What is a natural number? History, scope, properties. Natural numbers What are the names of the components of multiplication
The simplest number is natural number. They are used in Everyday life for counting items, i.e. to calculate their number and order.
What is a natural number: natural numbers name the numbers that are used for counting items or to indicate the serial number of any item from all homogeneous items.
Integersare 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 largest natural number. When counting the number zero is not used, so zero is a natural number.
natural series of numbers is the sequence of all natural numbers. Write natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...
In natural numbers, each number is one more than the previous one.
How many numbers are in the natural series? The natural series is infinite, there is no largest natural number.
Decimal since 10 units of any category form 1 unit of the highest order. positional so how the value of a digit depends on its place in the number, i.e. from the category where it is recorded.
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 andetc. Each of the digits of the class is called itsdischarge.
Comparison of natural numbers.
Of the 2 natural numbers, the number that is called earlier in the count is less. For example, number 7 less 11 (written like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .
Table of digits and classes of numbers.
1st class unit |
1st unit digit 2nd place ten 3rd rank hundreds |
2nd class thousand |
1st digit units of thousands 2nd digit tens of thousands 3rd rank hundreds of thousands |
3rd grade millions |
1st digit units million 2nd digit tens of millions 3rd digit hundreds of millions |
4th grade billions |
1st digit units billion 2nd digit tens of billions 3rd digit hundreds of billions |
Numbers from the 5th grade and above are large numbers. Units of the 5th class - trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions. Basic properties of natural numbers.
Actions on natural numbers. 4. Division of natural numbers is an operation inverse to multiplication. If b ∙ c \u003d a, That Division formulas: 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 Numeric 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. Entries where the equals sign concatenates 2 numeric expressions is numerical equalities. Equality has a left side and a right side. The order in which arithmetic operations are performed. Addition and subtraction of 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, then they are performed sequentially from left to right. When expressions consist of actions of only the first and second degree, then the actions are first performed second degree, and then - actions of the first degree. When there are parentheses in the expression, the actions in the parentheses are performed first. For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21. |
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Definition. Integers- these are the numbers that are used for counting: 1, 2, 3, ..., n, ...
The set of natural numbers is usually denoted by the symbol N(from lat. naturalis- natural).
Natural numbers in the decimal number system are written using ten digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The set of natural numbers is ordered set, i.e. for any natural numbers m and n, one of the following relations is true:
- or m = n (m is equal to n ),
- or m > n (m is greater than n ),
- or m< n (m меньше n ).
- Least natural number - unit (1)
- There is no largest natural number.
- Zero (0) is not a natural number.
Of the neighboring natural numbers, the number that is to the left of the number n is called the previous number n, and the number to the right is called following n.
Operations on natural numbers
Closed operations on natural numbers (operations resulting in natural numbers) include the following arithmetic operations:
- Addition
- Multiplication
- Exponentiation a b , where a is the base of the power and b is the exponent. If the base and exponent are natural numbers, then the result will be a natural number.
In addition, two more operations are considered. From a formal point of view, they are not operations on natural numbers, since their result will not always be a natural number.
- Subtraction(At the same time, the reduced must be greater than the subtracted)
- Division
Classes and ranks
Discharge - the position (position) of a digit in a number entry.
The lowest rank is the one on the right. The high order is the leftmost one.
Example:
5 - units, 0 - tens, 7 - hundreds,
2 - thousands, 4 - tens of thousands, 8 - hundreds of thousands,
3 - million, 5 - tens of millions, 1 - hundreds of millions
For ease of reading, natural numbers are divided into groups of three digits each, starting from the right.
Class- a group of three digits into which the number is divided, starting from the right. The last class can be three, two, or one digit.
- The first class is the class of units;
- The second class is the class of thousands;
- The third class is the class of millions;
- The fourth class is the class of billions;
- The fifth class is the class of trillions;
- The sixth class is the quadrillion (quadrillion) class;
- The seventh class is the class of quintillions (quintillions);
- The eighth class is the sextillion class;
- The ninth class is the class of septillons;
Example:
34 - billion 456 million 196 thousand 45
Comparison of natural numbers
Comparison of natural numbers with different number of digits
Among natural numbers, the one with more digits is greaterComparing natural numbers with the same number of digits
Compare numbers bit by bit, starting with the most significant digit. More than that, which has more units in the highest digit of the same name
Example:
3466 > 346 - since the number 3466 consists of 4 digits, and the number 346 consists of 3 digits.
34666 < 245784 - because 34666 has 5 digits and 245784 has 6 digits.
Example:
346 667 670 52 6 986
346 667 670 56 9 429
The second natural number with the same number of digits is greater because 6 > 2.
In mathematics, there are several different sets of numbers: real, complex, integer, rational, irrational, ... In our Everyday life we most often use natural numbers, as we encounter them when counting and when searching, indicating the number of objects.
In contact with
What numbers are called natural
From ten digits, you can write down absolutely any existing sum of classes and ranks. Natural values are those which are used:
- When counting any items (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, since the set of integer values is not limited.
Attention! Natural numbers are obtained by counting objects or by designating their quantity.
Absolutely any number can be decomposed and represented as bit 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. In the set diagram, they would be in each other, since the set of naturals 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 mathematical schools, the smallest value of N counted as a unit, since the absence of objects is considered empty.
But in foreign mathematical schools, for example, in French, it is considered natural. The presence of zero in the series facilitates the proof some theorems.
A set of values N that includes zero is called extended and is denoted by the symbol N0 (zero index).
Series of natural numbers
An N row 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 the convenience 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, one can find two intervals to which the sequence N belongs:
from zero to plus infinity, including the ends, and from one to plus infinity, including the ends, that is, all positive whole answers.
N sets of digits can be either even or odd. Consider the concept of oddness.
Odd (any odd ones end 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 dividing even N by 2, there will be no remainder, that is, the result is a whole answer. For example, 50:2=25, 100:2=50, 3456:2=1728.
Important! A numerical series of N cannot consist only of even or odd values, since they must alternate: an even number is always followed by an odd number, then an even number again, and so on.
N properties
Like all other sets, N has its own special properties. 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 are a sequence, i.e. 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 always comes out natural meaning.
- In calculations, you can use permutation and combination.
- Each subsequent value cannot be less than the previous one. Also in the N series, the following law will operate: if the number A is less than B, then in the number series there will always be a C, for which the equality is true: A + C \u003d B.
- If we take two natural expressions, for example, A and B, then one of the expressions will be true for them: A \u003d 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: B-A less S-A.
Attention! All of the above inequalities are also valid in the opposite direction.
What are the components of a multiplication called?
In many simple and even complex tasks, finding the answer depends on the ability of schoolchildren.
In order to quickly and correctly multiply and be able to solve inverse problems, you need to know the components of multiplication.
15. 10=150. In this expression, 15 and 10 are factors, and 150 is a product.
Multiplication has properties that are necessary when solving problems, equations and inequalities:
- Rearranging the factors does not change the final product.
- To find the unknown factor, you need to divide the product by the known factor (valid for all factors).
For example: 15 . X=150. Divide the product by a known factor. 150:15=10. Let's do a check. 15 . 10=150. According to this principle, even complex linear equations(if you simplify them).
Important! The product can consist of more than just two factors. For example: 840=2 . 5. 7. 3. 4
What are natural numbers in mathematics?
Discharges and classes of natural numbers
Conclusion
Let's summarize. N is used when counting or indicating the number of items. The number of natural sets of digits is infinite, but it includes only integer and positive sums of digits and classes. Multiplication is also necessary for to count things, as well as for solving problems, equations and various inequalities.
The history of natural numbers began in primitive times. Since ancient times, people have counted objects. For example, in trade, a commodity account was needed, or in construction, a material account. Yes, even in everyday life, too, I had to count things, products, livestock. At first, numbers were used only for counting in life, in practice, but later, with the development of mathematics, they became part of science.
Integers are the numbers that we use when counting objects.
For example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ....
Zero is not a natural number.
All natural numbers, or let's call the set of natural numbers, is denoted by the symbol N.
Table of natural numbers.
natural row.
Natural numbers written in ascending order in a row form natural series or series of natural numbers.
Properties of the natural series:
- The smallest natural number is one.
- In the natural series, the next number is greater than the previous one by one. (1, 2, 3, …) Three dots or three dots are used if it is impossible to complete the sequence of numbers.
- The natural series has no maximum number, it is infinite.
Example #1:
Write the first 5 natural numbers.
Solution:
Natural numbers start with one.
1, 2, 3, 4, 5
Example #2:
Is zero a natural number?
Answer: no.
Example #3:
What is the first number in the natural series?
Answer: the natural number starts with one.
Example #4:
What is the last number in the natural series? What is the largest natural number?
Answer: The natural number starts from one. Each next number is greater than the previous one by one, so the last number does not exist. There is no largest number.
Example #5:
Does the unit in the natural series have a previous number?
Answer: no, because one is the first number in the natural series.
Example #6:
Name the next number in the natural series after the numbers: a) 5, b) 67, c) 9998.
Answer: a) 6, b) 68, c) 9999.
Example #7:
How many numbers are in the natural series between the numbers: a) 1 and 5, b) 14 and 19.
Solution:
a) 1, 2, 3, 4, 5 - three numbers are between the numbers 1 and 5.
b) 14, 15, 16, 17, 18, 19 - four numbers are between the numbers 14 and 19.
Example #8:
Name the previous number after the number 11.
Answer: 10.
Example #9:
What numbers are used to count objects?
Answer: natural numbers.
Natural numbers are one of the oldest mathematical concepts.
In the distant past, people did not know numbers, and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.
The number of objects was compared with parts of the body, for example, with the fingers on the hand, and they said: "I have as many nuts as there are fingers on the hand."
Over time, people realized that five nuts, five goats and five hares have common property- their number is five.
Remember!
Integers are numbers, starting with 1, obtained when counting objects.
1, 2, 3, 4, 5…
smallest natural number — 1 .
largest natural number does not exist.
When counting, the number zero is not used. Therefore, zero is not considered a natural number.
People learned to write numbers much later than to count. First of all, they began to represent the unit with one stick, then with two sticks - the number 2, with three - the number 3.
| — 1, || — 2, ||| — 3, ||||| — 5 …
Then special signs appeared for designating numbers - the forerunners of modern numbers. The numbers we use to write numbers originated in India about 1,500 years ago. The Arabs brought them to Europe, so they are called Arabic numerals.
There are ten digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits can be used to write any natural number.
Remember!
natural series is the sequence of all natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …
In the natural series, each number is greater than the previous one by 1.
The natural series is infinite, there is no largest natural number in it.
The counting system we use is called decimal positional.
Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the value of a digit depends on its place in the notation of a number, that is, on the digit in which it is written.
Important!
The classes following the billion are named according to the Latin names of numbers. Each next unit contains a thousand previous ones.
- 1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
- 1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
- 1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)
However, physicists have found a number that surpasses the number of all atoms (the smallest particles of matter) in the entire universe.
This number has a special name - googol. A googol is a number that has 100 zeros.
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