Carry counting system

in the light of How data is represented in a computer and its arithmetic and logical operations Of ⽅ Law , A carry counting system is proposed , The computer makes ⽤ Machine language ⾔ Namely ⼆ Base bit , except ⼆ There are other hexadecimal digits for ⼈ Generic language ⾔ The base digit of the count , In order to better realize arithmetic ⾔ The interworking of needs to enter ⾏ Hexadecimal conversion .

Basic concepts ：

often ⽤ Hexadecimal digits of

1.  ⼆ Base number Binary： Only by 0 and 1 Composed of , stay ⼀ individual ⽐ There are only two situations that can be represented on the special bit , On the two into one , The data ends with B or （）2 Express .

2.  ⼋ Base number Octal： from 0～7 common 8 A number indicates , Meet ⼋ Into the ⼀, The data ends with （）8 Express .

3.  ⼗ Base number Decimal： from 0～9 common 10 A number means , Meet ⼗ Into the ⼀, The data ends with D or （）10 Express .

4.  ⼗ Hex Hexadecimal： from 0～9,A,B,C,D,E,F common 16 A number and a word ⺟ constitute , Meet ⼗ 6、 ... and （F） Into the ⼀, The data ends with （）H or 0x Prefix usage .

5.  A weight （ Bit weight or weight ）： Hexadecimal number the real value corresponding to the position of each value , For integers , The left digit is the hexadecimal multiple of its right adjacent digit （ Such as binary integer , The weight of the left bit is the integer of the right bit 2 times , The decimal system is 10 times ）. The position of each number reflects the weight , For any binary weight, the basic formula is as follows ：

 

Hexadecimal digit weight and conversion

• ⼆ Hexadecimal weight table

 

 1. ⼆ Turn into the system ⼗ Base number

according to ⼆ Every... Of the hexadecimal weight table ⼀ Bit corresponding weight and bit weight decimal formula directly convert each ⽐ Special position 1 Add the weights of to get ⼗ Base result .

example ： about ⼆ Base number 10010010.110 turn ⼗ Hexadecimal number

According to the right expansion summation method ： take ⼆ The hexadecimal number is written as the expansion of the weighted coefficient , According to ⼗ Base addition, direct summation .

1*2^7+1*2^4+1*2^1+1*2^(-1)+1*2^(-2) = 146.75

You can also directly remember each ⼆ Base number ⽐ The weight of the bit to which the special bit belongs is ⽽ Direct additive （ For one byte 8 The rule of binary integer weight of bits is... From right to left 1 2 4 8 16 32 64 128）.

   2. ⼆ Turn into the system ⼋ Base number

⼋ The system consists of 0～7⼋ A numerical , By every three ⼆ Binary phase combination , Every time Three Can mean 2^3=8 common ⼋ A data in one case .

example ： about ⼆ Hexadecimal number 11001111.01111 turn ⼋ Hexadecimal number

According to every three digits ⼀ individual ⼋ Decimal number principle ：

⼆ Base number ：011 001 111. 011 110

⼋ Base number ：3 1 7. 3 6

Be careful , The integer part is less than three digits on the left （⾼ position ） repair 0,⼩ The number of parts is not enough. Three are on the right （ Low position ） repair 0.

  3. ⼆ Turn into the system ⼗ Hex

And ⼋ Base is similar to ,⼆ Base digits Every time 4 A for ⼀ Group , Every time 4 Bits can indicate 16 One of these values , Then add the weights of each part .

example ：⼆ Hexadecimal number 1111000010.01101

According to every four digits ⼀ individual ⼗ Principle of hex numbers ：

Binary system ：0011  1100  0010 . 0110  1000

Hexadecimal ：3  C   2.  6  8

So the end result is 3C2.68H.  Same as ⼋ Base number ⼀ sample , The integer part is not enough. Four digits are on the left （⾼ position ） repair 0,⼩ The number of parts is not enough. Four digits are on the right （ Low position ） repair 0; Every time ⼀ A byte can be divided into two ⼗ Six base number .

 

•  ⼋ Hexadecimal weight

Binary octal correspondence table

  1. ⼋ Base number Count 17.36 turn ⼆ Hexadecimal number

According to the table ：

⼋ Base number ：1 7. 3 6

⼆ Base number ：001 111.011 110

You can get ⼆ Hexadecimal number 1111.01111

  2. ⼋ Base number Count 251.5 turn ⼗ Base number

Right expansion summation method ：

2*8^2+5*8^1+1*8^0+5*8^(-1) = 168.625

  3. ⼋ Base number Count 13.40 turn ⼗ Six base number

⼋ Turn into the system ⼗ Hex has no ⼀ Step in place ⽅ Law , It can only be mediated by intermediate hexadecimal numbers , Such as ⼆ Base or ⼗ Hexadecimal number , Then enter again ⾏ transformation .

⼋ Base number ：1 3. 4 0

⼆ Base number ：001 011. 100 000

⼗ Base number ：1*8^1+3*8^0+4*8^(-1) = 11.5（ The law of expansion by right ）

⼗ Hex ：B.8

•  ⼗ Hexadecimal weight （Decimalism）

Such as the bits ,⼗ position , One hundred, etc , The weight base of each digital position is 10, That is, the integer part from the lower right to the left ⾼ There are 10^0, 10^1, 10^2 etc. .⼀ Like writing numbers ⽅ The law is ：

 

 1. ⼗ Decimal conversion ⼆ Base number ,⼀ There are two kinds of ⽅ Law ：

•  Integral part Of The method of removing base and taking remainder ,⼩ Count Part of the Rounding by base .

example ：⼗ Hexadecimal number 75.3 turn ⼆ Base number

⼗ Base number ： Integral part ——75,⼩ Several parts ——0.3

explain ： Integers 75 Divide by 2, Will get every time The remainder is written on the right , Be a businessman 0 when , The remainder is from bottom to top In turn ⼆ The low order of hexadecimal to ⾼ position .

 ⼩ Count 0.3 Multiply sequentially 2, The results obtained each time Take out the integer bits , Calculation The result is from bottom to top In turn ⼆ Base number ⼩ The low order of the number to ⾼ position . Be careful , because ⼆ The binary weight is fixed , May encounter ⼗ Base number ⼩ The number multiplication base rounding method can never return to 0 The situation of , namely ⼆ The addition of hexadecimal weights can only approximate the ⼩ The number , be accurate ⼆ Base digits ⼀ like only take 5 position .

income ⼆ Hexadecimal integers and ⼩ The final number is 1001011.010

•  Rounding up

For the given ⼗ Hexadecimal numbers are not very ⼤, It can be done by ⼆ The digits of the binary digit weight table are pieced together and added ⽽ have to .

example ：⼗ Hexadecimal number 260.75 turn ⼆ Base number

No ⽤ Division based remainder method and multiplication based rounding method , Byte lookup ⼆ Hexadecimal weight table , It can be seen from the reference that 1*2^8+1*2^2+1*2^(-1)+1*2^(-2) = 100000100.11

notes ： The ⽅ Law The number of calculation digits is not much ⼤ Of ⼗ Turn into the system ⼆ Base number Faster ⽅ then .

  2. ⼗ Base number Count 167 turn ⼋ Base number

Integer part ⽤⽅ Law and transformation ⼆ Base integer The method of removing base and taking remainder identical , Just the cardinal part of the divisor r Change into 8.

And ⼆ The decimal remainder is the same , From bottom to top ⼋ Base number ⾼ Bit to low .

  3. ⼗ Decimal conversion ⼗ Hex Count two ⽅ Law ：

•  Integral part divided by 16 Base division and remainder method and ⼩ The multiplication basis rounding method of the number part （ Same as decimal to binary ）

example ： take ⼗ Hexadecimal number 4966.494 Turn into ⼗ Six base number

explain ：

1. Integer part and ⼩ Separate several parts , The integer part is continuously divided by 16, Turn the remainder from bottom to top （⼗ Hex ⾼ Go low ） Write , And meet 16 Into the 1.

2. ⼩ Multiply several parts continuously 16, Take the integer of the result , The integer of the subsequent multiplier returns to 0 The state of , Continue to multiply . Round it up and down （ Or from left to right ） In order of ⼗ Hex goes from low to ⾼ Row , Encountered a problem that cannot be multiplied by an integer ⼩ The number takes the first 6 position .

3. You can also put ⼗ Base to zero ⼆ Base number , Then pass every 4 Position as ⼀ Group converted to ⼗ Hex .

⼗ Hex （Hexadecimalism）

  1. ⼗ Hex to ⼆ Base number

example ：⼗ Six base number AE86.1 turn ⼆ Base number

⼗ Hex ：A E 8 6. 1

⼆ Base number ：1010 1110 1000 0110.0001

  2. ⼗ Hex to ⼋ Base number

You can start with 16 to decimal , Turn eight or . Sixteen turns ⼆ Re turn ⼋, No, ⼀ Step into place . The specific method is the same as above .

  3. ⼗ Hex to ⼗ Base number —— It can be obtained by weighted expansion method

The weighted expansion is consistent with the above method .

Each hexadecimal number represents ⽅ To sum up ：

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