What is a natural number examples. Natural numbers. Natural number series
Natural numbers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. These are the numbers:
This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite number of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to specify it, because there is an infinite number of natural numbers.
The sum of natural numbers is a natural number. So, adding 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 the natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers is not always a natural number. If for natural numbers a and b
where c is a natural number, this means that a is 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 a natural number by which the first number is divisible by a whole.
Every natural number is divisible by one and itself.
Prime natural numbers are divisible only by one and themselves. Here we mean divided entirely. Example, numbers 2; 3; 5; 7 is only divisible by one 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;
Integers
Integers are the natural numbers, zero, and the opposites of the natural numbers.
The opposite of 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 whole numbers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
From the examples it is clear that any integer is a periodic fraction with period 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 imagine the number 3,(6) from the previous example as such a fraction.
In mathematics, there are several different sets of numbers: real, complex, integer, rational, irrational, ... In our everyday life We most often use natural numbers, since we encounter them when counting and when searching, designating the number of objects.
What numbers are called natural numbers?
From ten digits you can write absolutely any existing sum of classes and ranks. Natural values are considered to be those which are used:
- When counting any objects (first, second, third, ... fifth, ... tenth).
- When indicating the number of items (one, two, three...)
N values are always integer and positive. There is no largest N because the set of integer values is unlimited.
Attention! Natural numbers are obtained when counting objects or when indicating their quantity.
Absolutely any number can be decomposed and presented in the form of digit terms, for example: 8.346.809=8 million+346 thousand+809 units.
Set N
The set N is in the set real, integer and positive. On the diagram of sets, they would be located in each other, since the set of natural ones is part of them.
The set of natural numbers is denoted by the letter N. This set has a beginning, but no end.
There is also an extended set N, where zero is included.
Smallest natural number
In most math schools, the smallest value of N is considered a unit, since the absence of objects is considered emptiness.
But in foreign mathematical schools, for example in French, it is considered natural. The presence of zero in the series makes the proof easier some theorems.
A series of values N that includes zero is called extended and is denoted by the symbol N0 (zero index).
Series of natural numbers
N series is a sequence of all N sets of digits. This sequence has no end.
The peculiarity of the natural series is that the next number will differ by one from the previous one, that is, it will increase. But the meanings cannot be negative.
Attention! For ease of counting, there are classes and categories:
- Units (1, 2, 3),
- Tens (10, 20, 30),
- Hundreds (100, 200, 300),
- Thousands (1000, 2000, 3000),
- Tens of thousands (30,000),
- Hundreds of thousands (800.000),
- Millions (4000000), etc.
All N
All N are in the set of real, integer, non-negative values. They are theirs integral part.
These values go to infinity, they can belong to the classes of millions, billions, quintillions, etc.
For example:
- Five apples, three kittens,
- Ten rubles, thirty pencils,
- One hundred kilograms, three hundred books,
- A million stars, three million people, etc.
Sequence in N
In different mathematical schools you can find two intervals to which the sequence N belongs:
from zero to plus infinity, including ends, and from one to plus infinity, including ends, that is, everything positive integer answers.
N sets of digits can be either even or odd. Let's consider the concept of oddity.
Odd (any odd number ends in the numbers 1, 3, 5, 7, 9.) with two have a remainder. For example, 7:2=3.5, 11:2=5.5, 23:2=11.5.
What does even N mean?
Any even sums of classes end in numbers: 0, 2, 4, 6, 8. When even N is divided by 2, there will be no remainder, that is, the result is the whole answer. For example, 50:2=25, 100:2=50, 3456:2=1728.
Important! A number series of N cannot consist only of even or odd values, since they must alternate: even is always followed by odd, followed by even again, etc.
Properties N
Like all other sets, N has its own special properties. Let's consider the properties of the N series (not extended).
- The value that is the smallest and that does not follow any other is one.
- N represent a sequence, that is, one natural value follows another(except for one - it is the first).
- When we perform computational operations on N sums of digits and classes (add, multiply), then the answer it always turns out natural meaning.
- Permutation and combination can be used in calculations.
- Each subsequent value cannot be less than the previous one. Also in the N series the following law will apply: if the number A is less than B, then in the number series there will always be a C for which the equality holds: A+C=B.
- If we take two natural expressions, for example A and B, then one of the expressions will be true for them: A = B, A is greater than B, A is less than B.
- If A is less than B, and B is less than C, then it follows that that A is less than C.
- If A is less than B, then it follows that: if we add the same expression (C) to them, then A + C is less than B + C. It is also true that if these values are multiplied by C, then AC is less than AB.
- If B is greater than A, but less than C, then it is true: B-A is less than C-A.
Attention! All of the above inequalities are also valid in the opposite direction.
What are the components of multiplication called?
In many simple and even complex problems, finding the answer depends on the students’ skills
The simplest number is natural number. They are used in everyday life for counting objects, i.e. to calculate their number and order.
What is a natural number: natural numbers name the numbers that are used to counting items or to indicate the serial number of any item from all homogeneous items.
Natural numbersare numbers starting from one. They are formed naturally when counting.For example, 1,2,3,4,5... -first natural numbers.
Smallest natural number- one. There is no greatest natural number. When counting the number Zero is not used, so zero is a natural number.
Natural number series is the sequence of all natural numbers. Writing natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...
In the natural series, each number is greater than the previous one by one.
How many numbers are there in the natural series? The natural series is infinite; the largest natural number does not exist.
Decimal since 10 units of any digit form 1 unit of the highest digit. Positionally so how the meaning of a digit depends on its place in the number, i.e. from the category where it is written.
Classes of natural numbers.
Any natural number can be written using 10 Arabic numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
To read natural numbers, they are divided, starting from the right, into groups of 3 digits each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andso on. Each of the class digits is called itsdischarge.
Comparison of natural numbers.
Of 2 natural numbers, the smaller is the number that is called earlier when counting. For example, number 7 less 11 (write like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .
Table of digits and classes of numbers.
|
1st class unit |
1st digit of the unit 2nd digit tens 3rd place hundreds |
|
2nd class thousand |
1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands |
|
3rd class millions |
1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions |
|
4th class billions |
1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions |
|
Numbers from 5th grade and above are considered large numbers. Units of the 5th class are trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions. Basic properties of natural numbers.
Operations on natural numbers. 4. Division of natural numbers is the inverse operation of multiplication. If b ∙ c = a, That
Formulas for division: a: 1 = a a: a = 1, a ≠ 0 0: a = 0, a ≠ 0 (A∙ b) : c = (a:c) ∙ b (A∙ b) : c = (b:c) ∙ a Numerical expressions and numerical equalities. A notation where numbers are connected by action signs is numerical expression. For example, 10∙3+4; (60-2∙5):10. Records where 2 numeric expressions are combined with an equal sign are numerical equalities. Equality has left and right sides. The order of performing arithmetic operations. Adding and subtracting numbers are operations of the first degree, while multiplication and division are operations of the second degree. When a numerical expression consists of actions of only one degree, they are performed sequentially from left to right. When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree. When there are parentheses in an expression, the actions in the parentheses are performed first. For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21. |
Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment her victorious march around the world began. Each stage of development introduced something new - elementary counting evolved, transformed into differential and integral calculus, centuries passed, 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 began
Natural numbers appeared along with the first mathematical operations. One spine, two spines, three spines... They appeared thanks to Indian scientists who developed the first positional
The word “positionality” means that the location of each digit in a number is strictly defined and corresponds to its rank. 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 Indian 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 number underlies the creation of the world along with the basic 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 integers and positive: 1, 2, 3, … + ∞. Zero is excluded. Used primarily to count items and indicate order.
What is it in mathematics? Peano's axioms
Field N is the basic one on which elementary mathematics is based. Over time, fields of integers, rational,
The work of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formality and prepared the way for further conclusions that went beyond the field area N.
What a natural number is was clarified earlier in simple language; below we will consider the mathematical definition based on the Peano 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.
- An 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 field N was the first for mathematical calculations, both the domains of definition and the ranges of values of a number of operations below belong to it. They may be closed or not. The main difference is that closed operations are guaranteed to leave the result within the set N, regardless of what numbers are involved. It is enough that they are natural. The outcome of other numerical interactions is no longer so clear 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 N field;
- 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 of “what is a natural number,” are as follows:
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 if the places of the terms are changed.”
- The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the N field.
- The combinational property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the N field.
- The matching property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the N field.
- distributive property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the N field.
Pythagorean table
One of the first steps in students’ knowledge of the entire structure of elementary mathematics after they have understood for themselves which numbers are called natural numbers is the Pythagorean table. It can be considered not only from the point of view of science, but also as a most 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 represent themselves, without taking into account orders (hundreds, thousands...). It is a table in which the row and column headings are numbers, and the contents of the cells where they intersect are 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 began first. Multiplication by 1 was excluded because the result was a multiplier of 1 or greater. Meanwhile, in the table with the naked eye you can notice a pattern: the product of numbers increases 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 obtain the desired product. This system is much more convenient than the one that was practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting by using a system that was 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 any less valuable in science. Natural number is the first thing a child learns when studying himself and the world around him. One finger, two fingers... Thanks to it, a person develops logical thinking, as well as the ability to determine the cause and deduce the effect, paving the way for great discoveries.
Natural numbers are familiar to humans and intuitive, because they surround us since childhood. In the article below we will give a basic understanding of the meaning of natural numbers and describe the basic skills of writing and reading them. The entire theoretical part will be accompanied by examples.
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General understanding of natural numbers
At a certain stage in the development of mankind, the task of counting certain objects and designating their quantity arose, which, in turn, required finding a tool to solve this problem. Natural numbers became such a tool. It is also clear that the main purpose of natural numbers is to give an idea of the number of objects or the serial number of a specific 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. Thus, a natural number can be voiced or depicted, which are natural ways of transmitting information.
Let's look at the basic skills of voicing (reading) and representing (writing) natural numbers.
Decimal notation of a natural number
Let us remember how the following characters are represented (we will indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . We call these signs numbers.
Now let’s take it as a rule that when depicting (recording) any natural number, only the indicated numbers 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 another in a line and there is always a digit other than zero on the left.
Let us indicate examples of the correct recording of natural numbers: 703, 881, 13, 333, 1,023, 7, 500,001. The spacing between numbers is not always the same; this will be discussed in more detail below when studying the classes of numbers. The given examples show that when writing a natural number, all the digits from the above series do not have to be present. Some or all of them may be repeated.
Definition 1
Records of the form: 065, 0, 003, 0791 are not records of natural numbers, because On the left is the number 0.
The correct recording of a natural number, made taking into account all the described requirements, is called decimal notation of a natural number.
Quantitative meaning of natural numbers
As already mentioned, natural numbers initially carry a quantitative meaning, among other things. Natural numbers, as a numbering tool, are discussed in the topic on comparing natural numbers.
Let's proceed to natural numbers, the entries of which coincide with the entries of digits, i.e.: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .
Let's imagine a certain object, for example, like this: Ψ. We can write down what we see 1 item. The natural number 1 is read as "one" or "one". The term "unit" also has another meaning: something that can be considered as a single whole. If there is a set, then any element of it can be designated as one. For example, out of a set of mice, any mouse is one; any flower from a set of flowers is one.
Now imagine: Ψ Ψ . We see one object and another object, i.e. in the recording it will be 2 items. The natural number 2 is read as “two”.
Further, by analogy: Ψ Ψ Ψ – 3 items (“three”), Ψ Ψ Ψ Ψ – 4 (“four”), Ψ Ψ Ψ Ψ Ψ – 5 (“five”), Ψ Ψ Ψ Ψ Ψ Ψ – 6 (“six”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 7 (“seven”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 8 (“eight”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 9 (“ nine").
From the indicated position, the function of a natural number is to indicate quantities items.
Definition 1
If the record of a number coincides with the record of the number 0, then such a number is called "zero". Zero is not a natural number, but it is considered along with other natural numbers. Zero denotes absence, i.e. zero items means none.
Single digit natural numbers
It is an obvious fact that when writing each of the natural numbers discussed above (1, 2, 3, 4, 5, 6, 7, 8, 9), we use one sign - one digit.
Definition 2
Single digit natural number– a natural number, which is written using one sign – one digit.
There are nine single-digit natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Two-digit and three-digit natural numbers
Definition 3Two-digit natural numbers- natural numbers, when writing which two signs are used - two digits. In this case, the numbers used can be either the same or different.
For example, the natural numbers 71, 64, 11 are two-digit.
Let's consider what meaning is contained in two-digit numbers. We will rely on the quantitative meaning of single-digit natural numbers that is already known to us.
Let's introduce such a concept as “ten”.
Let's imagine a set of objects that consists of nine and one more. In this case, we can talk about 1 ten (“one dozen”) objects. If you imagine one ten and one more, then we are talking about 2 tens (“two tens”). Adding one more to two tens, we get three tens. And so on: continuing to add one ten at a time, we will get four tens, five tens, six tens, seven tens, eight tens and, finally, nine tens.
Let's look at a two-digit number as a set of single-digit numbers, one of which is written on the right, the other on the left. The number on the left will indicate the number of tens in a natural number, and the number on the right will indicate the number of units. In the case where the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of two-digit natural numbers. There are 90 of them in total.
Definition 4
Three-digit natural numbers- natural numbers, when writing which three signs are used - three digits. The numbers can be different or repeated in any combination.
For example, 413, 222, 818, 750 are three-digit natural numbers.
To understand the quantitative meaning of three-digit natural numbers, we introduce the concept "hundred".
Definition 5
One hundred (1 hundred) is a set consisting of ten tens. A hundred and another hundred make 2 hundreds. Add one more hundred and get 3 hundreds. By gradually adding one hundred at a time, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.
Let's consider the notation of a three-digit number itself: the single-digit natural numbers included in it are written one after another from left to right. The rightmost single digit number indicates the number of units; the next single-digit number to the left is by the number of tens; the leftmost single digit number is in the number of hundreds. If the entry contains the number 0, it indicates the absence of units and/or tens.
Thus, the three-digit natural number 402 means: 2 units, 0 tens (there are no tens that are not combined into hundreds) and 4 hundreds.
By analogy, the definition of four-digit, five-digit, and so on natural numbers is given.
Multi-digit natural numbers
From all of the above, it is now possible to move on to the definition of multi-valued natural numbers.
Definition 6
Multi-digit natural numbers– natural numbers, when writing which two or more characters are used. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.
One thousand is a set that includes ten hundred; one million consists of a thousand thousand; one billion – one thousand million; one trillion – one thousand billion. Even larger sets also have names, but their use is rare.
Similar to the principle above, we can consider any multi-digit natural number as a set of single-digit natural numbers, each of which, being in a certain place, indicates the presence and number of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions , hundreds of millions, billions and so on (from right to left, respectively).
For example, the multi-digit number 4,912,305 contains: 5 units, 0 tens, three hundreds, 2 thousand, 1 ten thousand, 9 hundred thousand and 4 million.
To summarize, we looked at the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the notation of a multi-digit natural number are a designation of the number of units in each of such sets.
Reading natural numbers, classes
In the theory above, we indicated the names of natural numbers. In Table 1 we indicate how to correctly use the names of single-digit natural numbers in speech and in letter writing:
| Number | Masculine | Feminine | Neuter |
|
1 |
One |
One |
One |
| Number | Nominative case | Genitive | Dative | Accusative case | Instrumental case | Prepositional |
| 1 2 3 4 5 6 7 8 9 |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Semi Eight Nine |
Alone Two Three Four Five Six Semi Eight Nine |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Family Eight Nine |
About one thing About two About three About four Again About six About seven About eight About nine |
To correctly read and write two-digit numbers, you need to memorize the data in Table 2:
| Number |
Masculine, feminine and neuter gender |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
| Number | Nominative case | Genitive | Dative | Accusative case | Instrumental case | Prepositional |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty Sixty Seventy Eighty Ninety |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten Eleven twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty sixty Seventy Eighty nineteen |
About ten About eleven About twelve About thirteen About fourteen About fifteen About sixteen About seventeen About eighteen About nineteen About twenty About thirty Oh magpie About fifty About sixty About seventy About eighty Oh ninety |
To read other two-digit natural numbers, we will use the data from both tables; we will consider this with an example. Let's say we need to read the two-digit natural number 21. This number contains 1 unit and 2 tens, i.e. 20 and 1. Turning to the tables, we read the indicated number as “twenty-one”, while the conjunction “and” between the words does not need to be pronounced. Let's say we need to use the indicated number 21 in a certain 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 not twenty-one apples.”
Let us give another example for clarity: the number 76, which is read as “seventy-six” and, for example, “seventy-six tons.”
| Number | Nominative | Genitive | Dative | Accusative case | Instrumental case | Prepositional |
| 100 200 300 400 500 600 700 800 900 |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Semistam Eight hundred Nine hundred |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
Oh hundred About two hundred About three hundred About four hundred About five hundred About six hundred About the seven hundred About eight hundred About nine hundred |
To fully read a three-digit number, we also use the data from all of the indicated tables. For example, given the natural number 305. This number corresponds to 5 units, 0 tens and 3 hundreds: 300 and 5. Taking the table as a basis, we read: “three hundred and five” or in declension by case, for example, like this: “three hundred and five meters.”
Let's read one more number: 543. According to the rules of the tables, the indicated number will sound like this: “five hundred forty-three” or in declension according to cases, for example, like this: “there are no five hundred forty-three rubles.”
Let's move on to the general principle of reading multi-digit natural numbers: to read a multi-digit number, you need to divide it from right to left into groups of three digits, and the leftmost group can have 1, 2 or 3 digits. Such groups are called classes.
The rightmost class is the class of units; then the next class, to the left - the class of thousands; further – the class of millions; then comes the class of billions, followed by the class of trillions. The following classes also have a name, but natural numbers consisting of a large number of characters (16, 17 and more) are rarely used in reading, and it is quite difficult to perceive them by ear.
To make the recording easier to read, classes are separated from each other by a small indentation. For example, 31,013,736, 134,678, 23,476,009,434, 2,533,467,001,222.
| Class trillion |
Class billions |
Class millions |
Class of thousands | Unit class |
| 134 | 678 | |||
| 31 | 013 | 736 | ||
| 23 | 476 | 009 | 434 | |
| 2 | 533 | 467 | 001 | 222 |
To read a multi-digit number, we call the numbers that make it up one by one (from left to right by class, adding the name of the class). The name of the class of units is not pronounced, and those classes that make up three digits 0 are also not pronounced. If one 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 will be read as “fifty-four” or 001 as “one”.
Example 1
Let's look at the reading of the number 2,533,467,001,222 in detail:
We read the number 2 as a component of the class of trillions - “two”;
By adding the name of the class, we get: “two trillion”;
We read the next number, adding the name of the corresponding class: “five hundred thirty-three billion”;
We continue by analogy, reading the next class to the right: “four hundred sixty-seven million”;
In the next class we see two digits 0 located on the left. According to the above reading rules, digits 0 are discarded and do not participate in reading the record. Then we get: “one thousand”;
We read the last class of units without adding its name - “two hundred twenty-two”.
Thus, the number 2 533 467 001 222 will sound like this: two trillion five hundred thirty-three billion four hundred sixty-seven million one thousand two hundred twenty-two. Using this principle, we will read the other given numbers:
31,013,736 – thirty-one million thirteen thousand seven hundred thirty-six;
134 678 – one hundred thirty-four thousand six hundred seventy-eight;
23 476 009 434 – twenty-three billion four hundred seventy-six million nine thousand four hundred thirty-four.
Thus, the basis for correctly reading multi-digit numbers is the skill of dividing a multi-digit number into classes, knowledge of the corresponding names and understanding of the principle of reading two- and three-digit numbers.
As is already clear from all of the above, its value depends on the position at which the digit appears in the notation of a number. That is, for example, the number 3 in the natural number 314 indicates the number of hundreds, namely 3 hundreds. The number 2 is the number of tens (1 ten), and the number 4 is the number of units (4 units). In this case, we will say that the number 4 is in the ones place and is the value of the ones place in the given number. The number 1 is in the tens place and serves as the value of the tens place. The number 3 is located in the hundreds place and is the value of the hundreds place.
Definition 7
Discharge- this is the position of a digit in the notation of a natural number, as well as the value of this digit, which is determined by its position in a given number.
The categories have their own names, we have already used them above. From right to left there are digits: units, tens, hundreds, thousands, tens of thousands, etc.
For ease of remembering, you can use the following table (we indicate 15 digits):
Let’s clarify this detail: the number of digits in a given multi-digit number is the same as the number of characters in the number’s notation. For example, this table contains the names of all digits for a number with 15 digits. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient to hear.
With the help of such a table, it is possible to develop the skill of determining the digit by writing a given natural number into the table so that the rightmost digit is written in the units digit and then in each digit one by one. For example, let’s write the multi-digit natural number 56,402,513,674 like this:
Pay attention to the number 0, located in the tens of millions digit - it means the absence of units of this digit.
Let us also introduce the concepts of the lowest and highest digits of a multi-digit number.
Definition 8
Lowest (junior) rank of any multi-digit natural number – the units digit.
Highest (senior) category of any multi-digit natural number – the digit corresponding to the leftmost digit in the notation of a given number.
So, for example, in the number 41,781: the lowest digit is the ones digit; The highest rank is the rank of tens of thousands.
Logically it follows that it is possible to talk about the seniority of the digits relative to each other. Each subsequent digit when moving from left to right is lower (younger) than the previous one. And vice versa: when moving from right to left, each next digit is higher (older) than the previous one. For example, the thousands place is older than the hundreds place, but younger than the millions place.
Let us clarify that when solving some practical examples, it is not the natural number itself that is used, but the sum of the digit terms of a given number.
Briefly about the decimal number system
Definition 9Notation- a method of writing numbers using signs.
Positional number systems– those in which the value of a digit in a number depends on its position in the number’s notation.
According to this definition, we can say that, while studying natural numbers and the way they are written above, we used the positional number system. The number 10 plays a special place here. We count in tens: ten units make a ten, ten tens will unite into a hundred, etc. The number 10 serves as the base of this number system, and the system itself is also called decimal.
In addition to it, there are other number systems. For example, computer science uses the binary system. When we keep track of time, we use the sexagesimal number system.
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