The need for counting became obvious to man from the very beginning of the formation of primitive society. Their own number systems, with specific digital symbols, were formed in all isolated centers of civilization: in Egypt and Ancient Babylon, in China and India, among the South American Indians and in ancient Greece. Mathematics has gone from the simplest counting of objects to solving the most complex theorems of topology. Moreover, the history of the number zero covers only a tiny part of this period.

Numbers and figures

From the Latin nullis (“no”) comes the word denoting one of the most important mathematical concepts. It includes not only a symbol - a number that helps to keep count and write down mathematical operations. It's a whole concept. The absence of any quantity, emptiness, beginning and infinity - the philosophical attitude towards these concepts was different in different eras, in different systems worldview.

Positional number systems

In prehistoric times, fingers and toes helped with calculations. The division of numbers into fives and tens, the origin of the decimal is connected precisely with this. Later, to facilitate these operations, notches on wood and animal bones, notches on stones, and pebbles were used. shells and other small items. Each such element represents a specific number. The most practical numerical models have a similar nature. Such systems are called positional - the meaning of digits when writing numbers is determined by their position or digit.

An example of a system that is opposite in approach and is still used today is the method of writing numbers that has come down from the times Ancient Rome. It uses letters to denote units, tens, hundreds

Abacus

A counting board consisting of recesses corresponding to certain ranks into which pebbles or beads are placed is familiar to cultures different nations and eras. Other varieties of abacus are also known - ropes with knots or cords with beads. The next step in the development of such a device was the abacus, which was used before the advent of calculators.

The history of the number zero is the process of the emergence of a mathematical concept and the beginning of the use of the symbol denoting it. Both the abacus and the abacus are, in a sense, a means of visualization. An empty space in the corresponding recess or a missing knuckle on the abacus made the abstract concept of zero visual. The symbol denoting it first appeared among the mathematicians and astronomers of Ancient Babylon.

Babylonian sign of the void

The civilization born between the Tigris and Euphrates rivers adopted a numerical system inherited from the ancient Sumerians. It was positional - the meaning of the numbers depended on their position relative to other numbers. Developed 4-5 thousand years BC. e., it was built on the number 60. The mathematical calculations used by ancient Babylonian engineers and astronomers therefore looked quite cumbersome and inconvenient. To successfully operate with numbers, it was necessary to remember by heart or have in front of your eyes the results of multiplying all numbers from 1 to 60.

The number zero, or the sign adopted by the Babylonians to indicate rank, looked like two wedges or arrows placed at an angle. This symbol was integral part number and did not participate in arithmetic operations - it was impossible to add or multiply by it.

Overseas zero

Regardless of the mathematicians of Mesopotamia, the Indians of Central America - the Mayans and Incas - introduced their zero into use. What both number systems had in common was that they did not develop the idea of ​​zero as a number.

Ancient American civilization left the world many achievements in intellectual sphere. The complex calendar systems of the Mayans and Incas are the result of centuries of experience in astronomical observations and complex mathematical calculations. But never in their equations was the number zero present as a number influencing the result of mathematical operations.

Antique look

Their main legacy was their achievements in geometry and astronomy. Numbers in their representation are segments with a beginning, an end and a certain length. Zero is a number that has no practical value in this case. A segment with zero length had no meaning in ancient mathematics and philosophy.

One of the main tenets of Aristotle's teachings is the phrase Natura abhorret vacuum - "Nature abhors a vacuum." Infinity, nothingness, non-existence - these categories did not fit into the ancient universe. Therefore, the modern meaning of the question “what number is 0” was unattainable for Archimedes, Pythagoras or Euclid, although a symbol similar to zero is found in the tables of the great astronomer Ptolemy. He wrote the letter “Omicron” (the first letter in the word οὐδέν - “nothing”) in empty cells.

The birthplace of zero is India

What did Indian mathematicians invent? Mahavira (850), Brahmagupta (1114), Aryabhata (476) - authors of treatises in which the modern system writing numbers and rules for basic arithmetic operations. Historians believe that the decimal number system was borrowed by the Indians from the Chinese, and its positional character from the Babylonians. It is believed that the symbol of zero was also borrowed by the Indians from the works of Ptolemy.

The first mathematician to formulate a complete numerical system, which still remains unchanged and serves the majority of humanity, was Khwarizmi Muhammad ben Musa (787-850), who lived in Baghdad. His Book of Indian Counting describes in detail nine Arabic numerals and the answer to the question: "Is 0 a number?" The mention of zero in this book is considered the first. The Latin translation of this work became widely known in Europe in the 12th century and marked the beginning of the spread of Eastern mathematical knowledge.

Unlike Europeans, Eastern philosophers were awed by eternity. Therefore, zero in the equations of ancient Indian scientists finally became not only a symbol of the absence of units in the corresponding category, but also natural number, affecting the result of calculations. Adding zero, multiplying by 0 - all this acquired the meaning of meaningful mathematical operations.

The very writing of numbers from 1 to 0 acquired its final form also thanks to ancient Indian mathematical treatises, and those symbols that are usually called Arabic in Europe are called Indian by the Arabs themselves.

The history of the number “zero” is reflected in the etymology of basic mathematical terms. The word “digit” has Arabic roots and comes from the word “al-sifr”, which means “empty, zero”. The English “zero” vaguely resembles “zephyr” - the wind from the east - it was from the East that a finally formed, rational and convenient numerical system came to Europe.

in Europe

One of the main European propagandists of the Arabic digital system was the famous Italian mathematician Leonardo Fibonacci. His work “The Book of Abacus” (1202) introduced European scientists to the symbols and rules with which the Arabs wrote down mathematical operations. The first to appreciate the convenience and rationality of the Eastern mathematical model were those who were accustomed to daily handling of numbers - bankers and traders. They quickly adopted the number system and writing of numbers from Arab merchants. But in scientific practice In Europe, this knowledge became firmly established only after 4 centuries, replacing the ancient system adopted by European mathematicians.

Zero gained importance with the introduction into scientific use of the rectangular coordinate system, proposed in the 17th century by Rene Descartes. Zero, located in the center, has acquired the meaning of a visible and visually understandable reference point for the three coordinate axes.

In Russia, zero was introduced into practice through the efforts of Leonty Magnitsky, the author of the famous textbook “Arithmetic, that is, the science of numbers” (1703).

Properties of zero

Zero, which demarcates positive and negative numbers, has unique mathematical properties. This is an even, unsigned natural integer. Adding with zero and subtracting zero has no effect on the number, but multiplying by 0 gives zero. Division by zero is considered a meaningless operation, which, if performed in computer program can cause significant harm to the system.

It was in the attempt to divide by 0 that the meaning of the failure in the computer system of the US Navy cruiser Yorktown turned out to be, which occurred in the fall of 1997 and led to the unauthorized shutdown of the propulsion system. An incorrect attitude to the number meaning “nothing” turned a powerful warship into a helpless, motionless target.

The significance of this number increased significantly with the development of science. Zero appears in areas not only purely mathematical. The hearing threshold in acoustics is taken to be 0. What number is at the beginning of the scale of many measuring instruments is known to schoolchildren: 0 on the Celsius scale is the freezing point of water, the beginning of longitude is the prime meridian, etc.

Binary notation, which served as the basis for the creation of modern computing devices, is a positional number system with a base of two. This means that all data entered into computer systems is encoded using a combination of two characters - one and zero.

The role of computers in modern world becomes decisive for all aspects of life, which means that the history of the number zero, without which their appearance would have been impossible, continues.

Therefore, natural numbers begin with one, and the smallest of them is the number 1. It follows that zero is not a natural number. Natural numbers are needed for counting. Nobody counts zeros. As for Russia, it is generally accepted that zero is not a natural number, since it is not used in counting. The smallest natural number is one.

INTEGERS

These numbers express the measures of a finite number of individual objects, and also express the order of uniform unidirectional counting (each natural number has its “own” place - a unique number). In Russian mathematical literature, zero is usually not included in the set of natural numbers. The natural numbers are completely ordered: any subset of them will have a minimal element. Already for integers, this definition is violated exactly in the “other half of the cases”, when the negative is subtracted, giving a number greater than the first.

To read natural numbers, they are divided, starting from the right, into groups of 3 digits each.

Zero in the sense that it contains no information. 0 1 is one bit of information.

The concept of a natural number in mathematics is one of the basic ones. What numbers are called natural numbers? First natural numbers: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11, etc.

When counting, the number zero is not used. The natural series is infinite; there is no greatest natural number in it.

5th class units are trillions, 6th class units are quadrillions, 7th class units are quintillions, 8th class units are sextillions, 9th class units are eptillions.

Walking zero kilometers means not moving. Running at zero kilometers per hour means standing still. Yes, in general, it does not cause any difficulties. Adding or subtracting zero means neither adding nor subtracting anything.

Kuznetsov's large explanatory dictionary (2009) lists both forms: zero, zero - as equivalent.

This is due to the fact that the function of two variables at a point has an irremovable discontinuity.

Work plan: Find in the dictionary what “number” is. 2. The lowest, bad score (pre-revolutionary. For you, perhaps, he is a nonentity, a zero. Chekhov. Two zeros (colloquial joke) - a restroom. Dahl's Dictionary. Zero - m. zero; a numeral sign meaning nothing, nothing (0); but placed after another digit (on the right), raises it by ten, multiplies it by ten.

Belonging to natural numbers

If someone thinks that zero belongs to the natural numbers, that's okay. If a number crumbles into units without noise or dust, such a number is natural. Can a fractional number be a natural number? This is math for blondes and should be a) accurate, b) accessible. In Russia, zero is usually not included in the set of natural numbers, unless otherwise specified. Already in ancient times, 0 was considered a number. But 0 is not a number. 0 is the absence of a number, and not only, but the All-All. It’s just that in mathematics they use 0 as a number according to the logic of mathematics, because every result mathematical operation can be taken as a number and can be treated as a number.

In general, this question is not worth a damn: these are just subtleties of terminology. Interesting. And for whom is it logical to count from scratch?

Commutativity of multiplication.

The absence of any quantity, emptiness, beginning and infinity - the philosophical attitude towards these concepts was different in different eras, in different worldview systems. Such systems are called positional - the meaning of digits when writing numbers is determined by their position or digit. Therefore, the modern meaning of the question “what number is 0” was unattainable for Archimedes, Pythagoras or Euclid, although a symbol similar to zero is found in the tables of the great astronomer Ptolemy. Therefore, zero in the equations of ancient Indian scientists finally became not only a symbol of the absence of units in the corresponding digit, but also a natural number influencing the result of calculations. The very writing of numbers from 1 to 0 acquired its final form also thanks to ancient Indian mathematical treatises, and those symbols that are usually called Arabic in Europe are called Indian by the Arabs themselves.

The natural series is constructed in such a way that each next number is 1 (one) greater than the previous one. The natural series of numbers can be easily represented visually. To do this, go to the “Table of Natural Numbers” page, where natural numbers from 1 (one) to 120 (one hundred and twenty) are presented. If you add the toes, arms and legs of your neighbors to recount, you get a very a large number of numbers, and all these numbers will be natural numbers. Why are negative numbers not natural numbers?

The set of all natural numbers is usually denoted by a symbol (from the Latin naturalis - natural). This would significantly complicate the further construction and application of the theory, since in most constructions zero, like the empty set, is not something separate. One of the advantages of natural zero is that it forms a semigroup with identity. If the base and exponent are natural, then the result will be a natural number.

If a(\displaystyle a) and b(\displaystyle b) are natural numbers, then the result will be a natural number.

IN Ancient Greece the number 0 was not known. In the astronomical tables of Claudius Ptolemy, empty cells were designated by the symbol ο (the letter omicron, from ancient Greek. Without zero, the decimal positional notation of numbers, invented in India, would have been impossible. The first zero code was found in the Indian record of 876, it has the form familiar to us mug. In Europe for a long time 0 was considered a conventional symbol and was not recognized as a number; even in the 17th century Wallis wrote: “Zero is not a number.”

To the question: Is 0 a natural number? given by the author Botos777 the best answer is There are two approaches to defining natural numbers, differing by adding zero to natural numbers. In Russian school programs In mathematics, it is not customary to classify zero as a natural number.
Source: (number)

Answer from rostra[guru]
No.

Answer from Serega[expert]
No

Answer from Neurosis[newbie]
Natural numbers are numbers that arise naturally when counting (both in the sense of enumeration and in the sense of calculus) of objects. There are two approaches to defining natural numbers, differing by adding zero to natural numbers. Accordingly, natural numbers are defined as

* numbers used when listing (numbering) items: 1, 2, 3, ... (first, second, third, etc.). This definition is generally accepted in most countries, including Russia.
* numbers used to indicate the number of items: 0, 1, 2, ... (no items, one item, two items, etc.). This definition was popularized in the works of Bourbaki, where the natural numbers are defined as cardinalities of finite sets.

Negative and non-integer numbers are not natural numbers. The set of natural numbers is usually denoted by math.

There are infinitely many natural numbers. For any natural number there is a natural number greater than it.

Answer from self-preservation[newbie]

Answer from Iuslan Garayev[newbie]
no 0 is not a natural number

Answer from Artyom Artemenko[newbie]
No

Answer from Oliya Nikulina[newbie]
No

Answer from Matvey Sokolov[active]
yes, natural

Answer from Lena[newbie]
not but

Answer from Madina Mikailova[newbie]
0 is not a natural number

Answer from Pavel Mishchenko[newbie]
In the French tradition, dating back to the works of N. Bourbaki, in contrast to other mathematical schools, numbers that express the number of objects in a group are considered natural. Therefore, in this tradition, the smallest natural number is considered to be zero (“0”), and not one, and, accordingly, French mathematicians, unlike others, recognize zero as a natural number. This approach is also motivated by the set-theoretic model of the natural series, in which zero is identified with the empty set (O), and the operation of moving to the next one produces a set consisting of all previous natural numbers (represented by sets):
0 ? O

3? (O, (O), (O, (O)))

4 ? (O, (O), (O, (O)), (O, (O), (O, (O))))

etc. P.S. IN Russian Federation Zero is not considered a natural number.

Answer from Alexandra[newbie]
zero is not a natural number

Answer from Anton Yashagin[newbie]
no is not

Answer from Dima Kovelin[newbie]
there are 2 approaches to natural numbers
counting (numbering) items (first, second, third, ...);
designation of the number of items (no items, one item, two items, ...).
In the first case, zero will not be a natural number.
in the second case it will be.

Answer from Olesya Makushchenko[newbie]

Team "Plusiki"

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Project “Why is zero not a natural number?” Completed by a 5th grade student of the Municipal Educational Institution “Secondary School No. 100 named after. S.E. Tsvetkova” Andreeva Elizaveta Supervisor: teacher of mathematics and computer science Papsheva Victoria Vladimirovna Novokuznetsk, 2010

Project goal: Find proof that zero is not a natural number.

Tasks: Find a definition of what “number” and “natural numbers” are, using dictionaries and the Internet. Find the definition of "zero" using various sources of information. Prove that you cannot divide by zero. Find where "zero" is used.

Team "Plusiki" - participants in the telecommunications project "Numbers Rule the World"

Work plan: Find in the dictionary what “number” is. Find in the dictionary what a “natural number” is. Find the definition of the word "zero". Find out why you can't divide by zero. Where can you find "zero"? Answer the question “Why is zero not a natural number?”

What is a "number"? “A number is a sign expressing quantity, a figure.” “In order to describe a collection of homogeneous objects, it is necessary to indicate which objects and how many there are. For example, there are five pencils on this table, there are seven chairs in this room, there are two hundred and thirty-six books in this closet... The words: five, seven, two hundred and thirty-six... are numbers.” “Number is the basic concept of mathematics - the quantity by which counting is done.”

What is a "number"? “NUMBER, one of the basic concepts of mathematics; originated in ancient times and gradually expanded and generalized... NUMBER, a grammatical category indicating the number of objects denoted by a given word..."

What is a "number"? One apple Three apples Thirty two apples

What is a "natural number"? Natural numbers are numbers that arise naturally when counting. For any natural number, there is only one next one. One is the smallest natural number, since there is no natural number for which it would be the next one.

What is a "natural number"? Natural numbers, numbers that we use when counting: one, two, three, four, ...

What is "zero"? Zero is a number that denotes the absence of something, emptiness. The number 0 means that there are no place units in the number. 102; 150; 1,023; 120 125 ; 105,000

What is "zero"? In mathematics, zero is a real number; adding or subtracting it does not change anything. Multiplying any element by zero gives zero. This is an inherent property of zero, part of its definition. a + 0 = a a – 0 = a a 0 = 0

What is "zero"? Encyclopedic Dictionary Dictionary of Symbols Explanatory Dictionary of the Russian Language, ed. D. N. Ushakova Dahl's Dictionary. Ozhegov's Dictionary New explanatory dictionary of the Russian language. Author T. F. Efremova.

Why can't you divide by zero? If a: b=c , then b c=a . Let's say we divide the number 10 by 0. (a=10, b=0) We must find a number (c-?) that when multiplied by 0 will give 10. (C 0 = 10?) But: 1 0=0 2 0=0... C 0=0... You can’t divide by zero!

Where can you find “zero”? Zero-zero: 1) when indicating time, means: exactly, exactly, not a minute earlier or later. 2) (sport.) denotes a draw outcome of a game or competition. - absolute zero Zero attention. Doesn't pay any attention (not paying attention). Zero without a stick. About someone of no value, of little importance. Be equal to zero Be very insignificant, almost none. Start from scratch Start something without prior preparation.

Where can you find “zero”? Reduce to zero. Make it completely insignificant, destroy it. Cut to zero (zero) Cut bald. Mouth to zero. Meaning: shut your mouth, demand to be silent. example text: Mouth to zero! So if you don’t know, then your mouth is zero. Mouth to zero! Talk to me again here. Zero option. Allegorically about returning to the original state of affairs; about the equal or compromise position of the parties.

Why is zero not a natural number? Initially, the definition of a natural number came from numbering, and the object zero does not exist, so zero is not a natural number.

Why is zero not a natural number? “Do you use zero when counting?”

Literature: Vilenkin N.Ya. Mathematics 5th grade. – M.: Mnemosyne, 2008. http://ru.wikipedia.org/wiki/Zero_(group) http://www.bibliotekar.ru/encSlov/index.htm http://www.dict.t-mm .ru/dal/ http://www.zastavki.com/rus/3D-graphics/wallpaper-7012-9.htm http://elenakosilova.narod.ru/studia3/math/translatio/zero.htm

Thank you for your attention!

Encyclopedic Dictionary Zero - (from the Latin nullus - no) - the number 0, the addition (or subtraction) of which to any number does not change the latter: (a+0) = (-0+a) = a; the product of any number by zero gives zero: a??0 = 0 ? a = 0. Division by zero is impossible. In modern mathematics, the concept of zero (zero element) is considered in algebraic structures of a more general nature (for example, algebraic fields). Return to the list of dictionaries

Dictionary of symbols Zero. Means non-existence, nothingness, unmanifested, boundless, eternal, lack of quality and quantity. In Taoism, zero symbolizes emptiness and nothingness. In Buddhism it is emptiness and immateriality. In the teachings of Kabbalah, zero is limitlessness, limitless light, one. For Pythagoras, zero is the perfect form, monad, source and scope for everything. In Islam it is a symbol of the Essence of the Divine. Zero also represents the Cosmic Egg, the primary androgyne, completeness. Depicted as an empty circle, it indicates both the absence of death and the absolute life found within the circle. Zero has the same symbolism as a circle. When it is depicted as an ellipse, its sides symbolize ascent and descent, unfolding and enfolding. Return to the list of dictionaries

Explanatory Dictionary of the Russian Language, ed. D. N. Ushakova ZERO zero, m. 1. Digital sign: 0. Ѓ Lack of value (mat.). 2. Lowest, bad grade (pre-revolutionary school grade). 3. transfer A person of no importance (colloquial). We honor everyone with zeros, and ourselves with ones. Pushkin. For you, perhaps, he is a nonentity, a zero. Chekhov. Two zeros (colloquial joke) - restroom. Reduce (reduce) to zero - turn into nothing, lose meaning. Return to the list of dictionaries

Dahl's Dictionary. Zero - m. zero; numeral sign meaning nothing, nothing (0); but placed after another digit (on the right), raises it by ten, multiplies it by ten. Count from zero using a thermometer. The zero, the number on top and on the side, means degrees. A zero under a zero, next to the number, means a percentage, from a hundred. Multiply anything with any zero, everything will be zero. Zero sign, zero. Zero flag, naval in signals, meaning zero. The fraction of zero number is the largest. Return to the list of dictionaries

Ozhegov's Dictionary 1) Zero - A real number, the addition of which does not change any number 2) Zero - About insignificant, insignificant, nothing significant person 3) Zero - The digital sign “0”, denoting such a number, as well as, as part of digital notations, the absence of units of any digit Return to the list of dictionaries

New explanatory and word-formative dictionary of the Russian language. Author T. F. Efremova. Zero m. (as well as zero) 1) The digital sign “0”, indicating the absence of a value (added to any number on the right increases it tenfold). 2) An agreed quantity from which the calculation of similar quantities (time, temperature, etc.) begins. 3) The lowest score for assessing knowledge and behavior at school (in the Russian state until 1917). 4) Sth. infinitesimal, insignificant. 5) transfer An insignificant person of no importance. Return to the list of dictionaries

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Telecommunication project in mathematics

"Numbers rule the world"

“The first number is measurable only by one”

Euclid

“Why is zero not a natural number?”

Let's figure it out first: what is a number? Let's turn to dictionaries.

So in explanatory dictionary edited by S.I. Ozhegov and N.Yu. Shvedova can find such a definition of the number. “Number is the basic concept of mathematics - the quantity by which counting is done.”

In the explanatory dictionary of V.I. Dahl,number is a sign expressing quantity, a figure.

But in encyclopedic dictionary edited by F.A. Brockhaus, I.A. Efron can find such a definition of number.“In order to describe a collection of homogeneous objects, it is necessary to indicate which objects and how many there are. For example, there are five pencils on this table, there are seven chairs in this room, there are two hundred and thirty-six books in this closet... The words: five, seven, two hundred and thirty-six... are numbers.”

The Modern Explanatory Dictionary states that “NUMBER, one of the basic concepts of mathematics; originated in ancient times and gradually expanded and generalized... NUMBER, a grammatical category indicating the number of objects denoted by a given word..."

One apple

Three apples

Thirty two apples

So, we figured out what a NUMBER is!

Let's now try to find the definition of natural numbers.

Let's turn to dictionaries again and try to look for the answer on the Internet.

Integers – numbers that arise naturally when counting.

For any natural number, there is only one next one. One is the smallest natural number, since there is no natural number for which it would be the next one.

So, natural numbers, numbers that we use when counting: one, two, three, four, ...

What kind of number is “ZERO”?

Zero is a number that denotes the absence of something, emptiness.

Members of our team "Plusiki" found many more definitions of number zero and digits 0

Digit 0 means the absence of place units in the number notation.

Number zero means more than one.

What is zero ? This is a number due to which not a single example can be counted.

Zero in math – this is a real number, adding which does not change anything.

Multiplying any element by zero gives zero. This is an inherent property zero , part of its definition.

Division by zero is impossible, since it leads to a contradiction. What does the notation 5:0 mean? It means you need to find a number x that, when multiplied by zero the result was 5. There is no such number! It is impossible to find, since x·0=0 is always!

“Zero destroys every other number by which it is multiplied...”

F. Engels

Zero is nothing! Zero apples, that means not a single apple. Pass the zero kilometers means not moving. Run at speed zero kilometers per hour means standing still. Yes, in general, zero does not cause any difficulties. Add or subtract zero - this means nothing will be added and nothing will be taken away.

Zero - not a natural number (this is what I call numbers for numbering objects, although now the fashion has gone to “minus” (underground floors) and “zero” ones, but arithmetic “doesn’t worry” about this. Initially, the definition of a natural number came from numbering, and there is no null item, so zero not a natural number.

So to the question: “Is zero included in the natural numbers?”– You can ask a response question: “Do you use zero when counting?” - "No!". Therefore, zero is not a natural number.

See you again!

Team "Plusiki"

Municipal educational institution "Secondary school No. 100"

G. Novokuznetsk

Where does learning mathematics begin? Yes, that's right, from studying natural numbers and operations with them.Integers (fromlat. naturalis- natural; natural numbers) -numbers that occur naturally when counting (for example, 1, 2, 3, 4, 5, 6, 7, 8, 9...). The sequence of all natural numbers arranged in ascending order is called a natural series.

There are two approaches to defining natural numbers:

1. counting (numbering) items ( first, second, third, fourth, fifth"…);
2. natural numbers are numbers that arise when quantity designation items ( 0 items, 1 item, 2 items, 3 items, 4 items, 5 items ).

In the first case, the series of natural numbers begins with one, in the second - with zero. There is no consensus among most mathematicians on whether the first or second approach is preferable (that is, whether zero should be considered a natural number or not). The overwhelming majority of Russian sources traditionally adopt the first approach. The second approach, for example, is used in the worksNicolas Bourbaki , where the natural numbers are defined aspower finite sets .

Negative and integer (rational , real ,...) numbers are not considered natural numbers.

The set of all natural numbers usually denoted by the symbol N (fromlat. naturalis- natural). The set of natural numbers is infinite, since for any natural number n there is a natural number greater than n.

The presence of zero makes it easier to formulate and prove many theorems in natural number arithmetic, so the first approach introduces the useful concept extended natural range , including zero. The extended series is designated N 0 or Z 0 .

TOclosed operations (operations that do not derive a result from the set of natural numbers) on natural numbers include the following arithmetic operations:

• addition: term + term = sum;
• multiplication: factor × factor = product;
• exponentiation: a b , where a is the base of the degree, b is the exponent. If a and b are natural numbers, then the result will be a natural number.

Additionally, two more operations are considered (from a formal point of view, they are not operations on natural numbers, since they are not defined for allpairs of numbers (sometimes exist, sometimes not)):

• subtraction: minuend - subtrahend = difference. In this case, the minuend must be greater than the subtrahend (or equal to it, if we consider zero to be a natural number)
• division with remainder: dividend / divisor = (quotient, remainder). The quotient p and the remainder r from dividing a by b are defined as follows: a=p*r+b, with 0<=r

It should be noted that the operations of addition and multiplication are fundamental. In particular,