Natural value. What is a natural number? History, scope, properties

Integers is 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!

Integers- 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.

Integers

Natural numbers are those numbers that are used for counting various items or in order to indicate the serial number of any item among similar or homogeneous ones.

You can write natural numbers using the first ten digits:

To write simple natural numbers, it is customary to use the positional decimal number system, where the value of any digit is determined by its place in the record.

Natural numbers are the simplest numbers that we often use in Everyday life. With the help of these numbers we carry out calculations, count objects, determine their quantity, order and number.

We begin to get acquainted with natural numbers from the very beginning. early childhood, therefore they are familiar and natural for each of us.

General understanding of natural numbers

Natural numbers are intended to carry information about the number of objects, their serial number and the set of objects.

A person uses natural numbers, since they are available to him both at the level of perception and at the level of reproduction. When we voice any natural number, we easily catch it by ear, and when we depict a natural number, we see it.

All natural numbers are arranged in ascending order and form number series, starting with the smallest natural number, which is one.

If we have decided on the smallest natural number, then the largest will be more difficult, since such a number does not exist because the series of natural numbers is infinite.

When we add one to a natural number, we end up with the number that comes after the given number.

A number such as 0 is not a natural number, but only serves to designate the number “zero” and means “not a single one.” 0 means that there are no units of this series in decimal notation.

All natural numbers are denoted by the capital Latin letter N.

Historical background on the notation of natural numbers

In ancient times, people did not yet know what a number was or how to count the number of objects. But even then the need for counting arose, and the man came up with a way to count the fish caught, picked berries etc.

A little bit later, ancient man He came to the conclusion that it was easier to write down the quantity he needed. For these purposes primitive people They began to use pebbles, and then sticks, which were preserved in Roman numerals.

The next moment in the development of the number system was the use of letters of the alphabet in the designation of certain numbers.

The first number systems include the Indian decimal system and the Babylonian sexagesimal system.

The modern number system, although called Arabic, is, in fact, one of the Indian variants. True, in its number system there is no number zero, but the Arabs added it, and the system acquired its current form.

Decimal number system

We have already become familiar with natural numbers and learned to write them using ten digits. You also already know that writing numbers using signs is called a number system.

The meaning of a digit in a number depends on its position and is called positional. That is, when writing natural numbers, we use the positional number system.

This system is based on digits and decimals. In the decimal number system, the basis for its construction will be the numbers from 0 to 9.

A special place in such a system is given to the number 10, since basically counting is done in tens.

Table of classes and ranks:

So, for example, 10 units are combined into tens, then into hundreds, thousands, and the like. Therefore, the number 10 is the base of the number system and is called the decimal number system.

Integers They are very familiar and natural to us. And this is not surprising, since acquaintance with them begins from the first years of our life on an intuitive level.

The information in this article creates a basic understanding of natural numbers, reveals their purpose, and instills the skills of writing and reading natural numbers. For better understanding of the material, the necessary examples and illustrations are provided.

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Natural numbers – general representation.

The following opinion is not without sound logic: the emergence of the task of counting objects (first, second, third object, etc.) and the task of indicating the number of objects (one, two, three objects, etc.) led to the creation of a tool for solving it, this were the instrument integers.

From this sentence it is clear the main purpose of natural numbers– carry information about the number of any items or the serial number of a given item in the set of items under consideration.

In order for a person to use natural numbers, they must be in some way accessible to both perception and reproduction. If you voice each natural number, then it will become perceptible by ear, and if you depict a natural number, then it can be seen. These are the most natural ways, allowing you to convey and perceive natural numbers.

So let’s begin to acquire the skills of depicting (writing) and voicing (reading) natural numbers, while learning their meaning.

Decimal notation of a natural number.

First we need to decide what we will start from when writing natural numbers.

Let's remember the images of the following characters (we will show them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . The images shown are a recording of the so-called numbers. Let's immediately agree not to turn over, tilt, or otherwise distort the numbers when recording.

Now let’s agree that in the notation of any natural number only the indicated digits can be present and no other symbols can be present. Let us also agree that the digits in the notation of a natural number have the same height, are arranged in a line one after another (with almost no indentation) and on the left there is a digit other than the digit 0 .

Here are some examples correct entry natural numbers: 604 , 777 277 , 81 , 4 444 , 1 001 902 203, 5 , 900 000 (please note: the indents between numbers are not always the same, more about this will be discussed when reviewing). From the above examples it is clear that the notation of a natural number does not necessarily contain all of the digits 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ; some or all of the digits involved in writing a natural number may be repeated.

Posts 014 , 0005 , 0 , 0209 are not records of natural numbers, since there is a digit on the left 0 .

Writing a natural number, made taking into account all the requirements described in this paragraph, is called decimal notation of a natural number.

Further we will not distinguish between natural numbers and their writing. Let us explain this: further in the text we will use phrases like “given a natural number 582 ", which will mean that a natural number is given, the notation of which has the form 582 .

Natural numbers in the sense of the number of objects.

The time has come to understand the quantitative meaning that the written natural number carries. The meaning of natural numbers in terms of numbering of objects is discussed in the article comparison of natural numbers.

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

Let's imagine that we opened our eyes and saw some object, for example, like this. In this case, we can write down what we see 1 item. The natural number 1 is read as " one"(declension of the numeral “one”, as well as other numerals, we will give in paragraph), for the number 1 another name has been adopted - “ unit».

However, the term “unit” is multi-valued, in addition to the natural number 1 , call something considered as a whole. For example, any one item from their many can be called a unit. For example, any apple from a set of apples is a unit, any flock of birds from a set of flocks of birds is also a unit, etc.

Now we open our eyes and see: . That is, we see one object and another object. In this case, we can write down what we see 2 subject. Natural number 2 , reads " two».

Likewise, - 3 subject (read " three» subject), - 4 (« four") subject, - 5 (« five»), - 6 (« six»), - 7 (« seven»), - 8 (« eight»), - 9 (« nine") items.

So, from the considered position, natural numbers 1 , 2 , 3 , …, 9 indicate quantity items.

A number whose notation coincides with the notation of a digit 0 , called " zero" The number zero is NOT a natural number, however, it is usually considered together with natural numbers. Remember: zero means the absence of something. For example, zero items is not a single item.

In the following paragraphs of the article we will continue to reveal the meaning of natural numbers in terms of indicating quantities.

Single digit natural numbers.

Obviously, the recording of each of the natural numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 consists of one character - one number.

Definition.

Single digit natural numbers– these are natural numbers, the writing of which consists of one sign - one digit.

Let's list all single-digit natural numbers: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . There are nine single-digit natural numbers in total.

Two-digit and three-digit natural numbers.

First, let's define two-digit natural numbers.

Definition.

Two-digit natural numbers– these are natural numbers, the recording of which consists of two signs - two digits (different or the same).

For example, a natural number 45 – two-digit numbers 10 , 77 , 82 also two-digit, and 5 490 , 832 , 90 037 – not two-digit.

Let's figure out what meaning two-digit numbers carry, while we will build on the quantitative meaning of single-digit natural numbers that we already know.

To begin with, let's introduce the concept ten.

Let's imagine this situation - we opened our eyes and saw a set consisting of nine objects and one more object. In this case they talk about 1 ten (one dozen) items. If one ten and another ten are considered together, then they speak of 2 tens (two dozen). If we add another ten to two tens, we will have three tens. Continuing this process, we will get four tens, five tens, six tens, seven tens, eight tens, and finally nine tens.

Now we can move on to the essence of two-digit natural numbers.

To do this, let's look at a two-digit number as two single-digit numbers - one is on the left in the notation of a two-digit number, the other is on the right. The number on the left indicates the number of tens, and the number on the right indicates the number of ones. Moreover, if there is a digit on the right side of a two-digit number 0 , then this means the absence of units. This is the whole point of two-digit natural numbers in terms of indicating quantities.

For example, a two-digit natural number 72 corresponds 7 dozens and 2 units (that is, 72 apples is a set of seven dozen apples and two more apples), and the number 30 answers 3 dozens and 0 there are no units, that is, units that are not combined into tens.

Let’s answer the question: “How many two-digit natural numbers are there?” Answer: them 90 .

Let's move on to the definition of three-digit natural numbers.

Definition.

Natural numbers whose notation consists of 3 signs – 3 numbers (different or repeating) are called three-digit.

Examples of natural three-digit numbers are 372 , 990 , 717 , 222 . Integers 7 390 , 10 011 , 987 654 321 234 567 are not three-digit.

To understand the meaning inherent in three-digit natural numbers, we need the concept hundreds.

The set of ten tens is 1 hundred (one hundred). A hundred and a hundred is 2 hundreds. Two hundred and another hundred is three hundred. And so on, we have four hundred, five hundred, six hundred, seven hundred, eight hundred, and finally nine hundred.

Now let's look at a three-digit natural number as three single-digit natural numbers, following each other from right to left in the notation of a three-digit natural number. The number on the right indicates the number of units, the next number indicates the number of tens, and the next number indicates the number of hundreds. Numbers 0 in writing a three-digit number means the absence of tens and (or) units.

Thus, a three-digit natural number 812 corresponds 8 hundreds, 1 ten and 2 units; number 305 - three hundred ( 0 tens, that is, there are no tens not combined into hundreds) and 5 units; number 470 – four hundreds and seven tens (there are no units not combined into tens); number 500 – five hundreds (there are no tens not combined into hundreds, and no units not combined into tens).

Similarly, one can define four-digit, five-digit, six-digit, etc. natural numbers.

Multi-digit natural numbers.

So, let's move on to the definition of multi-valued natural numbers.

Definition.

Multi-digit natural numbers- these are natural numbers, the notation of which consists of two or three or four, etc. signs. In other words, multi-digit natural numbers are two-digit, three-digit, four-digit, etc. numbers.

Let's say right away that a set consisting of ten hundred is one thousand, a thousand thousand is one million, a thousand million is one billion, a thousand billion is one trillion. A thousand trillion, a thousand thousand trillion, and so on can also be given their own names, but there is no particular need for this.

So what is the meaning behind multi-digit natural numbers?

Let's look at a multi-digit natural number as single-digit natural numbers following one after another from right to left. The number on the right indicates the number of units, the next number is the number of tens, the next is the number of hundreds, then the number of thousands, then the number of tens of thousands, then hundreds of thousands, then the number of millions, then the number of tens of millions, then hundreds of millions, then – the number of billions, then – the number of tens of billions, then – hundreds of billions, then – trillions, then – tens of trillions, then – hundreds of trillions and so on.

For example, a multi-digit natural number 7 580 521 corresponds 1 unit, 2 dozens, 5 hundreds, 0 thousands, 8 tens of thousands, 5 hundreds of thousands and 7 millions.

Thus, we learned to group units into tens, tens into hundreds, hundreds into thousands, thousands into tens of thousands, and so on, and found out that the numbers in the notation of a multi-digit natural number indicate the corresponding number of the above groups.

Reading natural numbers, classes.

We have already mentioned how single-digit natural numbers are read. Let's learn the contents of the following tables by heart.

How are the remaining two-digit numbers read?

Let's explain with an example. Let's read the natural number 74 . As we found out above, this number corresponds to 7 dozens and 4 units, that is, 70 And 4 . We turn to the tables we just recorded, and the number 74 we read it as: “Seventy-four” (we do not pronounce the conjunction “and”). If you need to read a number 74 in the sentence: "No 74 apples" ( Genitive), then it will sound like this: “There are no seventy-four apples.” Another example. Number 88 - This 80 And 8 , therefore, we read: “Eighty-eight.” And here is an example of a sentence: “He is thinking about eighty-eight rubles.”

Let's move on to reading three-digit natural numbers.

To do this we will have to learn a few more new words.

It remains to show how the remaining three-digit natural numbers are read. In this case, we will use the skills we have already acquired in reading single-digit and double-digit numbers.

Let's look at an example. Let's read the number 107 . This number corresponds 1 hundred and 7 units, that is, 100 And 7 . Turning to the tables, we read: “One hundred and seven.” Now let's say the number 217 . This number is 200 And 17 , therefore, we read: “Two hundred and seventeen.” Likewise, 888 - This 800 (eight hundred) and 88 (eighty eight), we read: “Eight hundred eighty eight.”

Let's move on to reading multi-digit numbers.

To read, the record of a multi-digit natural number is divided, starting from the right, into groups of three digits, and in the leftmost such group there may be either 1 , or 2 , or 3 numbers. These groups are called classes. The class on the right is called class of units. The class following it (from right to left) is called class of thousands, next class – million class, next - billion class, next comes trillion class. You can give the names of the following classes, but natural numbers, the notation of which consists of 16 , 17 , 18 etc. signs are usually not read, since they are very difficult to perceive by ear.

Look at examples of dividing multi-digit numbers into classes (for clarity, classes are separated from each other by a small indent): 489 002 , 10 000 501 , 1 789 090 221 214 .

Let's put the written down natural numbers in a table that makes it easy to learn how to read them.

To read a natural number, we call its constituent numbers by class from left to right and add the name of the class. At the same time, we do not pronounce the name of the class of units, and also skip those classes that make up three digits 0 . If the class entry has a number on the left 0 or two digits 0 , then we ignore these numbers 0 and read the number obtained by discarding these numbers 0 . Eg, 002 read as “two”, and 025 - as in “twenty-five.”

Let's read the number 489 002 according to the given rules.

We read from left to right,

• read the number 489 , representing the class of thousands, is “four hundred eighty-nine”;
• add the name of the class, we get “four hundred eighty nine thousand”;
• further in the class of units we see 002 , there are zeros on the left, we ignore them, therefore 002 read as "two";
• there is no need to add the name of the unit class;
• in the end we have 489 002 - “four hundred eighty-nine thousand two.”

Let's start reading the number 10 000 501 .

• On the left in the class of millions we see the number 10 , read “ten”;
• add the name of the class, we have “ten million”;
• then we see the entry 000 in the thousand class, since all three digits are digits 0 , then we skip this class and move on to the next one;
• class of units represents number 501 , which we read “five hundred and one”;
• Thus, 10 000 501 - ten million five hundred one.

Let's do this without detailed explanation: 1 789 090 221 214 - “one trillion seven hundred eighty nine billion ninety million two hundred twenty one thousand two hundred fourteen.”

So, the basis of the skill of reading multi-digit natural numbers is the ability to divide multi-digit numbers into classes, knowledge of the names of classes and the ability to read three-digit numbers.

The digits of a natural number, the value of the digit.

In writing a natural number, the meaning of each digit depends on its position. For example, a natural number 539 corresponds 5 hundreds, 3 dozens and 9 units, therefore, the figure 5 in writing the number 539 determines the number of hundreds, digit 3 – the number of tens, and the digit 9 - number of units. At the same time they say that the figure 9 costs in units digit and number 9 is unit digit value, number 3 costs in tens place and number 3 is tens place value, and the figure 5 - V hundreds place and number 5 is hundreds place value.

Thus, discharge- on the one hand, this is the position of a digit in the notation of a natural number, and on the other hand, the value of this digit, determined by its position.

The categories are given names. If you look at the numbers in the notation of a natural number from right to left, then they will correspond to the following digits: units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, and so on.

It is convenient to remember the names of categories when they are presented in table form. Let's write down a table containing the names of 15 categories.

Note that the number of digits of a given natural number is equal to the number of characters involved in writing this number. Thus, the recorded table contains the names of the digits of all natural numbers, the recording of which contains up to 15 characters. The following ranks also have their own names, but they are very rarely used, so there is no point in mentioning them.

Using a table of digits it is convenient to determine the digits of a given natural number. To do this, you need to write this natural number into this table so that there is one digit in each digit, and the rightmost digit is in the units digit.

Let's give an example. Let's write down a natural number 67 922 003 942 into the table, and the digits and meanings of these digits will become clearly visible.

The number in this number is 2 stands in the units place, digit 4 – in the tens place, digit 9 – in the hundreds place, etc. You should pay attention to the numbers 0 , located in the tens of thousands and hundreds of thousands categories. Numbers 0 in these digits means the absence of units of these digits.

It is also worth mentioning the so-called lowest (junior) and highest (most significant) digit of a multi-digit natural number. Lowest (junior) rank of any multi-digit natural number is the units digit. The highest (most significant) digit of a natural number is the digit corresponding to the rightmost digit in the recording of this number. For example, the low-order digit of the natural number 23,004 is the units digit, and the highest digit is the tens of thousands digit. If in the notation of a natural number we move by digits from left to right, then each subsequent digit lower (younger) previous one. For example, the rank of thousands is lower than the rank of tens of thousands, and even more so the rank of thousands is lower than the rank of hundreds of thousands, millions, tens of millions, etc. If in the notation of a natural number we move by digits from right to left, then each subsequent digit taller (older) previous one. For example, the hundreds digit is older than the tens digit, and even more so, older than the units digit.

In some cases (for example, when performing addition or subtraction), it is not the natural number itself that is used, but the sum of the digit terms of this natural number.

Briefly about the decimal number system.

So, we got acquainted with natural numbers, the meaning inherent in them, and the way to write natural numbers using ten digits.

In general, the method of writing numbers using signs is called number system. The meaning of a digit in a number notation may or may not depend on its position. Number systems in which the value of a digit in a number depends on its position are called positional.

Thus, the natural numbers we examined and the method of writing them indicate that we use a positional number system. It should be noted that the number has a special place in this number system 10 . Indeed, counting is done in tens: ten units are combined into a ten, a dozen tens are combined into a hundred, a dozen hundreds are combined into a thousand, and so on. Number 10 called basis given number system, and the number system itself is called decimal.

In addition to the decimal number system, there are others, for example, in computer science the binary positional number system is used, and we encounter the sexagesimal system when we're talking about about measuring time.

Bibliography.

• Mathematics. Any textbooks for 5th grade of general education institutions.

Natural numbers and their properties

Natural numbers are used to count objects in life. When writing any natural number, the numbers $0,1,2,3,4,5,6,7,8,9$ are used.

A sequence of natural numbers, each next number in which is $1$ greater than the previous one, forms a natural series, which begins with one (since one is the smallest natural number) and has no highest value, i.e. infinite.

Zero is not considered a natural number.

Properties of the succession relation

All properties of natural numbers and operations on them follow from four properties of succession relations, which were formulated in 1891 by D. Peano:

One is a natural number that does not follow any natural number.

Each natural number is followed by one and only one number

Every natural number other than $1$ follows one and only one natural number

The subset of natural numbers containing the number $1$, and together with each number the number following it, contains all natural numbers.

If the entry of a natural number consists of one digit, it is called single-digit (for example, $2,6.9$, etc.), if the entry consists of two digits, it is called double-digit (for example, $12,18,45$), etc. Similarly. Two-digit, three-digit, four-digit, etc. In mathematics, numbers are called multi-valued.

Property of addition of natural numbers

Commutative property: $a+b=b+a$

The sum does not change when the terms are rearranged

Combinative property: $a+ (b+c) =(a+b) +c$

To add the sum of two numbers to a number, you can first add the first term, and then, to the resulting sum, add the second term

Adding zero does not change the number, and if you add any number to zero, you get the added number.

Properties of Subtraction

Property of subtracting a sum from a number $a-(b+c) =a-b-c$ if $b+c ≤ a$

In order to subtract a sum from a number, you can first subtract the first term from this number, and then the second term from the resulting difference.

The property of subtracting a number from the sum $(a+b) -c=a+(b-c)$ if $c ≤ b$

To subtract a number from a sum, you can subtract it from one term and add another term to the resulting difference.

If you subtract zero from a number, the number will not change

If you subtract it from the number itself, you get zero

Properties of Multiplication

Communicative $a\cdot b=b\cdot a$

The product of two numbers does not change when the factors are rearranged

Conjunctive $a\cdot (b\cdot c)=(a\cdot b)\cdot c$

To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor

When multiplied by one, the product does not change $m\cdot 1=m$

When multiplied by zero, the product is zero

When there are no parentheses in the product notation, multiplication is performed in order from left to right

Properties of multiplication relative to addition and subtraction

Distributive property of multiplication relative to addition

$(a+b)\cdot c=ac+bc$

In order to multiply a sum by a number, you can multiply each term by this number and add the resulting products

For example, $5(x+y)=5x+5y$

Distributive property of multiplication relative to subtraction

$(a-b)\cdot c=ac-bc$

In order to multiply the difference by a number, multiply the minuend and subtrahend by this number and subtract the second from the first product

For example, $5(x-y)=5x-5y$

Comparison of natural numbers

For any natural numbers $a$ and $b$, only one of three relations can be satisfied: $a=b$, $a

The number that appears earlier in the natural series is considered smaller, and the number that appears later is considered larger. Zero is less than any natural number.

Example 1

Compare the numbers $a$ and $555$, if it is known that there is a certain number $b$, and the following relations hold: $a

Solution: Based on the specified property, because by condition $a

in any subset of natural numbers containing at least one number there is a smallest number

In mathematics, a subset is a part of a set. A set is said to be a subset of another if each element of the subset is also an element of the larger set

Often, to compare numbers, they find their difference and compare it with zero. If the difference is greater than $0$, but the first number more than the second, if the difference is less than $0$, then the first number is less than the second.

Rounding natural numbers

When full precision is not needed or is not possible, numbers are rounded, that is, they are replaced by close numbers with zeros at the end.

Natural numbers are rounded to tens, hundreds, thousands, etc.

When rounding a number to tens, it is replaced by the nearest number consisting of whole tens; such a number has the digit $0$ in the units place

When rounding a number to the nearest hundred, it is replaced by the nearest number consisting of whole hundreds; such a number must have the digit $0$ in the tens and ones place. Etc

The numbers to which this is rounded are called the approximate value of the number with an accuracy of the indicated digits. For example, if you round the number $564$ to tens, we find that you can round it down and get $560$, or with an excess and get $570$.

Rule for rounding natural numbers

If to the right of the digit to which the number is rounded there is a digit $5$ or a digit greater than $5$, then $1$ is added to the digit of this digit; otherwise this figure is left unchanged

All digits located to the right of the digit to which the number is rounded are replaced with zeros

Integers– numbers that are used to count objects . Any natural number can be written using ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This type of number is called decimal

The sequence of all natural numbers is called natural next to .

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...

The most small natural number is one (1). In the natural series, each next number is 1 greater than the previous one. Natural series endless, there is no largest number in it.

The meaning of a digit depends on its place in the number record. For example, the number 4 means: 4 units if it is on last place in writing the number (in units place); 4 ten, if she is in second to last place (in the tens place); 4 hundreds, if she is in third place from the end (V hundreds place).

The number 0 means absence of units of this category in the decimal notation of a number. It also serves to designate the number “ zero" This number means "none". The score 0:3 in a football match means that the first team did not score a single goal against the opponent.

Zero do not include to natural numbers. And indeed, counting objects never starts from scratch.

If the notation of a natural number consists of one sign – one digit, then it is called unambiguous. Those. unambiguousnatural number– a natural number, the notation of which consists of one sign – one digit. For example, the numbers 1, 6, 8 are single digits.

Double digitnatural number– a natural number, the notation of which consists of two characters – two digits.

For example, the numbers 12, 47, 24, 99 are two-digit numbers.

Also according to the number of characters in given number give names to other numbers:

numbers 326, 532, 893 – three-digit;

numbers 1126, 4268, 9999 – four-digit etc.

Two-digit, three-digit, four-digit, five-digit, etc. numbers are called multi-digit numbers .

To read multi-digit numbers, they are divided, starting from the right, into groups of three digits each (the leftmost group may consist of one or two digits). These groups are called classes.

Million– this is a thousand thousand (1000 thousand), it is written 1 million or 1,000,000.

Billion- that's 1000 million. It is written as 1 billion or 1,000,000,000.

The first three digits on the right make up the class of units, the next three – the class of thousands, then come the classes of millions, billions, etc. (Fig. 1).

Rice. 1. Millions class, thousands class and units class (from left to right)

The number 15389000286 is written in the bit grid (Fig. 2).

Rice. 2. Bit grid: number 15 billion 389 million 286

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