Definition
Natural numbers are called numbers intended for counting objects. To record natural numbers, 10 Arabic numerals (0–9) are used, which form the basis of the decimal number system generally accepted for mathematical calculations.
Sequence of natural numbers
The natural numbers make up a series starting at 1 and covering the set of all positive integers. Such a sequence consists of numbers 1,2,3, ... . This means that in the natural series:
- There is a smallest number and no largest.
- Each next number is greater than the previous one by 1 (the exception is the unit itself).
- As the numbers go to infinity, they grow indefinitely.
Sometimes 0 is also introduced into a series of natural numbers. This is permissible, and then they talk about extended natural series.
Classes of natural numbers
Each digit of a natural number expresses a certain digit. The last one is always the number of units in the number, the one before it is the number of tens, the third from the end is the number of hundreds, the fourth is the number of thousands, and so on.
- in the number 276: 2 hundreds, 7 tens, 6 units
- in the number 1098: 1 thousand, 9 tens, 8 ones; the hundreds place is absent here, since it is expressed as zero.
For large and very large numbers, you can see a steady trend (if you examine the number from right to left, that is, from the last digit to the first):
- the last three digits in the number are units, tens and hundreds;
- the previous three are units, tens and hundreds of thousands;
- the three in front of them (i.e. the 7th, 8th and 9th digits of the number, counting from the end) are units, tens and hundreds of millions, etc.
That is, every time we are dealing with three digits, meaning units, tens and hundreds of a larger name. Such groups form classes. And if you have to deal with the first three classes in everyday life more or less often, then others should be listed, because not everyone remembers their names by heart.
- The 4th class, following the class of millions and representing numbers of 10-12 digits, is called a billion (or a billion);
- 5th grade - trillion;
- 6th grade - quadrillion;
- 7th grade - quintillion;
- 8th grade - sextillion;
- 9th grade - septillion.
Addition of natural numbers
The addition of natural numbers is an arithmetic operation that allows you to get a number that contains as many units as there are in the numbers added together.
The sign of addition is the "+" sign. Added numbers are called terms, the result is called the sum.
Small numbers are added (summed up) orally, in writing such actions are written in a line.
Multi-digit numbers, which are difficult to add in the mind, are usually added in a column. For this, the numbers are written one under the other, aligned with the last digit, that is, they write the units digit under the units digit, the hundreds digit under the hundreds digit, and so on. Next, you need to add the digits in pairs. If the addition of digits occurs with a transition through a ten, then this ten is fixed as a unit above the digit on the left (that is, following it) and is added together with the digits of this digit.
If not 2, but more numbers are added to the column, then when summing up the digits of the category, not 1 dozen, but several, may be redundant. In this case, the number of such tens is transferred to the next digit.
Subtraction of natural numbers
Subtraction is an arithmetic operation, the reverse of addition, which boils down to the fact that, given the amount and one of the terms, you need to find another - an unknown term. The number that is being subtracted from is called the minuend; the number that is being subtracted is the subtrahend. The result of the subtraction is called the difference. The sign that denotes the operation of subtraction is "-".
In the transition to addition, the subtrahend and the difference turn into terms, and the reduced into the sum. Addition usually checks the correctness of the subtraction performed, and vice versa.
Here 74 is the minuend, 18 is the subtrahend, 56 is the difference.
A prerequisite for subtracting natural numbers is the following: the minuend must necessarily be greater than the subtrahend. Only in this case the resulting difference will also be a natural number. If the subtraction action is carried out for an extended natural series, then it is allowed that the minuend is equal to the subtrahend. And the result of subtraction in this case will be 0.
Note: if the subtrahend is equal to zero, then the subtraction operation does not change the value of the minuend.
Subtraction of multi-digit numbers is usually done in a column. Write down the numbers in the same way as for addition. Subtraction is performed for the corresponding digits. If it turns out that the minuend is less than the subtrahend, then one is taken from the previous (located on the left) digit, which, after the transfer, naturally turns into 10. This ten is summed up with the figure of the reduced given digit and then subtracted. Further, when subtracting the next digit, it is necessary to take into account that the reduced has become 1 less.
Product of natural numbers
The product (or multiplication) of natural numbers is an arithmetic operation, which is finding the sum of an arbitrary number of identical terms. To record the operation of multiplication, use the sign "·" (sometimes "×" or "*"). For example: 3 5=15.
The action of multiplication is indispensable when it is necessary to add a large number of terms. For example, if you need to add the number 4 7 times, then multiplying 4 by 7 is easier than doing this addition: 4+4+4+4+4+4+4.
The numbers that are multiplied are called factors, the result of multiplication is the product. Accordingly, the term "work" can, depending on the context, express both the process of multiplication and its result.
Multi-digit numbers are multiplied in a column. For this number is written in the same way as for addition and subtraction. It is recommended to write first (above) which of the 2 numbers, which is longer. In this case, the multiplication process will be simpler, and therefore more rational.
When multiplying in a column, the digits of each of the digits of the second number are sequentially multiplied by the digits of the 1st number, starting from its end. Having found the first such work, they write down the number of units, and keep the number of tens in mind. When multiplying the digit of the 2nd number by the next digit of the 1st number, the number that is kept in mind is added to the product. And again they write down the number of units of the result obtained, and remember the number of tens. When multiplying by the last digit of the 1st number, the number obtained in this way is written down in full.
The results of multiplying the digits of the 2nd digit of the second number are written in the second row, shifting it 1 cell to the right. Etc. As a result, a "ladder" will be obtained. All the resulting rows of numbers should be added (according to the rule of addition in a column). Empty cells should be considered filled with zeros. The resulting sum is the final product.
Note
- The product of any natural number by 1 (or 1 by a number) is equal to the number itself. For example: 376 1=376; 1 86=86.
- When one of the factors or both factors are equal to 0, then the product is equal to 0. For example: 32·0=0; 0 845=845; 0 0=0.
Division of natural numbers
Division is called an arithmetic operation, with the help of which, according to a known product and one of the factors, it can be found another - unknown - factor. Division is the inverse of multiplication and is used to check if a multiplication has been performed correctly (and vice versa).
The number that is being divided is called the divisible; the number by which it is divided is the divisor; the result of a division is called a quotient. The division sign is ":" (sometimes, less often - "÷").
Here 48 is the dividend, 6 is the divisor, and 8 is the quotient.
Not all natural numbers can be divided among themselves. In this case, division is performed with a remainder. It consists in the fact that for the divisor such a factor is selected so that its product by the divisor would be a number that is as close as possible in value to the dividend, but less than it. The divisor is multiplied by this factor and subtracted from the dividend. The difference will be the remainder of the division. The product of a divisor by a factor is called an incomplete quotient. Attention: the remainder must be less than the selected multiplier! If the remainder is larger, then this means that the multiplier is chosen incorrectly, and it should be increased.
We select a factor for 7. In this case, this number is 5. We find an incomplete quotient: 7 5 \u003d 35. Calculate the remainder: 38-35=3. Since 3<7, то это означает, что число 5 было подобрано верно. Результат деления следует записать так: 38:7=5 (остаток 3).
Multi-digit numbers are divided into a column. To do this, the dividend and divisor are written side by side, separating the divisor with a vertical and horizontal line. In the dividend, the first digit or the first few digits (on the right) are selected, which should be a number that is minimally sufficient to divide by a divisor (that is, this number must be greater than the divisor). For this number, an incomplete quotient is selected, as described in the rule of division with a remainder. The number of the multiplier used to find the partial quotient is written under the divisor. The incomplete quotient is written under the number that was divided, right-aligned. Find their difference. The next digit of the dividend is demolished by writing it next to this difference. For the resulting number, an incomplete quotient is again found by writing down the figure of the selected factor, next to the previous one under the divisor. Etc. Such actions are performed until the numbers of the dividend run out. After that, the division is considered complete. If the dividend and the divisor are divided entirely (without a remainder), then the last difference will give zero. Otherwise, the remainder number will be returned.
Exponentiation
Exponentiation is a mathematical operation that consists in multiplying an arbitrary number of identical numbers. For example: 2 2 2 2.
Such expressions are written as: a x,
where a is a number multiplied by itself x is the number of such factors.
Prime and composite natural numbers
Any natural number, except 1, can be divided by at least 2 numbers - one and itself. Based on this criterion, natural numbers are divided into prime and composite.
Prime numbers are numbers that are only divisible by 1 and itself. Numbers that are divisible by more than these 2 numbers are called composite numbers. A unit divisible solely by itself is neither prime nor compound.
Numbers are prime: 2,3,5,7,11,13,17,19, etc. Examples of composite numbers: 4 (divisible by 1,2,4), 6 (divisible by 1,2,3,6), 20 (divisible by 1,2,4,5,10,20).
Any composite number can be decomposed into prime factors. In this case, prime factors are understood to be its divisors, which are prime numbers.
An example of factorization into prime factors:
Divisors of natural numbers
A divisor is a number by which a given number can be divided without a remainder.
In accordance with this definition, simple natural numbers have 2 divisors, composite numbers have more than 2 divisors.
Many numbers have common divisors. The common divisor is the number by which the given numbers are divisible without a remainder.
- The numbers 12 and 15 have a common divisor 3
- The numbers 20 and 30 have common divisors 2,5,10
Of particular importance is the greatest common divisor (GCD). This number, in particular, is useful to be able to find for reducing fractions. To find it, it is required to decompose the given numbers into prime factors and present it as the product of their common prime factors, taken in their smallest powers.
It is required to find the GCD of the numbers 36 and 48.
Divisibility of natural numbers
It is far from always possible to determine “by eye” whether one number is divisible by another without a remainder. In such cases, the corresponding divisibility test is useful, that is, the rule by which in a matter of seconds you can determine whether it is possible to divide numbers without a remainder. The sign "" is used to indicate divisibility.
Least common multiple
This value (denoted LCM) is the smallest number that is divisible by each of the given ones. The LCM can be found for an arbitrary set of natural numbers.
LCM, like GCD, has a significant applied meaning. So, it is the LCM that needs to be found by reducing ordinary fractions to a common denominator.
The LCM is determined by factoring the given numbers into prime factors. For its formation, a product is taken, consisting of each of the occurring (at least for 1 number) prime factors represented to the maximum degree.
It is required to find the LCM of the numbers 14 and 24.
Average
The arithmetic mean of an arbitrary (but finite) number of natural numbers is the sum of all these numbers divided by the number of terms:
The arithmetic mean is some average value for a number set.
The numbers 2,84,53,176,17,28 are given. It is required to find their arithmetic mean.
Integers- numbers that are used to count objects . Any natural number can be written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Such a record of numbers is called decimal.
The sequence of all natural numbers is called natural side by side .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
Most small a natural number is one (1). In the natural series, each next number is 1 more than the previous one. natural series endless there is no largest number.
The meaning of a digit depends on its place in the notation of the number. For example, the number 4 means: 4 units, if it is in the last place in the number entry (in units place); 4 ten, if she is in last place (in the tens place); 4 hundreds, if it is in third place from the end (in hundreds place).
Digit 0 means lack of units of this category in the decimal notation of a number. It also serves to denote the number " zero". This number means "none". Score 0: 3 of a football match indicates that the first team did not score a single goal against the opponent.
Zero do not include to natural numbers. And indeed the counting of items never starts from scratch.
If a natural number has only one digit – one digit, then it is called unambiguous. Those. unambiguousnatural number- a natural number whose record consists of one sign – one digit. For example, the numbers 1, 6, 8 are single digits.
double digitnatural number- a natural number, the record of which consists of two characters - two digits.
For example, the numbers 12, 47, 24, 99 are double digits.
Also, according to the number of characters in a given number, names are given to other numbers:
numbers 326, 532, 893 - three-digit;
numbers 1126, 4268, 9999 - four-digit etc.
Two digits, three digits, four digits, five digits, 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 can consist of one or two digits). These groups are called classes.
Million is a thousand thousand (1000 thousand), it is written 1 million or 1,000,000.
Billion is 1000 million. It is recorded by 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 there are the classes of millions, billions, etc. (Fig. 1).
Rice. 1. Class of millions, class of thousands and class of units (from left to right)
The number 15389000286 is written in the bit grid (Fig. 2).
Rice. 2. Digit grid: number 15 billion 389 million 286
This number has 286 ones in the one class, zero ones in the thousands class, 389 ones in the millions class, and 15 ones in the billions class.
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 is less than C-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 students
Natural numbers are one of the oldest mathematical concepts.
In the distant past, people did not know numbers, and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.
The number of objects was compared with parts of the body, for example, with the fingers on the hand, and they said: "I have as many nuts as there are fingers on the hand."
Over time, people realized that five nuts, five goats and five hares have a common property - their number is five.
Remember!
Integers are numbers, starting with 1, obtained when counting objects.
1, 2, 3, 4, 5…
smallest natural number — 1 .
largest natural number does not exist.
When counting, the number zero is not used. Therefore, zero is not considered a natural number.
People learned to write numbers much later than to count. First of all, they began to represent the unit with one stick, then with two sticks - the number 2, with three - the number 3.
| — 1, || — 2, ||| — 3, ||||| — 5 …
Then special signs appeared for designating numbers - the forerunners of modern numbers. The numbers we use to write numbers originated in India about 1,500 years ago. The Arabs brought them to Europe, so they are called Arabic numerals.
There are ten digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits can be used to write any natural number.
Remember!
natural series is the sequence of all natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …
In the natural series, each number is greater than the previous one by 1.
The natural series is infinite, there is no largest natural number in it.
The counting system we use is called decimal positional.
Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the value of a digit depends on its place in the notation of a number, that is, on the digit in which it is written.
Important!
The classes following the billion are named according to the Latin names of numbers. Each next unit contains a thousand previous ones.
- 1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
- 1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
- 1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)
However, physicists have found a number that surpasses the number of all atoms (the smallest particles of matter) in the entire universe.
This number has a special name - googol. A googol is a number that has 100 zeros.
Integers
Natural numbers 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; five; 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.
