Analysis of PLC's commonly used number system and its conversion method

1. What is the carry count system

The number system is also called the counting system, which refers to the method of expressing values ​​with a set of fixed symbols and uniform rules. The method of counting according to the principle of carry is called the carry counting system. For example, in the decimal counting system, counting is carried out according to the principle of "every decimal one".

Commonly used carry counting system:

1. Decimal notation, there are 10 bases: 0 ~~ 9, every decimal is one;

2. Binary notation, there are 2 base numbers: 0 ~~ 1, and enter one every two;

3. Octal notation, there are 8 bases: 0 ~~ 7, and it is entered every eight;

4. Hexdecimal notation, with 16 bases: 0 ~~ 9, A, B, C, D, E, F (A=10, B=11, C=12, D=13, E=14,F=15), enter one every sixteen.

2. The base number and bit weight of the carry counting system

"Base" and "bit weight" are the two elements of the carry counting system.

1. Base:

The so-called base is the number of possible digits in each digit of the carry counting system. For example, there are ten digits "0", "1", "3",..., "9" in each digit of the decimal number, so the base is 10.

2. Position:

The so-called bit weight refers to the size of the weight of each digit of a value. For example, the bit weights of the decimal number 4567 from low to high are 100, 101, 102, and 103, respectively. because:

4567=4x103＋5x 102＋6x 101 ＋7x100

3. Representation of the position and power of numbers:

The numbers of any number system can be expressed as the sum of polynomials expanded by bit weights.

For example: 435.05 in decimal number can be expressed as:

435.05=4x102＋3x 101＋5x100＋0x10－1 ＋5x10－2

The characteristic of position power representation is: each item = a number of powers of a certain position X base number; and the size of the power is determined by the position of the number.

Three, binary numbers

Why use binary system in computer: Binary operation is simple, circuit is simple and reliable, and logic is strong.

1. Definition:

Counting is carried out according to the principle of "every two enters one", which is called a binary number, that is, when each bit is full of 2, it advances by one to the higher digit.

2. Features:

The digits of each number can only be two digits: 0 and 1. The largest digit in a binary number is 1, and the smallest digit is 0; the base is 2;

For example, and 00101011 are two binary numbers.

3. Bit weight representation of binary numbers:

(1101.101) 2 = 1x23 + 1x 22 + 0x 21 + 1x 20 + 1x2-1 + 0x 2-2 + 1x2-3

4. Operation rules of binary numbers:

Addition operation

① 0＋0＝0 ③ 1＋1＝10

② 0＋1=1＋0=1

Multiplication

① 0×0=0 ③ 1×1=1

② 0×1=1×0=0

Four, octal number

1. Definition:

Counting is carried out according to the principle of "every eight into one", which is called an octal number, that is, when each person counts up to 8 times, it advances by one to the higher digit.

2. Features:

The digits of each number can only contain eight digits: 0, 1, 2, 3, 4, 5, 6, and 7; the largest digit in octal numbers is 7, and the smallest digit is 0; the base is 8;

For example: (1347)8 and (62435)8 are two octal numbers.

3. The representation of the bit power of an octal number:

(107.13) 8 = 1x 82 + 0x 81 + 7x 80 + 1x8-1 + 3x 8-2

Five, hexadecimal number

1. Definition:

Counting is carried out according to the principle of "advancing every hexadecimal", which is called hexadecimal number, which means that every digit is incremented by one to the higher digit when 16 is counted.

2. Features:

The digits of each number can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F sixteen digits; hexadecimal number The largest number is F, which is 15, and the smallest number is 0; the base is 16;

For example: (109)16 and (2FDE)16 are two hexadecimal numbers.

3. Representation of position rights of hexadecimal numbers:

(109.13) 16 = 1x 162 + 0x161 + 9x 160 + 1x16-1 + 3x 16-2

(2FDE)16=2x 163＋15x 162＋13x 161＋14x 160

6. Correspondence between common counting systems

Binary numbers, octal numbers, hexadecimal numbers, and decimal numbers are the four commonly used number systems in modern digital systems. The corresponding relations between these several carry system number systems are listed in Table 1.

Table 1 Commonly used counting system number representation method

Seven, the conversion between number systems

1. Convert a decimal number to a non-decimal number

(1) Convert a decimal integer to a non-decimal integer

â‘ Why do you need to convert between number systems?

Converting numbers from one number system to another is called conversion between number systems.

Because decimal numbers are often used in daily life, but binary numbers are used in computers. Therefore, when using a computer, you must convert the input decimal number into a binary number that the computer can accept. After the computer runs, it converts the binary number into a decimal number that people are used to and outputs. These two conversion processes are completely automatically completed by the computer.

â‘¡ Conversion method

Turning a decimal integer into a non-decimal integer adopts the "remainder method", that is, the remainder is obtained after dividing the base.

Divide the decimal integer with the base of any decimal number one by one, until the quotient is 0, and then arrange the resulting remainder from bottom to top.

â‘¡Conversion method of decimal decimal number to non-decimal decimal number

The decimal fraction is converted into a non-decimal fraction using the "carry method", that is, the base number is multiplied to an integer.

Multiply the decimal fraction with other radix until the current value of the decimal equals 0 or meets the required precision. The integer part of the final product is arranged from top to bottom.

2. Convert a non-decimal number to a decimal number

The conversion of non-decimal numbers into decimal numbers adopts the "bit weight method", that is, the non-decimal numbers are expanded by bit weights and then summed.

3. Convert between two, eight, and decimal numbers

(1) Conversion method between binary number and octal number

â‘  When converting a binary number to an octal number, follow the method of "three digits and one digit".

Take the decimal point as the boundary, divide the integer part into groups of three digits from right to left. When the highest digit is less than three digits, add 0 to make up three digits; the decimal part is from left to right, every three digits are grouped, and the least significant digit is less than three digits. When, add 0 to make up three places. Then, the three-digit binary numbers of each group are expanded according to the weight and added to obtain an octal number.

â‘¡When converting an octal number into a binary number, the method of "one digit divided into three digits" is adopted.

That is, the number on each octal number is represented by the corresponding three-digit binary number.

â‘¢Conversion method between binary number and hexadecimal number

a. When converting a binary number to a hexadecimal number, follow the "four digits and one digit" method.

With the decimal point as the boundary, the integer part is divided into groups of four digits from right to left. When the highest digit is less than four digits, 0 is added to make up four digits; the decimal part is from left to right, when the least significant digit of each group of four digits is less than four digits , Add 0 to make up four places. Then, the four binary numbers of each group are expanded and added together to obtain a hexadecimal number.

b. When converting a hexadecimal number into a binary number, the method of "one digit divided by four" is adopted.

That is, the number on each hexadecimal number is represented by the corresponding four-digit binary number.

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