Number  Systems
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Number  Systems Number Systems Presentation Transcript

  • Unit 4: The Number Systems and Data Representation
  • Problem-solving using “computers”
    • Computers solve “computable” problems
    A Problem Describing The Problem in Math. “ Computing” The Corresponding Math. Problem Returning The Result Solution To The Problem Human problem-solving v.s. computer-based problem-solving
  • Transformation input think act input compute act   Binary system
  • The Number System
    • A number system is to enumerate the “states” of something
      • For example, money, days, month, year, minutes, hours, …
    • In human’s world, we have very good tools  
    …which counts 10 states
  • The Needs of Number Systems
    • However, there are some states are not Decimal. For example,
      • Days and weeks
      • Minutes and hours
      • Months and year
    • We need convenient representations for different “nature states”
  •  
  • Egyptian numerals (10-based) Source: http://www-gap.dcs.st-and.ac.uk/~history
  • Babylonians numerals (60-based) Source: http://www-gap.dcs.st-and.ac.uk/~history
  • 1*60 3 + 57*60 2 + 46*60 + 40*60 0 = 424000 Source: http://www-gap.dcs.st-and.ac.uk/~history
  • Chinese numerals (10-based) 4359 45698 Source: http://www-gap.dcs.st-and.ac.uk/~history 壹貳參肆伍陸柒捌玖拾佰仟萬億兆京…
  • The Number Systems
    • A number system specifies how/what “numbers” represent and operate.
    • A K-based number system uses N different symbols to represent N different entities or “states”
    • Each symbol is a “digit”
    • Multiple digits are used for describing M>N states
  • K-based Number System
    • 2-based: ON/OFF; Yes/No; T/F; 1/0; etc.
    • 3-based: A/B/C; True/False/Unknown; 0/1/2; etc.
    • 7-based: Mon/Tue/Wes/Thu/Fri/Sat/Sun; 0/1/2/3/4/5/6; etc.
    • 8-based: ; 0/1/2/3/4/5/6/7
  • K-based Number System
    • 10-based:  /  /  /  /  /  /  /  /  /  ; 0/1/2/3/4/5/6/7/8/9; etc.
    • 16-based:  /  /  /  /  /  /  /  /  /  /  /  ; 0/1/2/3/4/5/6/7/8/9/A/B/C/D/E/F; etc.
    State-1 State-2 State-3 State-4 State-5 State-6 State-7 State-8 State-9 State-10 State-11 State-12 State-13 State-14 State-15 State-16
  • K-based Number System
    • In the 16-based number system:
    • 0/1/2/3/4/5/6/7/8/9/A/B/C/D/E/F; etc.
    Why not “ 1 0 ” 2 digits
  • More States
    • In a 6-based number system
    •  : state 1 (empty state) (0)
    •  : state 2 (1 st )
    •  : state 3 (2 nd )
    •  : state 4 (3 rd )
    •  : state 5 (4 th )
    •  : state 6 (5 th )
    •  : state 7 (6 th )
    •  : state 8 (7 th )
    •  : state 9 (8 th )
    •  : state 10 (9 th )
    •  : state 11 (10 th )
    •  : state 12 (13 th )
  • More States
    • In a 6-based number system
    • 0: state 1 (empty state) (0)
    • 1: state 2 (1 st )
    • 2: state 3 (2 nd )
    • 3: state 4 (3 rd )
    • 4: state 5 (4 th )
    • 5: state 6 (5 th )
    • 10: state 7 (6 th )
    • 11: state 8 (7 th )
    • 12: state 9 (8 th )
    • 13: state 10 (9 th )
    • 14: state 11 (10 th )
    • 15: state 12 (13 th )
    • 20: state 13 (14 th )
  • The Number of Cases
    • Generally, in a K-based number system,
      • 1 digit describes K 1 cases
      • 2 digits describes K 2 cases
      • 3 digits describes K 3 cases
      • N digits describes K N cases
    • For example,
      • 2 3-based (0,1,2) digits: 00, 01, 02, 10, 11, 12, 20, 21, 22 (3 2 =9 cases)
      • 3 10-based (0,1,2,..,9) digits: 000, 001, 002, 003, 004, 005, 006,007, 008, 009, 010, 011,…,998, 999 (10 3 =1000 cases)
  • K-based vs The Decimal system (10-based)
    • In general, any K-based number N can be expressed as
    • Usually, N k is denoted as
      • (A p-1 A p-2 ….A 1 A 0 . A -1 A -2 ….A -q ) k
      • A p-1 :Most Significant Digit, MSD
      • A -q :Least Significant Digit, LSD
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Using octal and hex numbers
    • Computers use binary, but the numbers are too long and confusing for people
    • Translation between binary and octal or hex is easy
      • One octal digit equals three binary digits
      • 101101011100101000001011
      • 5 5 3 4 5 0 1 3
      • One hexadecimal digit equals four binary digits
      • 101101011100101000001011
      • B 5 C A 0 B
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • The binary number system
    • In modem computers, the binary number system is easier to be implemented.
    • Currently, they are implemented in silicon chips (VLSI)
      • Circuits for computation
    01111 00011 10010 15 3 + 10 + 2 18
  • The binary number system
    • The binary (base 2 or 2-based) number system uses two “binary digits, ” (abbreviation: bits) -- 0 and 1
    • A bit is a single two-valued quantity: yes or no, true or false, on or off, high or low, good or bad
    • One bit can distinguish between two cases: T, F (True/False; Yes/No)
    • Two bits can distinguish between four cases: TT, TF, FT, FF
    • Three bits can distinguish between eight cases: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF
    • In general, n bits can distinguish between 2 n cases
    • A byte is 8 bits, therefore 2 8 = 256 cases
  • Data Processing in Computers with the Binary System
    • Data processing in binary computers
      • Numeric data: +2, -5.5, 50000.235698, etc.
      • Alphanumeric data: characters, name, address, telephone number, etc.
    • Two main tasks:
      • All data are converted into proper binary formats (bits)
      • Rules for counting and computing these bits
  • Part I: Numeric Data
    • Numeric data:
      • integers, real
      • positive, negative
    • Representation of numeric data
      • Sign-magnitude
      • 1’s Complement
      • 2’s Complement
    • Negative/Positive in a n-bit integer I
    • I=(A n-1 A n-2 ….A 1 A 0 ) 2
      • A n-1 = “0”, I is positive
      • A n-1 =“1”, I is negative
  • Sign-Magnitude
    • An n-bit integer I=
      • Min: 00000…0 (n-1 0’s)=0
      • Max: 11111…1 (n-1 1’s)=2 n-1 -1
      • +0= (0000) 2 vs –0=(1000) 2
      • +3= (0011) 2 vs –3=(1011) 2
    • Disadvantages
      • 2 different 0’s (+0/-0 )
      • Not easy to be implemented in simple circuits (usually adder)
  • 1’s Complement
    • An n-bit integer I=
      • For positive integers: the same as that in sign-magnitude
      • For negative integers: complement their positive representations
      • For example: 4-bit integers:
        • +3=(0011) 2 , -3=(1100) 2
        • +0=(0000) 2 , -0=(1111) 2
    • Disadvantages
      • 2 different 0’s (+0/-0 )
      • Easy to be implemented in simple circuits (usually adder) but not as efficient as 2’s complement
  • 2’s Complement
    • An n-bit integer I=
      • For positive integers: the same as that in sign-magnitude
      • For negative integers: find their 1’s complement + 1
        • For example: 4-bit integers:
        • +3 = (0011) 2 , -3 = (1100) 2 +1 = (1101) 2
        • +0 = (0000) 2 , -0 = (1111) 2 + 1 = (0000) 2
        • There is only one “0”
  • Comparison: 4-bit representation of integers
  • Range of integers Suppose that we use a byte (8-bit) to represent an integer (00000000) 2 (00000000) 2 (10000000) 2 =-128 (01111111) 2 =+127 2‘s complement (11111111) 2 (00000000) 2 (10000000) 2 =-127 (01111111) 2 =+127 1‘s Complement (10000000) 2 (00000000) 2 (11111111) 2 =-127 (01111111) 2 =+127 Sign-Magnitude -0 +0 Max. (-) Max. (+) -(2 n-1 )~0 0~2 n-1 -1 2‘s complement -(2 n-1 -1)~0 0~2 n-1 -1 1‘s Complement -(2 n-1 -1)~0 0~2 n-1 -1 Sign-Magnitude Negative numbers Positive numbers
  • Note
    • Representation level
    • Implementation level
  • Other Binary Codes for Decimal Numbers
    • BCD code
    • 2421 code
    • Excess-3 code
    • 84-2-1 code
    • 4-bit for a number (0~9)
  • 8421 BCD Code
    • In the 8421 Binary Coded Decimal (BCD) representation each decimal digit is converted to its 4-bit pure binary equivalent.
    • Each decimal number maps to four bits and is weighted by its bit-position (each bit represents a number, 1, 2, 4, 8)
    • BCD code is also called as 8421 code
  • 4221 BCD Code
    • In 4221 BCD code, each bit is weighted by 4, 2, 2 and 1 respectively.
    • Unlike BCD coding there are no invalid representations.
    advantages
  • Excess-3 Code
    • Add 3 for each binary number
    • Examples
      • Excess-3 code of “2 10 ” = (0010) 2 +(0011) 2 =(0101) 2
      • Excess-3 code of “5 10 ” = (0101) 2 +(0011) 2 =(1000) 2
    • Advantages
  • 84-2-1 Code
    • 4-bit for each number (0~9) weighted (left to right) as 8, 4, -2, and -1 。
    • Example
      • 84-2-1 code of “3” = 0101 (0+4+0+(-1)=3)
      • 84-2-1 code of “5” =1011 (8+0+(-2)+(-1)=5)
  • Self-complementing code Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Gray Codes
    • In pure binary coding or 8421 BCD, counting from 7 (0111) to 8 (1000) requires 4 bits to be changed simultaneously. If this does not happen then various numbers could be momentarily generated during the transition so creating spurious numbers which could be read.
    • In gray coding, only one bit changes between subsequent numbers.
    • Generating gray codes: start with all 0s and then proceed by changing the least significant bit (LSB) which will bring about a new state.
    • Advantages: fast, relatively free from errors.
  • Gray Codes
    • Gray code is not unique, there are many possibilities to generate gray codes
    • For example,
      • G1={0=00 , 1=01 , 2=11 , 3=10}
      • G2={00=10 , 1=11 , 2=01 , 3=00}
    • Unique gray code: reflected Gray code
    • 從十進位 -> 反射葛雷碼
    Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • 從反射葛雷碼 -> 十進位數字 Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Floating-Point Representation
    • Integers are fixed-point numbers in binary computers
    • Floating-point literals are written with a decimal point: 8.5 -7.923 5.000
    • Real numbers are represented as “floating-point” numbers in binary system (all computers)
    • The representation of floating-point numbers are various in different CPU
    • In Intel 80486 CPU
      • Single Precision: 32 bits
      • Double Precision: 64 bits
      • Extended Precision: 80bits
  • Floating-point literals
    • Floating-point numbers may also be written in “scientific notation”– times a power of 10
    • We use E to represent “times 10 to the”
    • Example: 4.32E5 means 4.32 x 10 5
    http://www.nuvisionmiami.com/books/asm/workbook/floating_tut.htm
    • 範例
    Source: 計算機概論 , 王孝熙 著 , 東華書局 .
  • Example
  • Binary Arithmetic on Numeric Data
    • Binary addition
    • Binary subtraction
    • Binary multiplication
    • Binary division
    • To be discussed in Unit 5
  • Part II: Alphanumeric Data
    • Alphanumeric data: character, letter, symbol, digit)
    • Not for calculation, but for representation some “meanings”
    • Coding
      • ASCII(as-kee):America Standard Code for Information Interchange
      • EBCDIC(eb-ce-dick):Extended Binary Coded Decimal Interchange Code (used by IBM, UNIVAC mainframes)
      • BIG-5: for Chinese characters
      • Uni-code
  • ASCII Code
    • 7-bit for each character, 2 7 =128 combinations for 128 characters
  • Extended ACSII 8-bit for each character, 2 8 =256 combinations for 256 characters 。
  • Examples N: 4E=00101110 a: 61=01100001 t: 74=01110100 i: 69=01101001 o: 6F=01101111 n: 6E=01101110 a: 61=01100001 l: 6C=01101010
  • EBCDIC
    • 8-bit for each character
      • 4-bit Zone bits: identifying the code is for character, (un)signed number, or symbols
      • 4-bit Digit bits: for numbers 0~9 。
    http://www.natural-innovations.com/computing/asciiebcdic.html
  •  
  • Big5 Code
    • Used for Chinese characters
    • 1 character = 2 bytes (16 bits)
    • Example
  •  
  • Unicode
    • ASCII was very simplistic, and so was extended by adding 'extended' sets by various manufacturers. Apart from being confusing this was still restricted to 256 characters.
    • Now computers are more widely established around the world the need to show other characters such as Japanese and Chinese languages along with various symbols became more important.
    • 2 bytes (16-bit) for each unicode
  • Unicode samples
  • Binary Operations on Alphanumeric Data
    • Binary addition?
    • Binary subtraction?
    • Binary multiplication?
    • Binary division?
    • To be discussed in Unit 5
  • Part III: Extensions
    • Everything in the computer is stored as a pattern of bits
      • Binary distinctions are easy for hardware to work with
    • Numbers are stored as a pattern of bits
      • Computers use the binary number system
    • Characters are stored as a pattern of bits
      • One byte (8 bits) can represent one of 256 characters
    • So, is everything in the computer stored as a number?
      • No it isn’t, it’s stored as a bit pattern
      • There are many ways to interpret a bit pattern
  • Extension: Image representation
    • Nature creations are analog, not digital
    • Digitizing (sampling)
    • Resolution: 800*600, 1026*768
    Original 7*7 digitizing 14*14 digitizing
  • For Black/White Images Original 7*7 digitizing 14*14 digitizing 0000000 1111100 … 00000000000000 00000000000000 01111111110000 01111111111000 01111111111110 … 0000 0001 1111 00 … 0000 0000 0000 0000 0000 0000 0000 0111 1111 1100 0001 11111111100001111111111110… (01F0..) 2 (00000007FA1..) 2
  • For Colored Images
  • Extension: Sound Representation time AMP
  • Extension: Sampling Max. AMP/16 t 1 t 2 t 3 t 4 … 16=2 4 t 1 : 8=(1000) 2 t 2 : 9=(1001) 2 t 3 : 7=(0111) 2 t 4 : 2=(0010) 2 t 1 : 2=(0010) 2 … Quality vs sampling rate
  • Conclusion