A03: Digital Representation

 

ASCII (American Standard Code for Information Interchange)

• 7-bit character code
• values: 0-127

Representing Sound

• For the computer to process the sound, it needs to be converted into binary.
• Convert the captured sound unto digital signal using microphone.

Representing Number

• In a computer system numbers are represented by a string of bits called Binary Number.

Number System

           1. Decimal (Base 10) Number system

• 10 symbols: 0-9
• Uses positional notational
  
  

• 2. Binary (Base 2) Number system

• Binary number system has two symbols: 0 and 1, thus the base used is 2
  

  Decimal to binary: Divide the number repeatedly by 2 until we get '0' as the quotient and remainders are written in reverse.

  Example:

  (23)₁₀= ()₂

  23/2=11 r:1

  11/2=5 r:1           Read from below          
  

  5/2=2 r:1              10111

  2/2=1 r:0

  0/2=0 r:1

  (23)₁₀= (10111)₂

  Binary to decimal (Use the weighted sum method)

  Example:

  (10111)₂ = ( )₁₀

  =1*2⁴+1*2²+1*2¹+1*2⁰

  =16+4+2+1

  =23

  (10111)₂ = (23)₁₀
  
  

  3. Hexadecimal (Base 16) Number System

• Uses 16 symbols: 0-9,A-F

• 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
  
  Hexadecimal to Binary

  Example:

  (56)₁₆= (01010110)₂

  Binary to hexadecimal

  Divide the bits into 4

  (01010110)₂= ()₁₆

  0101                ]     0110

  =1*2²+1*2⁰                =1*2² + 1*2¹

  =4+1                    =4+2

  =5                          =6

  (111000)₂ = (56)₁₆

  Hexadecimal to decimal

  (56)₁₆= ()₁₀

  5*16¹+ 6*16⁰

  80+6

  86

  (56)₁₆= (86)₁₀

  Decimal to hexadecimal

  Convert the decimal into binary and then convert to hexadecimal

  (86)₁₀= ()₂

  86/2=43 r:0

  43/2=21 r:1

  21/2=10 r:1             read from below:

  10/2=5 r:0               1010110

  5/2=2 r:1

  2/2=1 r=0

  0/2=0 r=1

  (86)₁₀= (1010110)₂

  (1010110)₂= ()₁₆

  Divide into 4bits

  101                ]     0110

  =1*4 + 1*1        =1*4+1*2

  =5                       =6

  (1010110)₂= (56)₁₆
  
  

  4. Octadecimal (Base 8) Number system

• It uses 8 symbols: 0-7
  

  Octadecimal to Binary

  Example:

  (32)₈= (11010)₂

  Binary to Octadecimal

  Divide the bits into 3

  (11010)₂= ()₈

  010           ]     11

  =1*2¹             =1*2¹ + 1*2⁰

  =2                   =2+1

                          =3

  (11010)₂= (32)₈

   

  Octadecimal to decimal

  (32)₈= ()₁₀

  =3*8¹+ 2*8⁰

  =24+2

  =26

  (32)₈= (26)₁₀

   

  Decimal to Octadecimal

  Convert the decimal into binary and then convert to Octadecimal

  (26)₁₀= ()₂

  26/2=13 r:0

  13/2=6 r:1

  6/2=3 r:0                  read from below:

  3/2=1 r:1                  11010

  0/2=2 r:1

  (26)₁₀= (11010)₂

   

  (11010)₂= ()₈

  Divide into 3bits

      11           ]         010

  =1*2¹+1*2⁰        =1*2¹

  =3                        =2

  (1010110)₂= (32)₈
  
  
  
  

Number Representation- Unsigned

• _{If there are N bits in the binary number, the range of the number is: 0- 2^N-1}
  
  

  
  
  
  

Number Representation- Signed

• ^{The leftmost bit is used to indicate the sign:}
• 0 for positive
• 1 for negative
• Positive values have identical representation in all system.
• Negative Number have different representations
• Sign-and-magnitude
• 1's-complement
• 2's-complement
• 1's-complement
• Negative values are obtained by complementing each bit of the corresponding positive number.
  
  

• 2's-complement
• obtain by forming bit complement of the number, then add 1

• It is the most efficient way to carry out addition and subtraction operations.
• Addition
• Rules:
• 0+0=0
• 1+0=1
• 1+1=10 (binary for 2)
• 1+1+1=11 (binary for 3)

• Subtraction
• Rules:
• 0-0=0
• 1-0=1
• 1-1=0
• 0-1=1 (Borrow 1)
• For subtraction, first we convert it to addition by changing the signs of the bits.

• Range: -2^n-1 to 2^n-1-1
• 4bits: -8 to +7

• Overflow
• It occurs when the answer doesn't fit in the range given
• Overflow occurs when 2's-complementary numbers of same sign are added and the result has the opposite sign.
• (+A) + (+B) = -C
• (-A) + (-B)= +C

• Sign Extension
• If it is a positive bit numbers add zero in the front
• Ex: 001 can be written in five bits by adding 2 zeros in the front i.e., 00001
• If it is a negative bit numbers add one in the front
• Ex: 101 can be written in five bits by adding 2 ones in the front i.e., 11101
• Character Representation
• It uses 7-bit codes