Representation of base and number 2

Learning notes ：
Carry system ： It is a counting method that uses a set of fixed symbols and unified rules to represent the size of numerical values .
base ： The number of digits used in the number system
Position right ： Represents the weight at different positions

Carry counting system

• Decimal system
  Definition ：10 An ordered number ：0、1、2、3 …… 9 And a decimal symbol “."
  usage ： Add “ Dot into one “, Subtraction ” Take a step back “
  Any decimal number can be expressed as a polynomial with a power based on ten
  Binary system
  Definition ： Binary digital symbols ,0 and 1, The base number is 2
  usage ： On the two into one , take a temporary measure 2n
  octal
  Definition ： have 8 A different number symbol 0、1、2…… 7, The base number is 8
  usage ： Every eight enters one. , take a temporary measure 8n
  Hexadecimal
  Definition ： have 16 A different number symbol 0、1、2…… 9、A、B、C、D、E、F, The base number is 16
  usage ： Every 16 to one , take a temporary measure 16n

The representation of carry digits

Subscript marks ：（1110）₂
The suffix letter indicates ： Binary system （B）、 octal （Q）、 Decimal system （D）、 Hexadecimal （H）

Hexadecimal conversion

R Base number → Decimal system
The law of expansion by right ： Find the sum of the product of each bit and its weight

Decimal system →R Base number
Integral part ： Divide the base and take the remainder
The fractional part ： Multiply by Radix to round

Binary system → 8、 ... and / Hexadecimal
Centered on the decimal point , Divide to the left and right 3/4 position , In the middle of the 0 Don't omit . If the highest / The lowest position is insufficient 3/4 position , Separately supplement 0 that will do

8、 ... and / Hexadecimal → Binary system
General eight / Each number in hexadecimal is expanded to 3/4 Bit binary number , The order of arrangement and the position of the decimal point remain unchanged , And remove the redundant at both ends 0

octal → Hexadecimal
Take binary as the bridge

Signed digits

Number of machines ： Use in the machine , Number digitized with the sign
There are three representations of machine number ： Original code , Inverse code , Complement code （ The length of the number of machines is limited by the word length ）
Original code

• Definition ： Numerical bits are expressed in absolute values ,0（ Plus sign ）、1（ Minus sign ）. The original code is the absolute value of digital symbol bits plus numbers
  operation ：
  Same number ： Add the values
  Different sign ： Compare the absolute value , Large number minus decimal , Finally, choose the appropriate symbol

Inverse code

• Definition ： The value bits of positive and inverse codes are the same as the true value bits （ That is, the same as the original code ）, The sign bit of negative inverse code remains unchanged , The rest is reversed

Complement code

• Complement allows the sign bit to directly participate in the operation , Simplify operations , It can be reduced to plus
  Positive complement is the same as the original , A negative complement is a sign bit invariant , The rest is reversed , Finally add 1
  When the calculation result does not exceed the range of the machine , The addition of complement is still complement , Complement subtraction and change addition operation , Symbol bits participate in operations ,

The representation of numbers

Fixed point representation
Fixed-point number ： The decimal point position is fixed
Fixed point decimals ： The position of the decimal point is after the sign digit , Before the highest order of the significant value part

• Set the word length of a machine 8 position , A sign bit
  The maximum number of fixed-point decimals ：0.1111 111 Minimum non-zero positive number ：0.0000 001
  The maximum number of fixed-point positive numbers ：0111 1111 Minimum positive number ：0000 0001

overflow
underflow ： If the data is less than the minimum value that the fixed-point number can represent , Computer work 0 Handle
Overflow ： When the data is greater than the maximum value that the fixed-point number can represent , The computer cannot express

Floating point number means

• The range of values represented by fixed-point numbers is small , And it is easy to overflow during operation , Therefore, real numbers are more represented in a way similar to scientific counting , Floating point numbers
  General expression ：F=M*2^p
  Pure decimal M It's the mantissa , Precision of representation number ,P It's an order code , Determine the position of the decimal point , Represents the range of numbers