To remember how much harvest they harvested or how many stars there were in the sky, people came up with symbols. These symbols were different in different areas.
But with the development of trade, in order to understand the designations of another people, people began to use the most convenient symbols. For example, we use Arabic symbols. And they are called Arab because Europeans learned them from the Arabs. But the Arabs learned these symbols from the Indians.
The symbols that are used to write numbers are called in numbers .
The word number comes from the Arabic name for the number 0 (sifr). This is a very interesting figure. It is called insignificant and denotes the absence of something.
In the picture we see a plate with 3 apples on it and an empty plate with no apples on it. In the case of an empty plate, we can say that there are 0 apples on it.
The remaining numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9 are called meaningful .
Bit units
Notation the one we use is called decimal. Because it is precisely ten units of one category that constitute one unit of the next category.
We count in units, tens, hundreds, thousands, and so on. These are the digit units of our number system.
10 ones – 1 ten (10)
10 tens – 1 hundred (100)
10 hundreds – 1 thousand (1000)
10 times 1 thousand – 1 ten thousand (10,000)
10 tens of thousands – 100 thousand (100,000) and so on...
Place is the place of a digit in a number notation.
For example, among 12 two digits: the ones digit consists of 2 units, the tens place consists of one dozen.
We talked about how 0 is an insignificant number that means the absence of something. In numbers, the number 0 indicates the absence of ones in the digit.
In the number 190, the digit 0 indicates the absence of a ones place. In the number 208, the digit 0 indicates the absence of a tens place. Such numbers are called incomplete .
And numbers whose digits do not have zeros are called full .
The digits are counted from right to left:
It will be clearer if you depict the bit grid as follows:
- Among 2375 :
5 units of the first category, or 5 units
7 units of the second digit, or 7 tens
3 units of the third category, or 3 hundreds
2 units of the fourth category, or 2 thousand
This number is pronounced like this: two thousand three hundred seventy five
- Among 1000462086432
2 pieces
3 tens
8 tens of thousands
0 hundred thousand
2 units million
6 tens of millions
4 hundred million
0 units billion
0 tens of billions
0 hundred billion
1 unit trillion
This number is pronounced like this: one trillion four hundred sixty two million eighty six thousand four hundred thirty two .
- Among 83 :
3 units
8 tens
Pronounced like this: eighty three .
bit, call numbers consisting of units of only one digit:
For example, numbers 1, 3, 40, 600, 8000 - bit numbers, in such numbers there can be as many zeros (insignificant digits) as desired or not at all, but there is only one significant digit.
Other numbers, for example: 34, 108, 756 and so on, unbited , they are called algorithmic.
Non-digit numbers can be represented as a sum of digit terms.
For example, number 6734 can be represented like this:
6000 + 700 + 30 + 4 = 6734
Integers– natural numbers are numbers that are used to count objects. Plenty of everyone natural numbers sometimes called the natural series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, etc.
To write natural numbers, ten digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using them, you can write any natural number. This notation of numbers is called decimal.
The natural series of numbers can be continued indefinitely. There is no such number that would be the last, because you can always add one to the last number and you will get a number that is already greater than the one you are looking for. In this case, they say that there is no greatest number in the natural series.
Places of natural numbers
When writing any number using digits, the place in which the digit appears in the number is critical. For example, the number 3 means: 3 units, if it appears in the number on last place; 3 tens, if she is in the penultimate place in the number; 4 hundred if she is in third place from the end.
The last digit means the units place, the penultimate digit means the tens place, and the 3 from the end means the hundreds place.
Single and multi-digit numbers
If any digit of a number contains the digit 0, this means that there are no units in this digit.
The number 0 is used to denote the number zero. Zero is “not one”.
Zero is not a natural number. Although some mathematicians think differently.
If a number consists of one digit it is called single-digit, if it consists of two it is called two-digit, if it consists of three it is called three-digit, etc.
Numbers that are not single-digit are also called multi-digit.
Digit classes for reading large natural numbers
To read large natural numbers, the number is divided into groups of three digits, starting from the right edge. These groups are called classes.
The first three digits on the right edge make up the units class, the next three are the thousands class, and the next three are the millions class.
Million – one thousand thousand; the abbreviation million is used for recording. 1 million = 1,000,000.
A billion = a thousand million. For recording, use the abbreviation billion. 1 billion = 1,000,000,000.
Example of writing and reading
This number has 15 units in the class of billions, 389 units in the class of millions, zero units in the class of thousands, and 286 units in the class of units.
This number reads like this: 15 billion 389 million 286.
Read numbers from left to right. Take turns calling the number of units of each class and then adding the name of the class.
The digits in multi-digit numbers are divided from right to left into groups of three digits each. These groups are called classes. In each class, the numbers from right to left indicate the units, tens and hundreds of that class:
The first class on the right is called class of units, second - thousand, third - millions, fourth - billions, fifth - trillion, sixth - quadrillion, seventh - quintillions, eighth - sextillions.
To make it easier to read the notation of a multi-digit number, a small space is left between the classes. For example, to read the number 148951784296, we highlight the classes in it:
and read the number of units of each class from left to right:
148 billion 951 million 784 thousand 296.
When reading a class of units, the word units is usually not added at the end.
Each digit in the notation of a multi-digit number occupies a certain place - position. The place (position) in the record of a number on which the digit stands is called discharge.
The counting of digits goes from right to left. That is, the first digit on the right in a number is called the first digit, the second digit on the right is the second digit, etc. For example, in the first class of the number 148,951,784,296, digit 6 is the first digit, 9 is the second digit, 2 - third digit:
Units, tens, hundreds, thousands, etc. are also called bit units:
units are called units of the 1st category (or simple units)
tens are called units of the 2nd digit
hundreds are called 3rd digit units, etc.
All units except simple units are called constituent units. So, ten, hundred, thousand, etc. are composite units. Every 10 units of any rank constitutes one unit of the next (higher) rank. For example, a hundred contains 10 tens, a ten contains 10 prime ones.
Any compound unit compared to another unit smaller than it is called unit of the highest category, and in comparison with a unit greater than it is called unit of the lowest category. For example, a hundred is a higher-order unit relative to ten and a lower-order unit relative to a thousand.
To find out how many units of any digit there are in a number, you need to discard all the digits representing the units of lower digits and read the number expressed by the remaining digits.
For example, you need to find out how many hundreds there are in the number 6284, i.e. how many hundreds are in the thousands and hundreds of a given number together.
In the number 6284, the number 2 is in third place in the units class, which means there are two prime hundreds in the number. The next number to the left is 6, meaning thousands. Since every thousand contains 10 hundreds, 6 thousand contain 60 of them. In total, therefore, this number contains 62 hundreds.
The number 0 in any digit means the absence of units in this digit. For example, the number 0 in the tens place means the absence of tens, in the hundreds place - the absence of hundreds, etc. In the place where there is a 0, nothing is said when reading the number:
172 526 - one hundred seventy two thousand five hundred twenty six.
102 026 - one hundred two thousand twenty six.
They are all different. For example, 2, 67, 354, 1009. Let's look at these numbers in detail.
2 consists of one digit, so this number is called single digit. Another example of single-digit numbers: 3, 5, 8.
67 consists of two digits, so this number is called double digit number. Example of two-digit numbers: 12, 35, 99.
Three digit numbers consist of three numbers, for example: 354, 444, 780.
Four digit numbers consist of four digits, for example: 1009, 2600, 5732.
Two digits, three digits, four digits, five digits, six digits, etc. numbers are called multi-digit numbers.
Number digits.
Consider the number 134. Each digit of this number has its own place. Such places are called discharges.
The number 4 takes the place or place of ones. The number 4 can also be called a number first category.
The number 3 occupies the place or tens place. Or the number 3 can be called a number second class.
And the number 1 occupies the hundreds place. In another way, the number 1 can be called the number third category. The number 1 is the last digit of the glory of the number 134, so the number 1 can be called the highest digit. The highest digit is always greater than 0.
Every 10 units of any rank form a new unit of a higher rank. 10 units form one tens place, 10 tens form one hundreds place, ten hundreds form one thousand place, etc.
If there is no digit, then it will be replaced by 0.
For example: the number 208.
The number 8 is the first digit of units.
The number 0 is the second tens place. 0 means nothing in mathematics. From the record it follows that this number does not have tens.
The number 2 is the third hundreds place.
This parsing of a number is called digit composition of the number.
Classes.
Multi-digit numbers are divided into groups of three digits from right to left. Such groups of numbers are called classes. The first class on the right is called class of units, the second one is called class of thousands, third - million class, fourth - class of billions, fifth - trillion class, sixth – class quadrillion, seventh - class quintillions, eighth – class sextillions.
Unit class– the first class on the right from the end is three digits consisting of a units place, a tens place and a hundreds place.
Class of thousands– the second class consists of the category: units of thousands, tens of thousands and hundreds of thousands.
Million class– the third class consists of the category: units of millions, tens of millions and hundreds of millions.
Let's look at an example:
We have the number 13,562,006,891.
This number has 891 units in the units class, 6 units in the thousands class, 562 units in the millions class, and 13 units in the billions class.
13 billion 562 million 6 thousand 891.
Sum of bit terms.
Anything having different digits can be decomposed into sum of bit terms. Let's look at an example:
Let's write the number 4062 into digits.
4 thousand 0 hundreds 6 tens 2 units or in another way you can write
4062=4 ⋅1000+0 ⋅100+6 ⋅10+2
Next example:
26490=2 ⋅10000+6 ⋅1000+4 ⋅100+9 ⋅10+0
Because decimal system dead reckoning place number, then the number depends not only on the digits written in it, but also on the place where each digit is written.
Definition: The place where a digit is written in a number is called the digit of the number.
For example, a number consists of three digits: 1, 0 and 3. The place, or digit, notation system allows you to create three-digit numbers from these three digits: 103, 130, 301, 310 and two-digit numbers: 013, 031. The given numbers are arranged in order increasing: each previous number is less than the next one.
Consequently, the numbers that are used to write a number do not completely define this number, but only serve as a tool for writing it.
The number itself is constructed taking into account ranks, in which this or that digit is written, i.e. the required digit must also occupy Right place in writing the number.
Rule. Places of natural numbers are named from right to left from 1 to the larger number, each digit has its own number and place in the number record.
The most commonly used numbers have up to 12 digits. Numbers with more than 12 digits belong to the group of large numbers.
The number of places occupied by digits, provided that the largest digit is not 0, determines the digit capacity of the number. We can say about a number that it is: single-digit (single-digit), for example 5; two-digit (two-digit), for example 15; three-digit (three-digit), for example 551, etc.
In addition to the serial number, each of the digits has its own name: the units digit (1st), the tens digit (2nd), the hundreds digit (3rd), the units of thousands digit (4th), the tens of thousands digit (5th ) etc. Every three digits, starting from the first, are combined into classes. Every Class also has its own serial number and name.
For example, the first 3 category(from 1st to 3rd inclusive) - this is Class units with serial number 1; third Class- This Class million, it includes the 7th, 8th and 9th ranks.
Let us present the structure of the digit construction of a number, or a table of digits and classes.
The number 127 432 706 408 is twelve-digit and reads like this: one hundred twenty-seven billion four hundred thirty-two million seven hundred six thousand four hundred eight. This is a fourth grade multi-digit number. The three digits of each class are read as three-digit numbers: one hundred twenty-seven, four hundred thirty-two, seven hundred six, four hundred eight. To each class of a three-digit number the name of the class is added: “billions”, “millions”, “thousands”.
For the class of units, the name is omitted (implying “units”).
Numbers from 5th grade and above are considered large numbers. Large numbers are used only in specific branches of Knowledge (astronomy, physics, electronics, etc.).
Let us give an introduction to the names of the classes from the fifth to the ninth: the units of the 5th class are trillions, the 6th class are quadrillions, the 7th class are quintillions, the 8th class are sextillions, the 9th class are septillions.
