Natural numbers are familiar to humans and intuitive, because they surround us since childhood. In the article below we will give a basic understanding of the meaning of natural numbers and describe the basic skills of writing and reading them. The entire theoretical part will be accompanied by examples.

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General understanding of natural numbers

At a certain stage in the development of mankind, the task of counting certain objects and designating their quantity arose, which, in turn, required finding a tool to solve this problem. They became such a tool natural numbers. The main purpose of natural numbers is also clear - to give an idea of ​​the number of objects or the serial number of a specific object, if we're talking about about the multitude.

It is logical that for a person to use natural numbers, it is necessary to have a way to perceive and reproduce them. So, a natural number can be voiced or depicted, which is natural ways transfer of information.

Let's look at the basic skills of voicing (reading) and representing (writing) natural numbers.

Decimal notation of a natural number

Let us remember how the following characters are represented (we will indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . We call these signs numbers.

Now let’s take it as a rule that when depicting (recording) any natural number, only the numbers indicated without the participation of any other symbols. Let the digits when writing a natural number have the same height, are written one after another in a line and there is always a digit other than zero on the left.

Let us indicate examples of the correct recording of natural numbers: 703, 881, 13, 333, 1,023, 7, 500,001. The spacing between numbers is not always the same; this will be discussed in more detail below when studying the classes of numbers. The given examples show that when writing a natural number, all the digits from the above series do not have to be present. Some or all of them may be repeated.

Definition 1

Records of the form: 065, 0, 003, 0791 are not records of natural numbers, because On the left is the number 0.

The correct recording of a natural number, made taking into account all the described requirements, is called decimal notation of a natural number.

Quantitative meaning of natural numbers

As already mentioned, natural numbers initially carry a quantitative meaning, among other things. Natural numbers, as a numbering tool, are discussed in the topic on comparing natural numbers.

Let's proceed to natural numbers, the entries of which coincide with the entries of digits, i.e.: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .

Let's imagine a certain object, for example, like this: Ψ. We can write down what we see 1 item. The natural number 1 is read as "one" or "one". The term "unit" also has another meaning: something that can be considered as a single whole. If there is a set, then any element of it can be designated as one. For example, out of a set of mice, any mouse is one; any flower from a set of flowers is one.

Now imagine: Ψ Ψ . We see one object and another object, i.e. in the recording it will be 2 items. The natural number 2 is read as “two”.

Further, by analogy: Ψ Ψ Ψ – 3 items (“three”), Ψ Ψ Ψ Ψ – 4 (“four”), Ψ Ψ Ψ Ψ Ψ – 5 (“five”), Ψ Ψ Ψ Ψ Ψ Ψ – 6 (“six”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 7 (“seven”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 8 (“eight”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 9 (“ nine").

From the indicated position, the function of a natural number is to indicate quantities items.

Definition 1

If the record of a number coincides with the record of the number 0, then such a number is called "zero". Zero is not a natural number, but it is considered along with other natural numbers. Zero denotes absence, i.e. zero items means none.

Single digit natural numbers

It is an obvious fact that when writing each of the natural numbers discussed above (1, 2, 3, 4, 5, 6, 7, 8, 9), we use one sign - one digit.

Definition 2

Single digit natural number– a natural number, which is written using one sign – one digit.

There are nine single-digit natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9.

Two-digit and three-digit natural numbers

Definition 3

Two-digit natural numbers- natural numbers, when writing which two signs are used - two digits. In this case, the numbers used can be either the same or different.

For example, the natural numbers 71, 64, 11 are two-digit.

Let's consider what meaning is contained in two-digit numbers. We will rely on the quantitative meaning of single-digit natural numbers that is already known to us.

Let's introduce such a concept as “ten”.

Let's imagine a set of objects that consists of nine and one more. In this case, we can talk about 1 ten (“one dozen”) objects. If you imagine one ten and one more, then we are talking about 2 tens (“two tens”). Adding one more to two tens, we get three tens. And so on: continuing to add one ten at a time, we will get four tens, five tens, six tens, seven tens, eight tens and, finally, nine tens.

Let's look at a two-digit number as a set of single-digit numbers, one of which is written on the right, the other on the left. The number on the left will indicate the number of tens in a natural number, and the number on the right will indicate the number of units. In the case where the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of two-digit natural numbers. There are 90 of them in total.

Definition 4

Three-digit natural numbers- natural numbers, when writing which three signs are used - three digits. The numbers can be different or repeated in any combination.

For example, 413, 222, 818, 750 are three-digit natural numbers.

To understand the quantitative meaning of three-digit natural numbers, we introduce the concept "hundred".

Definition 5

One hundred (1 hundred) is a set consisting of ten tens. A hundred and another hundred make 2 hundreds. Add one more hundred and get 3 hundreds. By gradually adding one hundred at a time, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.

Let's consider the notation of a three-digit number itself: the single-digit natural numbers included in it are written one after another from left to right. The rightmost single digit number indicates the number of units; the next single-digit number to the left is by the number of tens; the leftmost single digit number is in the number of hundreds. If the entry contains the number 0, it indicates the absence of units and/or tens.

Thus, the three-digit natural number 402 means: 2 units, 0 tens (there are no tens that are not combined into hundreds) and 4 hundreds.

By analogy, the definition of four-digit, five-digit, and so on natural numbers is given.

Multi-digit natural numbers

From all of the above, it is now possible to move on to the definition of multi-valued natural numbers.

Definition 6

Multi-digit natural numbers– natural numbers, when writing which two or more characters are used. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.

One thousand is a set that includes ten hundred; one million consists of a thousand thousand; one billion – one thousand million; one trillion – one thousand billion. Even larger sets also have names, but their use is rare.

Similar to the principle above, we can consider any multi-digit natural number as a set of single-digit natural numbers, each of which, being in a certain place, indicates the presence and number of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions , hundreds of millions, billions and so on (from right to left, respectively).

For example, the multi-digit number 4,912,305 contains: 5 units, 0 tens, three hundreds, 2 thousand, 1 ten thousand, 9 hundred thousand and 4 million.

To summarize, we looked at the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the notation of a multi-digit natural number indicate the number of units in each of such sets.

Reading natural numbers, classes

In the theory above, we indicated the names of natural numbers. In Table 1 we indicate how to correctly use the names of single-digit natural numbers in speech and in letter writing:

Number	Masculine	Feminine	Neuter
1 2 3 4 5 6 7 8 9	One Two Three Four Five Six Seven Eight Nine	One Two Three Four Five Six Seven Eight Nine	One Two Three Four Five Six Seven Eight Nine

Number	Nominative case	Genitive	Dative	Accusative case	Instrumental case	Prepositional
1 2 3 4 5 6 7 8 9	One Two Three Four Five Six Seven Eight Nine	One Two Three Four Five Six Semi Eight Nine	Alone Two Three Four Five Six Semi Eight Nine	One Two Three Four Five Six Seven Eight Nine	One Two Three Four Five Six family Eight Nine	About one thing About two About three About four Again About six About seven About eight About nine

To correctly read and write two-digit numbers, you need to memorize the data in Table 2:

Number	Masculine, feminine and neuter gender
10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90	Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety

Number	Nominative case	Genitive	Dative	Accusative case	Instrumental case	Prepositional
10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90	Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety	Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty Sixty Seventy Eighty Ninety	Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty Sixty Seventy Eighty Ninety	Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety	Ten Eleven twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty sixty Seventy Eighty nineteen	About ten About eleven About twelve About thirteen About fourteen About fifteen About sixteen About seventeen About eighteen About nineteen About twenty About thirty Oh magpie About fifty About sixty About seventy About eighty Oh ninety

To read other two-digit natural numbers, we will use the data from both tables; we will consider this with an example. Let's say we need to read the two-digit natural number 21. This number contains 1 unit and 2 tens, i.e. 20 and 1. Turning to the tables, we read the indicated number as “twenty-one”, while the conjunction “and” between the words does not need to be pronounced. Let's say we need to use the specified number 21 in a certain sentence, indicating the number of objects in genitive case: “there are no 21 apples.” In this case, the pronunciation will sound like this: “there are not twenty-one apples.”

Let us give another example for clarity: the number 76, which is read as “seventy-six” and, for example, “seventy-six tons.”

Number	Nominative	Genitive	Dative	Accusative case	Instrumental case	Prepositional
100 200 300 400 500 600 700 800 900	One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred	hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred	hundred Two hundred Three hundred Four hundred Five hundred Six hundred Semistam Eight hundred Nine hundred	One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred	hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred	Oh hundred About two hundred About three hundred About four hundred About five hundred About six hundred About the seven hundred About eight hundred About nine hundred

To fully read a three-digit number, we also use the data from all of the indicated tables. For example, given the natural number 305. This number corresponds to 5 units, 0 tens and 3 hundreds: 300 and 5. Taking the table as a basis, we read: “three hundred and five” or in declension by case, for example, like this: “three hundred and five meters.”

Let's read one more number: 543. According to the rules of the tables, the indicated number will sound like this: “five hundred forty-three” or in declension according to cases, for example, like this: “there are no five hundred forty-three rubles.”

Let's move on to general principle reading multi-digit natural numbers: to read a multi-digit number, you need to divide it from right to left into groups of three digits, and the leftmost group can have 1, 2 or 3 digits. Such groups are called classes.

The rightmost class is the class of units; then the next class, to the left - the class of thousands; further – the class of millions; then comes the class of billions, followed by the class of trillions. The following classes also have a name, but the natural numbers consisting of large quantity characters (16, 17 or more) are rarely used in reading; it is quite difficult to perceive them by ear.

To make the recording easier to read, classes are separated from each other by a small indentation. For example, 31,013,736, 134,678, 23,476,009,434, 2,533,467,001,222.

Class trillion	Class billions	Class millions	Class of thousands	Unit class
			134	678
		31	013	736
	23	476	009	434
2	533	467	001	222

To read a multi-digit number, we call the numbers that make it up one by one (from left to right by class, adding the name of the class). The name of the class of units is not pronounced, and those classes that make up three digits 0 are also not pronounced. If one class contains one or two digits on the left, then they are not used in any way when reading. For example, 054 will be read as “fifty-four” or 001 as “one”.

Example 1

Let's look at the reading of the number 2,533,467,001,222 in detail:

We read the number 2 as a component of the class of trillions - “two”;

By adding the name of the class, we get: “two trillion”;

We read the next number, adding the name of the corresponding class: “five hundred thirty-three billion”;

We continue by analogy, reading the next class to the right: “four hundred sixty-seven million”;

In the next class we see two digits 0 located on the left. According to the above reading rules, digits 0 are discarded and do not participate in reading the record. Then we get: “one thousand”;

We read the last class of units without adding its name - “two hundred twenty-two”.

Thus, the number 2 533 467 001 222 will sound like this: two trillion five hundred thirty-three billion four hundred sixty-seven million one thousand two hundred twenty-two. Using this principle, we will read the other given numbers:

31,013,736 – thirty-one million thirteen thousand seven hundred thirty-six;

134 678 – one hundred thirty-four thousand six hundred seventy-eight;

23 476 009 434 – twenty-three billion four hundred seventy-six million nine thousand four hundred thirty-four.

Thus, the basis correct reading multi-digit numbers is the ability to break a multi-digit number into classes, knowledge of the corresponding names and understanding of the principle of reading two- and three-digit numbers.

As is already clear from all of the above, its value depends on the position at which the digit appears in the notation of a number. That is, for example, the number 3 in the natural number 314 indicates the number of hundreds, namely 3 hundreds. The number 2 is the number of tens (1 ten), and the number 4 is the number of units (4 units). In this case, we will say that the number 4 is in the ones place and is the value of the ones place in the given number. The number 1 is in the tens place and serves as the value of the tens place. The number 3 is located in the hundreds place and is the value of the hundreds place.

Definition 7

Discharge- this is the position of a digit in the notation of a natural number, as well as the value of this digit, which is determined by its position in a given number.

The categories have their own names, we have already used them above. From right to left there are digits: units, tens, hundreds, thousands, tens of thousands, etc.

For ease of remembering, you can use the following table (we indicate 15 digits):

Let’s clarify this detail: the number of digits in a given multi-digit number is the same as the number of characters in the number’s notation. For example, this table contains the names of all digits for a number with 15 digits. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient to hear.

With the help of such a table, it is possible to develop the skill of determining the digit by writing a given natural number into the table so that the rightmost digit is written in the units digit and then in each digit one by one. For example, let’s write the multi-digit natural number 56,402,513,674 like this:

Pay attention to the number 0, located in the tens of millions digit - it means the absence of units of this digit.

Let us also introduce the concepts of the lowest and highest digits of a multi-digit number.

Definition 8

Lowest (junior) rank of any multi-digit natural number – the units digit.

Highest (senior) category of any multi-digit natural number – the digit corresponding to the leftmost digit in the notation of a given number.

So, for example, in the number 41,781: the lowest digit is the ones digit; The highest rank is the rank of tens of thousands.

Logically it follows that it is possible to talk about the seniority of the digits relative to each other. Each subsequent digit, when moving from left to right, is lower (younger) than the previous one. And vice versa: when moving from right to left, each next digit is higher (older) than the previous one. For example, the thousands place is older than the hundreds place, but younger than the millions place.

Let us clarify that when solving some practical examples It is not the natural number itself that is used, but the sum of the digit terms of a given number.

Briefly about the decimal number system

Definition 9

Notation– a method of writing numbers using signs.

Positional number systems– those in which the meaning of a digit in a number depends on its position in the number record.

According to this definition, we can say that, while studying natural numbers and the way they are written above, we used the positional number system. The number 10 plays a special place here. We count in tens: ten units make a ten, ten tens will unite into a hundred, etc. The number 10 serves as the base of this number system, and the system itself is also called decimal.

In addition to it, there are other number systems. For example, computer science uses the binary system. When we keep track of time, we use the sexagesimal number system.

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Natural numbers are one of the oldest mathematical concepts.

In the distant past, people did not know numbers, and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.

The number of objects was compared with parts of the body, for example, with fingers on a hand, and they said: “I have as many nuts as there are fingers on my hand.”

Over time, people realized that five nuts, five goats and five hares have common property- their number is five.

Remember!

Natural numbers- these are numbers, starting from 1, obtained by counting objects.

1, 2, 3, 4, 5…

Smallest natural number — 1 .

Largest natural number does not exist.

When counting, the number zero is not used. Therefore, zero is not considered a natural number.

People learned to write numbers much later than to count. First of all, they began to depict one with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then special signs appeared to designate numbers - the predecessors of modern numbers. The numerals we use to write numbers originated in India approximately 1,500 years ago. The Arabs brought them to Europe, which is why they are called Arabic numerals.

There are ten numbers in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these numbers you can write any natural number.

Remember!

Natural series is the sequence of all natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …

In the natural series, each number is greater than the previous one by 1.

The natural series is infinite; there is no greatest natural number in it.

The counting system we use is called decimal positional.

Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the meaning of a digit depends on its place in the number record, that is, on the digit in which it is written.

Important!

The classes following the billion are named according to the Latin names of numbers. Each subsequent unit contains a thousand previous ones.

• 1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
• 1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
• 1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)

However, physicists have found a number that exceeds the number of all atoms (the smallest particles of matter) in the entire Universe.

This number received a special name - googol. Googol is a number with 100 zeros.

Natural numbers– natural numbers are numbers that are used to count objects. The set of all natural numbers is sometimes called the natural series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, etc.

To write natural numbers, ten digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using them, you can write any natural number. This notation of numbers is called decimal.

The natural series of numbers can be continued indefinitely. There is no such number that would be the last, because you can always add one to the last number and you will get a number that is already greater than the one you are looking for. In this case, they say that there is no greatest number in the natural series.

Places of natural numbers

When writing any number using digits, the place in which the digit appears in the number is critical. For example, the number 3 means: 3 units, if it appears in the last place in the number; 3 tens, if she is in the penultimate place in the number; 4 hundred if she is in third place from the end.

The last digit means the units place, the penultimate digit means the tens place, and the 3 from the end means the hundreds place.

Single and multi-digit numbers

If any digit of a number contains the digit 0, this means that there are no units in this digit.

The number 0 is used to denote the number zero. Zero is “not one”.

Zero is not a natural number. Although some mathematicians think differently.

If a number consists of one digit it is called single-digit, if it consists of two it is called two-digit, if it consists of three it is called three-digit, etc.

Numbers that are not single-digit are also called multi-digit.

Digit classes for reading large natural numbers

To read large natural numbers, the number is divided into groups of three digits, starting from the right edge. These groups are called classes.

The first three digits on the right edge make up the units class, the next three are the thousands class, and the next three are the millions class.

Million – one thousand thousand; the abbreviation million is used for recording. 1 million = 1,000,000.

A billion = a thousand million. For recording, use the abbreviation billion. 1 billion = 1,000,000,000.

Example of writing and reading

This number has 15 units in the class of billions, 389 units in the class of millions, zero units in the class of thousands, and 286 units in the class of units.

This number reads like this: 15 billion 389 million 286.

Read numbers from left to right. Take turns calling the number of units of each class and then adding the name of the class.

The simplest number is natural number. They are used in everyday life for counting objects, i.e. to calculate their number and order.

What is a natural number: natural numbers name the numbers that are used to counting items or to indicate the serial number of any item from all homogeneous items.

Natural numbers- these are numbers starting from one. They are formed naturally when counting.For example, 1,2,3,4,5... -first natural numbers.

Smallest natural number- one. There is no greatest natural number. When counting the number Zero is not used, so zero is a natural number.

Natural number series is the sequence of all natural numbers. Writing natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...

In the natural series, each number is greater than the previous one by one.

How many numbers are there in the natural series? The natural series is infinite; the largest natural number does not exist.

Decimal since 10 units of any digit form 1 unit of the highest digit. Positionally so how the meaning of a digit depends on its place in the number, i.e. from the category where it is written.

Classes of natural numbers.

Any natural number can be written using 10 Arabic numerals:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

To read natural numbers, they are divided, starting from the right, into groups of 3 digits each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andso on. Each of the class digits is called itsdischarge.

Comparison of natural numbers.

Of 2 natural numbers, the smaller is the number that is called earlier when counting. For example, number 7 less 11 (written like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .

Table of digits and classes of numbers.

1st class unit	1st digit of the unit 2nd digit tens 3rd place hundreds
2nd class thousand	1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands
3rd class millions	1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions
4th class billions	1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions

Numbers from 5th grade and above refer to large numbers. Units of the 5th class are trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions. Basic properties of natural numbers. Commutativity of addition . a + b = b + a Commutativity of multiplication. ab = ba Associativity of addition. (a + b) + c = a + (b + c) Associativity of multiplication. Distributivity of multiplication relative to addition: Operations on natural numbers. 4. Division of natural numbers is the inverse operation of multiplication. If b ∙ c = a, That Formulas for division: a: 1 = a a: a = 1, a ≠ 0 0: a = 0, a ≠ 0 (A∙ b) : c = (a:c) ∙ b (A∙ b) : c = (b:c) ∙ a Numerical expressions and numerical equalities. A notation where numbers are connected by action signs is numerical expression. For example, 10∙3+4; (60-2∙5):10. Records where 2 numeric expressions are combined with an equal sign are numerical equalities. Equality has left and right sides. The order of performing arithmetic operations. Adding and subtracting numbers are operations of the first degree, while multiplication and division are operations of the second degree. When numeric expression consists of actions of only one degree, they are performed sequentially from left to right. When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree. When there are parentheses in an expression, the actions in the parentheses are performed first. For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21.

Numbers are an abstract concept. They are a quantitative characteristic of objects and can be real, rational, negative, integer and fractional, as well as natural.

The natural series is usually used when counting, in which quantity notations naturally arise. Acquaintance with counting begins in the very early childhood. What kid avoided funny rhymes that used elements of natural counting? "One, two, three, four, five... The bunny went out for a walk!" or "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the king decided to hang me..."

For any natural number, you can find another one greater than it. This set is usually denoted by the letter N and should be considered infinite in the direction of increase. But this set has a beginning - it is one. Although there are French natural numbers, the set of which also includes zero. But the main distinctive features of both sets is the fact that they do not include either fractional or negative numbers.

The need to count a variety of objects arose in prehistoric times. Then the concept of “natural numbers” was supposedly formed. Its formation occurred throughout the entire process of changing a person’s worldview and the development of science and technology.

However, they could not yet think abstractly. It was difficult for them to understand what the commonality of the concepts of “three hunters” or “three trees” was. Therefore, when indicating the number of people, one definition was used, and when indicating the same number of objects of a different kind, a completely different definition was used.

And it was extremely short. It contained only the numbers 1 and 2, and the count ended with the concepts of “many”, “herd”, “crowd”, “heap”.

Later, a more progressive and broader account was formed. An interesting fact is that there were only two numbers - 1 and 2, and the next numbers were obtained by adding.

An example of this was the information that has reached us about the numerical series of the Australian tribe. They had 1 for the word “Enza”, and 2 for the word “petcheval”. The number 3 therefore sounded like “petcheval-Enza”, and 4 sounded like “petcheval-petcheval”.

Most peoples recognized fingers as the standard of counting. Further development of the abstract concept of “natural numbers” followed the path of using notches on a stick. And then it became necessary to designate a dozen with another sign. The ancient people found our way out - they began to use another stick, on which notches were made to indicate tens.

The ability to reproduce numbers expanded enormously with the advent of writing. At first, numbers were depicted as lines on clay tablets or papyrus, but gradually other writing icons began to be used. This is how Roman numerals appeared.

Much later, they appeared that opened up the possibility of writing numbers with a relatively small set of characters. Today does not amount to special labor write down such huge numbers as the distance between planets and the number of stars. You just have to learn to use degrees.

Euclid in the 3rd century BC in the book “Elements” establishes the infinity of the numerical set, and Archimedes in “Psamita” reveals the principles for constructing the names of arbitrarily large numbers. Almost until the middle of the 19th century, people did not face the need for a clear formulation of the concept of “natural numbers”. The definition was required with the advent of the axiomatic mathematical method.

And in the 70s of the 19th century he formulated a clear definition of natural numbers based on the concept of set. And today we already know that natural numbers are all integers, starting from 1 to infinity. Young children, taking their first step in becoming acquainted with the queen of all sciences - mathematics - begin to study these very numbers.