Representation of unsigned and signed numbers

Fixed point and floating point representations of numbers

Character and Chinese character coding method

Data check code

 

Numerical data represent

data ： Signed number An unsigned number

Signed number ： Original code Complement code Inverse code

 

• Numerical data in a computer

Binary number （B）

Octal number （Q）

Decimal number （D）

Hexadecimal number （H)

 

Number system representation and conversion

 

• Unsigned and signed numbers

An unsigned number ： All binary bits of the whole machine word length represent numeric bits , Equal to the absolute value of a number

   The machine word length is n+1 The representation range of an unsigned number of bits 0~2n+1-1

Signed number ： just 、 negative

   The machine word length is n+1 The signed number of bits represents the range -2n+1~2n+1-1 8 position （-128-127）

Integers ： Sign bit Value bits .

decimal ： Sign bit . Value bits

• Original code representation

• Reverse code representation

• Complement representation

The original code and complement are interchangeable

X Another way to be negative ： The first of the last 1 And to the right 0 remain unchanged , On the left, take the opposite , Sign bit

remain unchanged .

[X] primary =10111001100

[X] back =11000110100

• Comparison of three coding systems

Complement and inverse sign bits can be treated as part of numeric bits , Operate with numeric bits ; However, the sign bit of the original code is not allowed to operate with the numeric bit , It has to be dealt with separately

 

Fixed point representation and floating point representation of machine numbers -- According to whether the position of the decimal point is fixed

Fixed point representation ： The position of all decimal points is fixed

• Fixed point decimals ：Xs.X1X2……Xn

Scope of representation ：X Maximum positive number =1-2-n   X Minimum positive number =2-n

Original code ：-（1-2-n）~1-2-n

Complement code ：-1~1-2-n

• Fixed point integers ：XsX1X2……Xn ( The machine word length has n+1 position ）

The original code indicates the range ：-（2n-1）~2n-1

The complement indicates the range ：-2n~2n-1

Floating point representation

Floating point numbers ： The position of the decimal point floats as needed

N=M×rE

1. r Cardinal number , Usually r=2
2. E It's a step code , Signed integers , It is often expressed by code shift or complement
3. M mantissa , Signed decimal number , It is often expressed by original code or complement code

• The bottom of a floating point number is implicit , Does not appear in the whole number of machines
• The sign bit of the order code is es, Order code e The size of is reflected in the number N The actual position of the decimal point in
• The sign bit of mantissa is ms, It is the sign bit of the entire floating point number , mantissa m Represents the precision of the floating point number

The representation range of floating-point numbers

• Normalize floating-point numbers ： In order to improve the operation accuracy , We need to make full use of the mantissa significand . That is, the highest digit of the mantissa must be a valid value .

1/2<=|M|<1

• When the mantissa is represented by a complement , The normalized floating-point number shall satisfy that the highest digit of the mantissa is different from the sign bit

Code shifting representation

In truth X Add an offset value （ constant ）, amount to X On the number axis, I shifted a distance in the positive direction .

• [X] move = Offset value +X
• Offset value =2n-1

characteristic ：

• highest “0” A negative number , highest “1” It means a positive number
• Shift to full 0 when , His corresponding truth value is the smallest , Shift to full 1 when , The corresponding truth value is the maximum
• The representation of truth zero is unique in code shift ,1000 0000
• The shift maps the truth value to a positive field , Therefore, the code shift can be regarded as an unsigned number , Compare sizes directly according to the unsigned number rule
• The code shift and complement of the same value are opposite except the highest bit , Everyone else is the same

The reason why order code adopts code shift

• The reason why the order code of floating point number adopts code shift , If the order code is large, its corresponding truth value is large , If the order code is small, its corresponding truth value is small
• Simplify the zero judgment circuit in the machine . When the order codes are all zero （ Everyone who shifts the code is 0 when , The corresponding order code value is the smallest ）, The mantissa is all zero , Indicates that the machine is zero

Fixed point 、 Floating point representation and fixed point representation 、 Floating point computer

Fixed point 、 The difference between floating point notation

1. If the word length is the same , The range represented by floating-point numbers is much larger than that of fixed-point numbers
2. Floating point precision decreases , The points on the number axis are arranged more sparsely
3. Floating point operations are more complex than fixed-point operations
4. Overflow handling ： Fixed point operation , The result of the operation is beyond the representation range of the number , There is an overflow , But in floating-point operations , The operation result exceeds the representation range of the mantissa and does not necessarily overflow , Overflow occurs when the order code exceeds the range that can be represented .

Fixed point machine and floating point machine

• Pointing machine ： Mainly fixed-point operation , Floating point operation is realized by software
• Pointing machine + Floating point software ： Floating point arithmetic unit is a unit specially used for floating point arithmetic
• Floating point machine ： With floating-point instructions and basic floating-point arithmetic unit

The representation of non numerical data

Non numeric data is character data , That is, the characters 、 character string 、 Various data such as image symbols and Chinese characters , It is not used to represent the size of the value

Representation of characters and strings

1. ASCII Character encoding

Represent a character in seven binary digits , Including ten binary digits （0~9）、52 English capital letters （A-Z,a-z）、34 A special symbol ,3 A control symbol , common 128 Characters

A computer usually stores one character per byte

1. String storage

Vector storage method ： Occupy a contiguous space in memory , Each byte stores a character code , All elements of the string are physically adjacent . What is actually stored in each byte is the of the corresponding character ASCII code .

Unified code

Unicode： Unified code , A character coding scheme that can accommodate all characters and symbols in the world .

Unicode The basic method ： Use one 16 A number of bits to represent each symbol , Can be said 65536 A different character or symbol . It is called the Basic Multilingual plane （BMP）, namely UCS-2 code .

USC-4, Can be said 100 More than 10000 custom characters

Representation of decimal and number strings

Coding of decimal numbers

Use four binary numbers to represent one decimal number , Called binary coded decimal numbers , abbreviation BCD code .

BCD The code has a binary form , It retains the characteristics of decimal numbers

Decimal string

1. Uncompressed decimal string ： A byte that holds a decimal number or symbol ASCII-7 code （ A byte =8 Bit binary number 1Byte=8bit）
2. Compressed decimal string ： One byte can hold two bits BCD A decimal number represented by a code , Save storage space , Convenient for direct decimal arithmetic operation

Examples of data representation in modern microcomputer systems

Most modern microcomputer systems use Intel Series microprocessors , structure ：IA-32 structure

IA-32 Structure basic data type ： byte （8 position ）、 word （16 position ）、 Two words （32 position ）、 Four words （64 position ）、 Double four characters （128 position ）

1. Unsigned integer ：
1. byte ：0~255
2. word ：0~65535
3. Two words ：0~2^32-1
4. Four words ：0~2^64-1
2. Signed integers
1. byte ：-128~127
2. word ：-32768~32767
3. Two words ：-2^31~2^31-1
4. Four words ：-2^63~2^63-1
3. Floating point numbers

And IEEE 754 The format specified in the standard directly corresponds to

1. The pointer is the address of the main memory unit ,IA-32 Structure defines two types of pointers ： Near pointer （32 position ） And the far pointer （64 position ）
2. String data ： Including bit string 、 Byte string 、 String and double string . A string can contain from one byte to 4GB The content of .
3. BCD Count ：IA-32 In structure BCD Code actually means 8421 code