What is a natural number? History, scope, properties

Mathematics was distinguished from the general philosophy around the sixth century BC. And from that moment began its victorious procession around the world. Each stage of development introduced something new - the elementary account evolved, transformed into a differential and integral calculus, centuries replaced, the formulas became more complicated, and the moment came when "the most complex mathematics began - all the numbers disappeared from it". But what was the basis?

The beginning of time

Natural numbers appeared on a par with the first mathematical operations. Once a spine, two roots, three roots ... They appeared thanks to Indian scientists who deduced the first positional number system. The word "positional" means that the location of each digit in the number is strictly defined and corresponds to its category. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundred, while the second number is only 4. The innovation of the Indians was picked up by the Arabs, who brought the numbers to the species that we know now.

In ancient times, numbers were given a mystical meaning, the greatest mathematician Pythagoras believed that the number underlies the creation of the world along with the main elements - fire, water, earth, air. If we consider everything from the mathematical side, then what is a natural number? The field of natural numbers is denoted by N and represents an infinite series of numbers that are integer and positive: 1, 2, 3, ... + ∞. Zero is excluded. It is used mainly for counting objects and ordering.

What is a natural number in mathematics? Axioms of Peano

The field N is the base field on which elementary mathematics is based. With the passage of time, fields of integer, rational, complex numbers were distinguished .

The works of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formalities and prepared the ground for further conclusions that went beyond the field of the field N. What is a natural number has been clarified earlier by a simple language, below is a mathematical definition based on the axioms of Peano.

• A unit is considered a natural number.
• The number that follows the natural number is natural.
• Before unity there is no natural number.
• If the number b follows both the number c and the number d, then c = d.
• The axiom of induction, which in turn shows that such a natural number: if some assertion that depends on the parameter is true for the number 1, then we assume that it works for the number n from the field of natural numbers N. Then the assertion is true for n = 1 from the field of natural numbers N.

Basic operations for the field of natural numbers

Since the field N was the first for mathematical calculations, it is to it that both the domain of definition and the range of values of a number of operations are referred to below. They are closed and no. The main difference is that closed operations are guaranteed to leave the result within the set of N, regardless of which numbers are involved. It is enough that they are natural. The outcome of the remaining numerical interactions is no longer so unambiguous and directly depends on what numbers are involved in the expression, since it may contradict the basic definition. So, closed operations:

• Addition - x + y = z, where x, y, z are included in the field N;
• Multiplication - x * y = z, where x, y, z are included in the field N;
• Exponentiation - x ^y , where x, y are included in the field N.

Other operations, the outcome of which may not exist in the context of the definition of "what is a natural number", are as follows:

• Subtraction - x - y = z. The field of natural numbers admits it only in the event that x is greater than y;
• Division - x / y = z. The field of natural numbers admits it only in the case when z is divisible by y without remainder, that is, completely.

Properties of numbers belonging to the field N

All further mathematical arguments will be based on the following properties, the most trivial, but from this no less important.

• The displacement property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or all known "the sum does not change from the change of the places of the summands".
• The displacement property of multiplication is x * y = y * x, where the numbers x, y are included in the field N.
• The combining property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the field N.
• The combining property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the field N.
• The distribution property is x (y + z) = x * y + x * z, where the numbers x, y, z are included in the field N.

Table of Pythagoras

One of the first steps in the students' knowledge of the entire structure of elementary mathematics after they have understood for themselves which numbers are called natural are the Pythagoras table. It can be viewed not only from the point of view of science, but also as a most valuable scientific monument.

This multiplication table underwent a number of changes over time: from it, zero was removed, and numbers from 1 to 10 denote themselves, without taking into account the orders (hundreds, thousands ...). It is a table in which the headings of rows and columns are numbers, and the contents of the cells of their intersection are equal to their product.

In the practice of teaching the last decades, it was necessary to memorize the Pythagorean table "in order", that is, first there was a memorization. Multiplication by 1 was eliminated, since the result was 1 or more multiplier. Meanwhile, in the table with the naked eye you can see the regularity: the product of numbers grows by one step, which is equal to the title of the line. Thus, the second factor shows us how many times to take the first, in order to obtain the desired product. This system is not more convenient than the one practiced in the Middle Ages: even realizing what a natural number is and how trivial it is, people managed to complicate their daily account using a system based on the powers of deuce.

A subset like the cradle of mathematics

At the moment, the field of natural numbers N is considered only as one of the subsets of complex numbers, but this does not make them less valuable in science. The natural number is the first thing a child learns by studying himself and the world around him. One finger, two fingers ... Thanks to him, a person develops logical thinking, as well as the ability to determine the cause and output a consequence, preparing the ground for greater discoveries.