconstruction of its foundations at the new and higher level, a strict theory of real digits was established in the works of Weierstrass, Dedekind, Cantor, Heine, and Méray.
From the standpoint of modern mathematics, the set of real numbers is a continuous ordered field. This determination, or an equivalent system of axioms, exactly defines the concept of a number in the sense that there is only one, up to isomorphism, a continuous ordered field. The set of real numbers has a standard designation – R («bold R»).
Types of Real Numbers
The real number is any positive number, a negative number or zero. They are divided into rational and irrational numbers.
The simplest numbers are positive integers 1, 2, and so on, that are used in the long run. The needs of the practice led to the formation of simple fractions, i.e., numbers of the form 1/2, 3/5, and so on. Much later Indians invented an important number 0, and in the beginning of our era, the Italians invented negative numbers.
Generally speaking, the concept of a number is one of the most uncertain concepts in mathematics.
A rational number is a number that can be represented as m/n, where m is an integer, and n is a natural number.
Irrational numbers are infinite and non-recurring decimals.
Rational and irrational numbers with the rules made for them are called real numbers.
Properties of Real Numbers
- Connection with rational numbers.
It is obvious that on the number line, rational numbers are arranged alternately with the real ones with the dense set of real numbers. The question arises, how often do real and rational numbers appear on the number line and is it possible to distinguish them from one another? The answer to this question is given by three lemmas, based mainly on the axiom of Archimedes.
Lemma 1: for any real number and any positive rational distance taken in advance exists a pair of rational numbers, separated by less than this distance, such that the real number is the interval between the rational numbers. This lemma says that any real number can be specified with accuracy on both sides to bring the rational numbers.
Lemma 2: There is a rational number between any two distinct real ones. An obvious consequence of this lemma is that there is a whole set of rational numbers between any two mismatched real ones. In addition, even more obvious is the fact that there is a real number between any two different rational numbers.
Lemma 3: The approximation of a real digit by a rational one, described in Lemma 1, identifies a real digit in a unique way.
These lemmas, first of all, say that the set of real digits is not as «tight» as compared with the set of rational numbers, as it may seem. This fact is especially clearly illustrated by the second lemma. All three lemmas are widely used to prove various theorems relating to the operations of addition and multiplication of real digits.
- Set-theoretic properties.
Initially, the real digits were a natural generalization of the rational numbers, but it was found for the first time that they have a property of nondenumerability, which means that the set of real digits can’t be numbered, that is, there is no bijection between the sets of real and natural numbers.
Construction of Real Digits
When using a constructive definition of a real digit on the basis of known mathematical objects (e.g., the set of rational numbers (Q), which are taken as set, new objects are being built, which, in a sense, reflect our intuitive understanding of the concept of a real digit. The essential difference between the real digits and these constructed objects is that the former, unlike the latter, are understood only intuitively, and yet are not strictly defined with a mathematical concept.
These objects are declared real digits. It is possible to do basic arithmetic operations with them.
Historically, the first rigorous definitions of real digits were constructive definitions. In 1872, three works were published at the same time: the theory of fundamental sequences by Cantor, Weierstrass theory (in the modern version – the theory of infinite decimals), and the theory of cuts in the area of rational numbers by Dedekind.
- Theory of fundamental sequences by Cantor.
In this approach, the real digit is regarded as the limit of a sequence of rational numbers. In order for the sequence of rational numbers to converge, the condition of Cauchy is imposed on it. The meaning of this condition is that the members of the sequence, starting with a certain number, will lie arbitrarily close to each other. The method of constructing the set of real digits via Cauchy sequences of rational numbers is a special case of construction of recharge an arbitrary metric space. As well as in the general case, the resulting completion of the set of real digits itself is already complete, that is, it contains all the fundamental sequences of its elements.
- The theory of infinite decimals.
A real digit is defined as an infinite decimal fraction.
- The theory of cuts in the area of rational numbers.
In Dedekind’s approach, the real digits are determined by the cuts in the set of rational numbers. A cut in the set of rational numbers (Q) is any partition of the set of all rational numbers into two non-empty classes – the lower A and the upper A’, so that each number of the lower class is strictly less than any number of the upper class.
Generalization of Real Numbers
The field of real digits (R) is constantly served in math as a source of generalizations in a variety of important practical directions. The options of generalized numerical systems presented below are directly connected to the field R.
- Complex numbers.
These numbers are especially fruitful in algebra and analysis.
- Interval numbers.
These numbers are used mainly in the theory of approximate calculation and probability theory.
- Non-standard analysis.
The analysis adds infinitely small and infinitely large numbers (of different orders) to the real numbers.
The mathematical model of the real digits is used everywhere in science and technology for the measurement of constantly changing variables. However, it is not its main application because the actual measured values always have a finite number of decimal signs, that is, they are rational numbers. The main purpose of this model is to serve as the basis for analytical methods. The enormous success of these methods over the past three centuries has shown that the model of real numbers in most cases adequately reflects the structure of continuous physical variables.
This, of course, does not mean that the real digit line is an exact image of the actual continuous value. For example, modern science doesn’t know yet whether space and time is discrete or divisible without limit. However, even in the second case, the model of real digits for these quantities should be regarded as approximate, since the concepts of space point and the moment of time are idealizations that have no real equivalent. This fundamental question is widely discussed in science, starting with the paradoxes of Zeno. This model is also approximate when applied to the quantities, which in classical physics were regarded as continuous, but in fact they turned out to be discrete (the quantized).
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