Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:
This is a natural series of numbers.
Zero is a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.
The sum of natural numbers is a natural number. So, the addition of natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers There is not always a natural number. If for natural numbers a and b
where c is a natural number, it means that a is evenly divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is the natural number by which the first number is evenly divisible.
Every natural number is divisible by 1 and itself.
Simple natural numbers are only divisible by 1 and themselves. Here we mean divided completely. Example, numbers 2; 3; 5; 7 is only divisible by 1 and itself. These are simple natural numbers.
One is not considered a prime number.
Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers consists of one, prime numbers and composite numbers.
The set of natural numbers is denoted by the Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab)c = a(bc);
distributive property of multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are natural numbers, zero and the opposite of natural numbers.
Numbers opposite to natural numbers are negative integers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are integers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
It can be seen from the examples that any integer is a periodic fraction with a period of zero.
Any rational number can be represented as a fraction m/n, where m is an integer and n is a natural number. Let's represent the number 3,(6) from the previous example as such a fraction.
Integers
Natural numbers are those numbers that are used to count various objects or to indicate the serial number of an object among similar or homogeneous ones.
Natural numbers can be written using the first ten digits:
To write simple natural numbers, it is customary to use a positional decimal calculus, 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 make calculations, count objects, determine their quantity, order and number.
We begin to get acquainted with natural numbers from early childhood, so they are familiar and natural for each of us.
General idea of natural numbers
Natural numbers are designed 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 voicing any natural number, we can easily catch it by ear, and having depicted a natural number, we see it.
All natural numbers are arranged in ascending order and form a number series starting with the smallest natural number, which is one.
If we have decided on the smallest natural number, then it will be more difficult with the largest, 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 follows the given number.
A number such as 0 is not a natural number, but only serves to denote the number "zero" and means "none". 0 means the absence of numbers of units of this series in the decimal notation.
All natural numbers are denoted by the capital Latin letter N.
Historical reference for the designation of natural numbers
In ancient times, people did not yet know what a number is and how to count the number of objects. But already then the need arose for counting, and the man figured out how to count the caught fish, the collected berries, etc.
A little later, the ancient man came to the conclusion that the amount he needed was easier to write down. For these purposes, primitive people began to use pebbles, and then sticks, which were preserved in Roman numerals.
The next moment in the development of the calculus system was the use of letters of the alphabet in the notation of some numbers.
The first systems of calculation include the decimal Indian system and the sexagesimal Babylonian.
The modern system of calculus, although called Arabic, is, in fact, one of the variants of the Indian one. True, in its system of calculation there is no number zero, but the Arabs added it, and the system acquired its current form.
Decimal system
We have already met natural numbers and learned how to write them using ten digits. You also already know that writing numbers using signs is called a number system.
The value of a digit in a number entry depends on its position and is called positional. That is, when writing natural numbers, we use the positional calculus.
This system is based on bit depth and decimal. In the decimal 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, the account is kept in tens.
Table of classes and categories:
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 calculus system and is called the decimal calculus system.
Natural numbers and their properties
Natural numbers are used to count objects in life. Any natural number uses the digits $0,1,2,3,4,5,6,7,8,9$
A sequence of natural numbers, each next number in which is $1$ greater than the previous one, forms a natural series , which starts with one (because one is the smallest natural number) and does not have the largest value, i.e. endless.
Zero is not considered a natural number.
Following relationship properties
All properties of natural numbers and operations on them follow from the four properties of sequence relations, which were formulated in $1891$ by D. Peano:
One is a natural number that does not follow any natural number.
Every 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 the natural numbers.
If the record of a natural number consists of one digit, it is called single-digit (for example, $2.6.9$, etc.), if the record consists of two digits, it is called double-digit (for example, $12.18.45$), etc. Similarly. Two-digit, three-digit, four-digit, etc. numbers are called multivalued in mathematics.
Addition property of natural numbers
Commutative property: $a+b=b+a$
The sum does not change when the terms are rearranged
Associative 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, the second term
Adding zero does not change the number, and if you add any number to zero, you get the added number.
subtraction properties
The property of subtracting the sum from the number $a-(b+c) =a-b-c$ if $b+c ≤ a$
In order to subtract the sum from a number, you can first subtract the first term from this number, and then from the resulting difference, the second term
The property of subtracting a number from the sum $(a+b) -c=a+(b-c)$ if $c ≤ b$
To subtract a number from the 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
Multiplication Properties
Displacement $a\cdot b=b\cdot a$
The product of two numbers does not change when the factors are rearranged
Associative $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
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 brackets in the product notation, the multiplication is performed in order from left to right
Properties of multiplication with respect to addition and subtraction
Distributive property of multiplication with respect to addition
$(a+b)\cdot c=ac+bc$
In order to multiply the sum by a number, you can multiply each term by this number and add the resulting products
For example, $5(x+y)=5x+5y$
The distributive property of multiplication with respect to subtraction
$(a-b)\cdot c=ac-bc$
In order to multiply the difference by a number, multiply the minuend and subtracted 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 the three relations $a=b$, $a
The smaller number is the one that appears earlier in the natural series, and the larger one that appears later. Zero is less than any natural number.
Example 1
Compare the numbers $a$ and $555$, if it is known that there is some number $b$, and the following relations hold: $a
Solution: Based on the specified property, because by condition $a
any subset of natural numbers containing at least one number has a smallest number
A subset in mathematics 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 is greater 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 not possible, the numbers are rounded off, that is, they are replaced by close numbers with zeros on the end.
Natural numbers are rounded up 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 hundreds, it is replaced by the nearest number consisting of whole hundreds; such a number should have the digit $0$ in the tens and ones place. Etc
The numbers to which the given is rounded are called the approximate value of the number with an accuracy of the specified digits. For example, if you round the number $564$ to tens, we get that you can round it with a disadvantage and get $560$, or with an excess and get $570$.
Rounding rule for natural numbers
If to the right of the digit to which the number is rounded is the figure $5$ or a figure 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 by zeros
Integers very familiar and natural to us. And this is not surprising, since acquaintance with them begins from the first years of our life at an intuitive level.
The information in this article creates a basic understanding of natural numbers, reveals their purpose, instills the skills of writing and reading natural numbers. For better assimilation of the material, the necessary examples and illustrations are given.
Page navigation.
Natural numbers are a general representation.
The following opinion is not devoid of sound logic: the appearance of the problem of counting objects (first, second, third object, etc.) and the problem of indicating the number of objects (one, two, three objects, etc.) led to the creation of a tool for its solution, this tool was integers.
This proposal shows main purpose of natural numbers- carry information about the number of any items or the serial number of a given item in the considered set of items.
In order for a person to use natural numbers, they must be accessible in some way, both for perception and for reproduction. If you sound 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 to convey and perceive natural numbers.
So let's start acquiring the skills of depicting (writing) and the skills of voicing (reading) natural numbers, while learning their meaning.
Decimal notation for a natural number.
First, we should decide on what we will build on when writing natural numbers.
Let's memorize the images of the following characters (we show them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . The images shown are a record of the so-called numbers. Let's agree right away not to flip, tilt, or otherwise distort the numbers when writing.
Now we agree that only the indicated digits can be present in the notation of any natural number and no other symbols can be present. We 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 indents), and on the left there is a digit that is different from the digit 0 .
Here are some examples of the correct notation of natural numbers: 604 , 777 277 , 81 , 4 444 , 1 001 902 203, 5 , 900 000 (note: the indents between the numbers are not always the same, more on this will be discussed when reviewing). From the above examples, it can be seen that 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.
Entries 014 , 0005 , 0 , 0209 are not records of natural numbers, since there is a digit on the left 0 .
The record of a natural number, performed 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 notation. Let us clarify this: further in the text, 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.
It's time to deal with the quantitative meaning that the recorded natural number carries. The meaning of natural numbers in terms of numbering objects is considered in the article comparison of natural numbers.
Let's start with natural numbers, the entries of which coincide with the entries of the digits, that is, with the numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and 9 .
Imagine that we opened our eyes and saw some object, for example, like this. In this case, we can write what we see 1 subject. 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 adopted another name - " unit».
However, the term "unit" is multi-valued; in addition to the natural number 1 , are called something that is considered as a whole. For example, any one item from their set can be called a unit. For example, any apple out of many apples is one, any flock of birds out of many flocks of birds is also one, and so on.
Now we open our eyes and see: That is, we see one object and another object. In this case, we can write what we see 2 subject. Natural number 2 , reads like " two».
Likewise, - 3
subject (read " three» subject), - 4
(« four"") of the subject, - 5
(« five»), 
- 6
(« six»), 
- 7
(« seven»), - 8
(« eight»), - 9
(« nine”) items.
So, from the considered position, the natural numbers 1 , 2 , 3 , …, 9 indicate amount items.
A number whose notation matches 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 the quantity.
single digit natural numbers.
Obviously, the record of each of the natural numbers 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 consists of one sign - one digit.
Definition.
Single digit natural numbers are natural numbers, the record 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.
Two-digit and three-digit natural numbers.
First, we give a definition of two-digit natural numbers.
Definition.
Two-digit natural numbers- these are natural numbers, the record of which is two characters - two digits (different or the same).
For example, a natural number 45 - two-digit, numbers 10 , 77 , 82 also two-digit 5 490 , 832 , 90 037 - not double digit.
Let's figure out what meaning two-digit numbers carry, while we will start from the quantitative meaning of single-digit natural numbers already known to us.
First, let's introduce the concept ten.
Let's imagine such a situation - we opened our eyes and saw a set consisting of nine objects and one more object. In this case, one speaks of 1 ten (one dozen) items. If one considers together one ten and one more ten, then one speaks of 2 tens (two tens). 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, consider 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 units. Moreover, if there is a digit on the right in the record 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 the amount.
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 united in tens.
Let's answer the question: "How many two-digit natural numbers exist"? Answer: them 90 .
We turn to the definition of three-digit natural numbers.
Definition.
Natural numbers whose notation consists of 3 signs - 3 digits (different or repeated) 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 digits.
To understand the meaning inherent in three-digit natural numbers, we need the concept hundreds.
A set of ten tens is 1 one hundred (one hundred). Hundred and 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, going one after another 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, the next number the number of hundreds. Numbers 0 in the record of a three-digit number means the absence of tens and (or) units.
Thus, a three-digit natural number 812 corresponds 8 hundreds 1 top ten and 2 units; number 305 - three hundred 0 tens, that is, tens not combined into hundreds, no) and 5 units; number 470 - four hundred and seven tens (there are no units that are not combined into tens); number 500 - five hundred (tens not combined into hundreds, and units not combined into tens, no).
Similarly, one can define four-digit, five-digit, six-digit, and so on. natural numbers.
Multivalued natural numbers.
So, we turn to the definition of multi-valued natural numbers.
Definition.
Multivalued natural numbers- these are natural numbers, the record 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 the 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-valued natural numbers?
Let's look at a multi-digit natural number as single-digit natural numbers following one after the other 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, the next is the number of thousands, the next is the number of tens of thousands, the next is hundreds of thousands, the next is the number of millions, the next is the number of tens of millions, the next is hundreds of millions, the next - 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 record of a multi-digit natural number indicate the corresponding number of the above groups.
Reading natural numbers, classes.
We have already mentioned how one-digit natural numbers are read. Let's learn the contents of the following tables by heart.
And how are the other two-digit numbers read?
Let's explain with an example. Reading a 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 just written, and the number 74 we read as: “Seventy-four” (we do not pronounce the union “and”). If you want to read a number 74 in the sentence: "No 74 apples" (genitive case), then it will sound like this: "There are no seventy-four apples." Another example. Number 88 - this is 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 already acquired skills in reading single-digit and double-digit numbers.
Let's take 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 is 800 (eight hundred) and 88 (eighty-eight), we read: "Eight hundred and eighty-eight."
We turn to reading multi-digit numbers.
For reading, the record of a multi-digit natural number is divided, starting from the right, into groups of three digits, while 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 unit class. The next class (from right to left) is called class of thousands, the next class is class of millions, next - class of billions, then goes trillion class. You can give the names of the following classes, but natural numbers, the record 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 splitting 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 recorded natural numbers in a table, according to which it is easy to learn how to read them.
To read a natural number, we call from left to right the numbers that make it up by class 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 record has a digit on the left 0 or two digits 0 , then ignore these numbers 0 and read the number obtained by discarding these digits 0 . For example, 002 read as "two", and 025 - like "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 and eighty-nine";
- add the name of the class, we get "four hundred eighty-nine thousand";
- further in the class of units we see 002 , zeros are on the left, we ignore them, therefore 002 read as "two";
- the unit class name need not be added;
- as a result we have 489 002 - four hundred and 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 , we read "ten";
- add the name of the class, we have "ten million";
- next we see the record 000 in the thousands class, since all three digits are digits 0 , then we skip this class and move on to the next one;
- unit class represents number 501 , which we read "five hundred and one";
- thus, 10 000 501 ten million five hundred and one.
Let's do it without detailed explanations: 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 break 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 value of each digit depends on its position. For example, a natural number 539 corresponds 5 hundreds 3 dozens and 9 units, hence the figure 5 in the number entry 539 defines the number of hundreds, a digit 3 is the number of tens, and the digit 9 - number of units. It is said that the number 9 stands in units digit and number 9 is unit digit value, number 3 stands in tens place and number 3 is tens place value, and the number 5 - in hundreds place and number 5 is hundreds place value.
In this way, discharge- this is, on the one hand, the position of the digit in the notation of a natural number, and on the other hand, the value of this digit, determined by its position.
The ranks have been given names. If you look at the numbers in the record of a natural number from right to left, then the following digits will correspond to them: units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, and so on.
The names of the categories are convenient to remember when they are presented in the form of a table. Let's write a table containing the names of 15 digits.
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 record of which contains up to 15 characters. The following digits also have their own names, but they are very rarely used, so it makes no sense to mention them.
Using the 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 take an example. Let's write a natural number 67 922 003 942 in the table, and the digits and the values of these digits will become clearly visible.
In the record of this number, the digit 2 stands in the units place, digit 4 - in the tens place, digit 9 - in the hundreds place, etc. Pay attention to the numbers 0 , which are in the digits of tens of thousands and hundreds of thousands. Numbers 0 in these digits means the absence of units of these digits.
We should also mention the so-called lowest (lowest) and highest (highest) category of a multivalued natural number. Lower (junior) rank any multi-valued natural number is the units digit. The highest (highest) digit of a natural number is the digit corresponding to the rightmost digit in the record of this number. For example, the least significant digit of the natural number 23004 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 next digit lower (younger) the previous one. For example, the digit of thousands is less than the digit of tens of thousands, especially the digit of thousands is less than the digit of hundreds of thousands, millions, tens of millions, etc. If, in the notation of a natural number, we move in digits from right to left, then each next digit higher (older) the previous one. For example, the hundreds digit is older than the tens digit, and even more so, it is older than the ones digit.
In some cases (for example, when performing addition or subtraction), not the natural number itself is used, but the sum of the bit terms of this natural number.
Briefly about the decimal number system.
So, we got acquainted with natural numbers, with 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 value of a digit in a number entry may or may not depend on its position. Number systems in which the value of a digit in a number entry depends on its position are called positional.
Thus, the natural numbers we have considered and the method of writing them indicate that we are using a positional number system. It should be noted that a special place in this number system has the number 10 . Indeed, the score is kept in tens: ten units are combined into a ten, ten tens are combined into a hundred, ten hundreds 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 it comes to measuring time.
Bibliography.
- Maths. Any textbooks for 5 classes of educational institutions.
