What numbers are natural. Studying the exact subject: natural numbers are what numbers, examples and properties

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 the fingers on the hand, and they said: "I have as many nuts as there are fingers on the hand."

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

Remember!

Integers are numbers, starting with 1, obtained when 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 represent the unit with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then special signs appeared for designating numbers - the forerunners of modern numbers. The numbers we use to write numbers originated in India about 1,500 years ago. The Arabs brought them to Europe, so they are called Arabic numerals.

There are ten digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits can be used to 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 largest 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 value of a digit depends on its place in the notation of a number, 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 next 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 surpasses the number of all atoms (the smallest particles of matter) in the entire universe.

This number has a special name - googol. A googol is a number that has 100 zeros.

Integers- 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. With the help of them, you can write any natural number. This notation is called decimal.

The natural series of numbers can be continued indefinitely. There is no number that would be the last, because one can always be added to the last number and one will get a number that is already greater than the desired one. In this case, we say that there is no greatest number in the natural series.

Digits of natural numbers

In writing any number using numbers, the place on which the number stands in the number is crucial. For example, the number 3 means: 3 units if it comes last in the number; 3 tens if it will be in the number in the penultimate place; 4 hundreds, if she will be in the number in third place from the end.

The last digit means the units digit, the penultimate one - the tens digit, 3 from the end - the hundreds digit.

Single and multiple digits

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

The number 0 stands for zero. Zero is "none".

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

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

Numbers that are not single digits are also called multiple digits.

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 from the right edge make up the units class, the next three the thousands class, the next three the millions class.

A million is a thousand thousand, for the record they use the abbreviation million 1 million = 1,000,000.

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

Write and Read Example

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

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

Read numbers from left to right. In turn, the number of units of each class is called and then the name of the class is added.

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

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General idea of natural numbers

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

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 are natural ways of conveying information.

Consider the basic skills of voicing (reading) and images (writing) of natural numbers.

Decimal notation of a natural number

Recall how the following characters are displayed (we indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . These characters are called numbers.

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

Let us indicate examples of the correct notation of natural numbers: 703, 881, 13, 333, 1023, 7, 500001. The indents between the digits are 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, it is not necessary to have all the digits from the above series. 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 notation 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, among other things, a quantitative meaning. Natural numbers, as a numbering tool, are discussed in the topic of comparing natural numbers.

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

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

Now imagine: Ψ Ψ . We see one object and another object, i.e. in the record 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 entry of a number matches the entry of the digit 0, then such a number is called "zero". Zero is not a natural number, but it is considered together with other natural numbers. Zero means no, 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, which are written using two signs - two digits. In this case, the numbers used can be either the same or different.

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

Consider the meaning of two-digit numbers. We will rely on the quantitative meaning of single-valued natural numbers already known to us.

Let's introduce such concept as "ten".

Imagine a set of objects, which consists of nine and one more. In this case, we can talk about 1 dozen ("one dozen") items. If you imagine one dozen and one more, then we will talk about 2 tens (“two tens”). Adding one more tens to two tens, we get three tens. And so on: continuing to add one dozen, we 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 the natural number, and the number on the right will indicate the number of ones. In the case when the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of natural two-digit numbers. There are 90 of them in total.

Definition 4

Three-digit natural numbers- natural numbers, which are written using three characters - 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-valued natural numbers, we introduce the concept "a hundred".

Definition 5

One hundred (1 hundred) is a set of ten tens. One hundred plus one hundred equals two hundred. Add another hundred and get 3 hundreds. Adding gradually one hundred, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.

Consider the record of a three-digit number itself: the single-digit natural numbers included in it are written one after the other from left to right. The rightmost single digit indicates the number of units; the next one-digit number to the left - by the number of tens; the leftmost single digit is the number of hundreds. If the number 0 is involved in the entry, it indicates the absence of units and / or tens.

So, 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.

Multivalued natural numbers

From all of the above, it is now possible to proceed to the definition of multivalued natural numbers.

Definition 6

Multivalued natural numbers- natural numbers, which are written using two or more characters. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.

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

Similarly 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 tens of thousands, 9 hundreds of thousands and 4 millions.

Summarizing, we examined the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the record of a multi-digit natural number are a designation of the number of units in each of such sets.

Reading natural numbers, classes

In the theory above, we denoted 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 alphabetic notation:

Number	masculine	Feminine	Neuter gender
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	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	to one two Trem 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 About two About three About four Again About six About seven About eight About nine

For competent reading and writing two-digit numbers, you need to learn the data in table 2:

Number	Masculine, feminine and neuter
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	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 Ninety	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 About ninety

To read other natural two-digit numbers, we will use the data from both tables, consider this with an example. Let's say we need to read a natural two-digit 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 union “and” between the words does not need to be pronounced. Suppose we need to use the indicated number 21 in some sentence, indicating the number of objects in the genitive case: "there are no 21 apples." In this case, the pronunciation will sound like this: “there are no twenty-one apples.”

Let's 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	Instrumental case	Prepositional
100 200 300 400 500 600 700 800 900	Hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundreds	Sta two hundred three hundred four hundred five hundred six hundred Seven hundred eight hundred nine hundred	Sta two hundred Tremstam 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 hundreds	Sta two hundred Three hundred four hundred five hundred six hundred seven hundred eight hundred Nine hundred	About a hundred About two hundred About three hundred About four hundred About five hundred About six hundred About seven hundred About eight hundred About nine hundred

To fully read a three-digit number, we also use the data of all the specified tables. For example, given a 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 cases, 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 and forty-three” or in case declension, for example, like this: “no five hundred and forty-three rubles.”

Let's move on to the general principle of reading multi-digit natural numbers: to read a multi-digit number, you need to break 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 extreme right 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 natural numbers consisting of a large number of characters (16, 17 and more) are rarely used in reading, it is quite difficult to perceive them by ear.

For convenience of perception of the record, the classes are separated from each other by a small indent. For example, 31 013 736 , 134 678 , 23 476 009 434 , 2 533 467 001 222 .

Class trillion	Class billion	Class million	Thousand class	Unit class
			134	678
		31	013	736
	23	476	009	434
2	533	467	001	222

To read a multi-digit number, we call in turn the numbers that make it up (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 the three digits 0 are also not pronounced. If one or two digits 0 are present on the left in one class, then they are not used in any way when reading. For example, 054 is read as "fifty-four" or 001 as "one".

Example 1

Let us examine in detail the reading of the number 2 533 467 001 222:

We read the number 2, as a component of the class of trillions - "two";

Adding the name of the class, we get: "two trillion";

We read the following 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 and sixty-seven million”;

In the next class, we see two digits 0 located on the left. According to the above read rules, the 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 can also 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 for the correct reading of 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 it already becomes clear from all of the above, its value depends on the position on which the digit stands in the record of the number. That is, for example, the number 3 in the natural number 314 denotes 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 units 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 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 discharges have their own names, we have already used them above. From right to left, the digits follow: units, tens, hundreds, thousands, tens of thousands, etc.

For convenience of memorization, 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 entry. For example, this table contains the names of all digits for a number with 15 characters. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient for listening.

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

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

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

Definition 8

Lowest (junior) rank any multi-valued natural number is the units digit.

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

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

It follows logically 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 digit is older than the hundreds digit, but younger than the millions digit.

Let us clarify that when solving some practical examples, not the natural number itself is used, but the sum of the bit 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 value of a digit in the number depends on its position in the notation of the number.

According to this definition, we can say that, while studying the natural numbers and the way they are written above, we used the positional number system. Number 10 plays a special place here. We keep counting in tens: ten units make ten, ten tens unite into a hundred, and so on. 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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Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment began her victorious march around the world. Each stage of development introduced something new - elementary counting evolved, transformed into differential and integral calculus, centuries changed, formulas became more and more confusing, and the moment came when "the most complex mathematics began - all numbers disappeared from it." But what was the basis?

The beginning of time

Natural numbers appeared along with the first mathematical operations. Once a spine, two spines, three spines ... They appeared thanks to Indian scientists who deduced the first positional

The word "positionality" means that the location of each digit in a 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 hundreds, while the second only 4. The Indians' innovation was picked up by the Arabs, who brought the numbers to the form that we know now.

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

What is in mathematics? Peano's axioms

The field N is the base field on which elementary mathematics relies. Over time, the fields of integers, rational,

The work of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formality and paved the way for further conclusions that went beyond the field N.

What is a natural number, was previously clarified in simple language, below we will consider a mathematical definition based on Peano's axioms.

One is considered a natural number.
The number that follows a natural number is a natural number.
There is no natural number before one.
If the number b follows both the number c and the number d, then c=d.
The axiom of induction, which in turn shows what a natural number is: if some statement that depends on a parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is also true for n =1 from the field of natural numbers N.

Basic operations for the field of natural numbers

Since the field N became the first for mathematical calculations, both the domains of definition and the ranges of values of a number of operations below refer to it. They are closed and not. The main difference is that closed operations are guaranteed to leave a result within the set N, no matter what 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 kind of numbers are involved in the expression, since it may contradict the main 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 N field;
exponentiation - x y , where x, y are included in the N field.

The remaining operations, the result of which may not exist in the context of the definition "what is a natural number", are the following:

Properties of numbers belonging to the field N

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

The commutative property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or the well-known "the sum does not change from a change in the places of the terms."
The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the field N.
The associative property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the field N.
The associative property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the field N.
distribution property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the field N.

Pythagorean table

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

This multiplication table has undergone a number of changes over time: zero has been removed from it, and numbers from 1 to 10 denote themselves, without taking into account 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 is equal to their product.

In the practice of teaching in recent decades, there has been a need to memorize the Pythagorean table "in order", that is, memorization went first. Multiplication by 1 was excluded because the result was 1 or greater. Meanwhile, in the table with the naked eye, you can see a pattern: 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 we need to take the first one in order to get the desired product. This system is much more convenient than the one practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting using a system based on powers of two.

Subset as 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. A 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 deduce the effect, paving the way for great discoveries.

Definition

Natural numbers are called numbers intended for counting objects. To record natural numbers, 10 Arabic numerals (0–9) are used, which form the basis of the decimal number system generally accepted for mathematical calculations.

Sequence of natural numbers

The natural numbers make up a series starting at 1 and covering the set of all positive integers. Such a sequence consists of numbers 1,2,3, ... . This means that in the natural series:

There is a smallest number and no largest.
Each next number is greater than the previous one by 1 (the exception is the unit itself).
As the numbers go to infinity, they grow indefinitely.

Sometimes 0 is also introduced into a series of natural numbers. This is permissible, and then they talk about extended natural series.

Classes of natural numbers

Each digit of a natural number expresses a certain digit. The last one is always the number of units in the number, the one before it is the number of tens, the third from the end is the number of hundreds, the fourth is the number of thousands, and so on.

in the number 276: 2 hundreds, 7 tens, 6 units
in the number 1098: 1 thousand, 9 tens, 8 ones; the hundreds place is absent here, since it is expressed as zero.

For large and very large numbers, you can see a steady trend (if you examine the number from right to left, that is, from the last digit to the first):

the last three digits in the number are units, tens and hundreds;
the previous three are units, tens and hundreds of thousands;
the three in front of them (i.e. the 7th, 8th and 9th digits of the number, counting from the end) are units, tens and hundreds of millions, etc.

That is, every time we are dealing with three digits, meaning units, tens and hundreds of a larger name. Such groups form classes. And if you have to deal with the first three classes in everyday life more or less often, then others should be listed, because not everyone remembers their names by heart.

The 4th class, following the class of millions and representing numbers of 10-12 digits, is called a billion (or a billion);
5th grade - trillion;
6th grade - quadrillion;
7th grade - quintillion;
8th grade - sextillion;
9th grade - septillion.

Addition of natural numbers

The addition of natural numbers is an arithmetic operation that allows you to get a number that contains as many units as there are in the numbers added together.

The sign of addition is the "+" sign. Added numbers are called terms, the result is called the sum.

Small numbers are added (summed up) orally, in writing such actions are written in a line.

Multi-digit numbers, which are difficult to add in the mind, are usually added in a column. For this, the numbers are written one under the other, aligned with the last digit, that is, they write the units digit under the units digit, the hundreds digit under the hundreds digit, and so on. Next, you need to add the digits in pairs. If the addition of digits occurs with a transition through a ten, then this ten is fixed as a unit above the digit on the left (that is, following it) and is added together with the digits of this digit.

If not 2, but more numbers are added to the column, then when summing up the digits of the category, not 1 dozen, but several, may be redundant. In this case, the number of such tens is transferred to the next digit.

Subtraction of natural numbers

Subtraction is an arithmetic operation, the reverse of addition, which boils down to the fact that, given the amount and one of the terms, you need to find another - an unknown term. The number that is being subtracted from is called the minuend; the number that is being subtracted is the subtrahend. The result of the subtraction is called the difference. The sign that denotes the operation of subtraction is "-".

In the transition to addition, the subtrahend and the difference turn into terms, and the reduced into the sum. Addition usually checks the correctness of the subtraction performed, and vice versa.

Here 74 is the minuend, 18 is the subtrahend, 56 is the difference.

A prerequisite for subtracting natural numbers is the following: the minuend must necessarily be greater than the subtrahend. Only in this case the resulting difference will also be a natural number. If the subtraction action is carried out for an extended natural series, then it is allowed that the minuend is equal to the subtrahend. And the result of subtraction in this case will be 0.

Note: if the subtrahend is equal to zero, then the subtraction operation does not change the value of the minuend.

Subtraction of multi-digit numbers is usually done in a column. Write down the numbers in the same way as for addition. Subtraction is performed for the corresponding digits. If it turns out that the minuend is less than the subtrahend, then one is taken from the previous (located on the left) digit, which, after the transfer, naturally turns into 10. This ten is summed up with the figure of the reduced given digit and then subtracted. Further, when subtracting the next digit, it is necessary to take into account that the reduced has become 1 less.

Product of natural numbers

The product (or multiplication) of natural numbers is an arithmetic operation, which is finding the sum of an arbitrary number of identical terms. To record the operation of multiplication, use the sign "·" (sometimes "×" or "*"). For example: 3 5=15.

The action of multiplication is indispensable when it is necessary to add a large number of terms. For example, if you need to add the number 4 7 times, then multiplying 4 by 7 is easier than doing this addition: 4+4+4+4+4+4+4.

The numbers that are multiplied are called factors, the result of multiplication is the product. Accordingly, the term "work" can, depending on the context, express both the process of multiplication and its result.

Multi-digit numbers are multiplied in a column. For this number is written in the same way as for addition and subtraction. It is recommended to write first (above) which of the 2 numbers, which is longer. In this case, the multiplication process will be simpler, and therefore more rational.

When multiplying in a column, the digits of each of the digits of the second number are sequentially multiplied by the digits of the 1st number, starting from its end. Having found the first such work, they write down the number of units, and keep the number of tens in mind. When multiplying the digit of the 2nd number by the next digit of the 1st number, the number that is kept in mind is added to the product. And again they write down the number of units of the result obtained, and remember the number of tens. When multiplying by the last digit of the 1st number, the number obtained in this way is written down in full.

The results of multiplying the digits of the 2nd digit of the second number are written in the second row, shifting it 1 cell to the right. Etc. As a result, a "ladder" will be obtained. All the resulting rows of numbers should be added (according to the rule of addition in a column). Empty cells should be considered filled with zeros. The resulting sum is the final product.

Note

The product of any natural number by 1 (or 1 by a number) is equal to the number itself. For example: 376 1=376; 1 86=86.
When one of the factors or both factors are equal to 0, then the product is equal to 0. For example: 32·0=0; 0 845=845; 0 0=0.

Division of natural numbers

Division is called an arithmetic operation, with the help of which, according to a known product and one of the factors, it can be found another - unknown - factor. Division is the inverse of multiplication and is used to check if a multiplication has been performed correctly (and vice versa).

The number that is being divided is called the divisible; the number by which it is divided is the divisor; the result of a division is called a quotient. The division sign is ":" (sometimes, less often - "÷").

Here 48 is the dividend, 6 is the divisor, and 8 is the quotient.

Not all natural numbers can be divided among themselves. In this case, division is performed with a remainder. It consists in the fact that for the divisor such a factor is selected so that its product by the divisor would be a number that is as close as possible in value to the dividend, but less than it. The divisor is multiplied by this factor and subtracted from the dividend. The difference will be the remainder of the division. The product of a divisor by a factor is called an incomplete quotient. Attention: the remainder must be less than the selected multiplier! If the remainder is larger, then this means that the multiplier is chosen incorrectly, and it should be increased.

We select a factor for 7. In this case, this number is 5. We find an incomplete quotient: 7 5 \u003d 35. Calculate the remainder: 38-35=3. Since 3<7, то это означает, что число 5 было подобрано верно. Результат деления следует записать так: 38:7=5 (остаток 3).

Multi-digit numbers are divided into a column. To do this, the dividend and divisor are written side by side, separating the divisor with a vertical and horizontal line. In the dividend, the first digit or the first few digits (on the right) are selected, which should be a number that is minimally sufficient for dividing by a divisor (that is, this number must be greater than the divisor). For this number, an incomplete quotient is selected, as described in the rule of division with a remainder. The number of the multiplier used to find the partial quotient is written under the divisor. The incomplete quotient is written under the number that was divided, right-aligned. Find their difference. The next digit of the dividend is demolished by writing it next to this difference. For the resulting number, an incomplete quotient is again found by writing down the figure of the selected factor, next to the previous one under the divisor. Etc. Such actions are performed until the numbers of the dividend run out. After that, the division is considered complete. If the dividend and the divisor are divided entirely (without a remainder), then the last difference will give zero. Otherwise, the remainder number will be returned.

Exponentiation

Exponentiation is a mathematical operation that consists in multiplying an arbitrary number of identical numbers. For example: 2 2 2 2.

Such expressions are written as: a x,

where a is a number multiplied by itself x is the number of such factors.

Prime and composite natural numbers

Any natural number, except 1, can be divided by at least 2 numbers - one and itself. Based on this criterion, natural numbers are divided into prime and composite.

Prime numbers are numbers that are only divisible by 1 and itself. Numbers that are divisible by more than these 2 numbers are called composite numbers. A unit divisible solely by itself is neither prime nor compound.

Numbers are prime: 2,3,5,7,11,13,17,19, etc. Examples of composite numbers: 4 (divisible by 1,2,4), 6 (divisible by 1,2,3,6), 20 (divisible by 1,2,4,5,10,20).

Any composite number can be decomposed into prime factors. The prime factors here are understood as its divisors, which are prime numbers.

An example of factorization into prime factors:

Divisors of natural numbers

A divisor is a number by which a given number can be divided without a remainder.

In accordance with this definition, simple natural numbers have 2 divisors, composite numbers have more than 2 divisors.

Many numbers have common divisors. The common divisor is the number by which the given numbers are divisible without a remainder.

The numbers 12 and 15 have a common divisor 3
The numbers 20 and 30 have common divisors 2,5,10

Of particular importance is the greatest common divisor (GCD). This number, in particular, is useful to be able to find for reducing fractions. To find it, it is required to decompose the given numbers into prime factors and present it as the product of their common prime factors, taken in their smallest powers.

It is required to find the GCD of the numbers 36 and 48.

Divisibility of natural numbers

It is far from always possible to determine “by eye” whether one number is divisible by another without a remainder. In such cases, the corresponding divisibility test is useful, that is, the rule by which in a matter of seconds you can determine whether it is possible to divide numbers without a remainder. The sign "" is used to indicate divisibility.

Least common multiple

This value (denoted LCM) is the smallest number that is divisible by each of the given ones. The LCM can be found for an arbitrary set of natural numbers.

LCM, like GCD, has a significant applied meaning. So, it is LCM that needs to be found by reducing ordinary fractions to a common denominator.

The LCM is determined by decomposing the given numbers into prime factors. For its formation, a product is taken, consisting of each of the occurring (at least for 1 number) prime factors represented to the maximum degree.

It is required to find the LCM of the numbers 14 and 24.

Average

The arithmetic mean of an arbitrary (but finite) number of natural numbers is the sum of all these numbers divided by the number of terms:

The arithmetic mean is some average value for a number set.

The numbers 2,84,53,176,17,28 are given. It is required to find their arithmetic mean.

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