Deep understanding of bit operations

What is base

​   The decimal system is also called Carry counting system , It's a human defined count method with carry （ There are counting methods without carry , For example, the original twine counting method , Commonly used in singing “ just ” Word count , And similar tally mark Count ）. For any system —X Base number , It means that the number operation on each bit is every X Into one . The decimal system is one in ten , Hexadecimal is every hexadecimal into one , Binary system is a binary system , And so on ,x The base is the square x carry .

​   So What is the essence of operation Well ？ I think it's counting . Break the inherent thought , We think of every number as Symbol , This is much more interesting , We can write our own radix , You can do your own calculations

Hexadecimal conversion

1、 Other decimal system to decimal system

Let's start at the lowest point , Extract the number in each bit , Multiplied by the base number ( digit -1) Power , Sum up

// 2^n  Binary for ：1{n individual 0} 2^n-1  Binary for ：{n individual 1}
 Binary to decimal ：	1011 = 1*2^0 + 1*2^1 + 0*2^2 + 1*2^3 = 11
 Octal to decimal ：	0123 = 3*8^0 + 2*8^1 + 1*8^2 = 83
 Hexadecimal to decimal ：	0x34A = 10*16^0 + 4*16^1 + 3*16^2 = 786

2、 Decimal to other bases

Divide the number by the base number to convert , Know Shang Wei 0 until , Then reverse the remainder

 Decimal to binary ：	56 = 2^5 + 2^4 + 2^3 = 111000
 Decimal to octal ：  156 = 0234
 Decimal to hexadecimal ：  256 = 0x164

3、 Binary to other bases

Group binary numbers into three bits ( Start at the bottom ), Convert to corresponding octal In hexadecimal, there are four digits

 Binary to octal ： 11 010 101 = 0325
 Binary to hexadecimal ： 1101 0101 = 0xD5

4、 Other base to binary

Put octal / Each hexadecimal bit is converted to a corresponding one 3/4 A binary number of bits

 Octal to binary ：	0123 = 0 001 010 011 = 1010011
 Hex to binary ：	0x156 = 0001 0101 0110 = 101010110

Original code 、 Inverse code 、 Complement code

• For signed numbers ：
  • The highest bit of binary is the signed bit ：0 It means a positive number ,1 It's a negative number
  • The original code of a positive number 、 Inverse code 、 The complement is the same
  • 0 The inverse of 、 The complements are all 0
  • The inverse of a negative number = Its original symbol bit remains unchanged , Reverse other bits
  • A negative complement = Its inverse +1
  • When the computer is working , They all operate in the form of complement

		1                  		     	 -1   
 Original code  0000 0001						 Original code  1000 0001
 Inverse code  0000 0001						 Inverse code  1111 1110
 Complement code  0000 0001						 Complement code  1111 1111

An operator

Operator	describe	The rules
&	Bitwise AND	Also for 1, The result is 1, Otherwise 0
|	Press bit or	There is one for 1, The result is 1, Otherwise 0
^	Bitwise XOR	When the result is different, the result is 1, Otherwise 0
<<	Move left	Move left N Bits are multiplied by 2 Of N Power
>>	Move left	Move right N Bits are divided by 2 Of N Power

The essence of bit operation

As long as it's an operation , All complement ！！！ The end of the operation needs to be converted into the source code

// 2 Complement code  3 Complement code 
2 & 3 = 0000 0010 & 0000 0011 = 0000 0010 = 2
2 | 3 = 0000 0010 | 0000 0011 = 0000 0011 = 3
2 ^ 3 = 0000 0010 ^ 0000 0011 = 0000 0001 = 1

// -2 Complement code  3 Complement code   result ( Complement code ) ->  Launch original code 
-2 & 3 = 1111 1110 & 0000 0011 = 0000010 =  2

//  Displacement operators   The arrow moves in which direction 
//  Move right ： The low overflow sign bit remains unchanged , And use the sign bit to fill the high bit of overflow 
//  Move left ： The sign bits remain the same , Low complement 0
//  Move left n position  =  Original value  * 2^n  Move right n position  =  Original value  / 4
1 >> 2 = 0  
1 << 2 = 0000 0100 = 4 

​   thus , Bit operation has been explained . We can find that the function of bit operation is really just doing Mark nothing more , Is worthy of the name Bitwise operation .