outline

Chapter 4

• The Hardware World

Binary number system

• Base 2 instead of 10
  • 2 digits, 0 and 1
  • BIT (Binary digIT)

Binary number system 2

• All information stored internally in binary form
  • letters, intergers, decimal numbers, graphic images
  • external/internal encoding/decoding

Internal representation

• letters A = 01000001
• integers 65 = 01000001
• different information but same internal code
• computer needs to know the data type to know how to interpret (assign meaning)

COMPUTER ARITHMETIC:Place Values

• number system with five symbols: 0, 1, 2, 3, and 4
• How would we count 7 coins?
• 1 stack & 2 coins

COMPUTER ARITHMETIC:Place Values 2

• How would 24 coins look?
• Or how about 25 coins?

COMPUTER ARITHMETIC:Place Values 3

• numerical representaion scheme:
  • 4 = stack-of-stacks
  • 1 = 1 stack of 5
  • 3 = single coins

COMPUTER ARITHMETIC:Place Values 4

• This idea of giving symbols different values that depends on where they are writen and using zeros to fill the empty positions is called the place-value system.

COMPUTER ARITHMETIC:Place Values 5

• The stack size of number system is called its base

COMPUTER ARITHMETIC:Place Values 6

• What number system (base) did we design?

COMPUTER ARITHMETIC:The Base-2 System - Binary

• Binary Addition:
  

  ZERO         F        0
  ONE          T        1
  TWO         TF       10
  THREE       TT       11
  FOUR       TFF      100
  FIVE       ???      ???

COMPUTER ARITHMETIC:The Base-2 System - Binary 2

• Binary Subtraction:
  

       24 - 5  =>     ?? + 4 = 24
  Ex:  10101 - 11 =>  ????? 
                      +  11
                      10101

COMPUTER ARITHMETIC:The Base-2 System - Binary 3

• Multiplication:
  

       100001
     _x   101_
       100001
      000000
   + 100001   

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion

• Ex: 1024₁₀; 6786₁₀

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 2

• Negative Numbers in Binary
• Three representation schemes are used:
  1. Signed Magnitude
     Left most bit is the sign bit (0 => + & 1 => -).
     Remaining bits hold absolute magnitude.

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 3

• Ex:     210 => 0000 00102
           -210 => 1000 00102
  

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 4

• 2. One's Compliment
     Sign bit is similar to (1). The magnitude is complimented.

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 5

• Ex:     410 => 0000 01002
           -410 => 1111 10112
  

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 6

• 3. Two's Compliment
     Sign bit same as (1).
     Magnitude is complimented first and a "1" is added to the complimented digits.

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 7

• Ex:     710 => 0000 01112
           -710 => 1111 10012
  

COMPUTER ARITHMETIC:Base₁₀ => Base₂ Conversion 8

• Ex:     7 + -3
   A - B = A + ~B + 1
  

COMPUTER ARITHMETIC: Octal Number System

• Base=8
• 8 Symbols: {0,1,2,3,4,5,6,7}

COMPUTER ARITHMETIC: Octal Number System - Octal to Decimal

• (a_n-1a_n-2 ... a₁a₀)₈ = (a_n-1x8^n-1+a_n-2x8^n-2+ ... +a₁x8¹+a₀x8⁰)₁₀
• Example:
  (127)₈ = (1x 8² + 2 x 8¹ + x 8⁰) = (64 + 16 +)₁₀ = (87)₁₀

COMPUTER ARITHMETIC: Octal Number System - Decimal to Octal

• Repeated division by 8.
  (Similar to principle to generate binary codes).
• Example: (213)₁₀ = (what)₈?

  1. 213/8 = 26(quotient),5(remainder) => lowest octal number = 5
  2. 26/8 = 3(quitient), 2(remainder) => second octal digit = 2
  3. 3/8 = 0(quotient), 3(remainder) => third octal digit = 3
  Stop, since quotient = 0
  Hence, (213)10 = (325)8
  

COMPUTER ARITHMETIC: Octal Number System - Octal to Binary

• Expand each octal digit to 3 binary bits.
• Example: (725)₈ = (111|010|101)₂

COMPUTER ARITHMETIC: Octal Number System - Binary to Octal

• Combine every 3 bits into one octal digit.
• Example: (110|010|011)₂ = (623)₈

COMPUTER ARITHMETIC: Hexadecimal Number System

• Base = 16
• 16 symbols: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A(=10), B(=11), C(=12), D(=13), E(=14), F(=15)}

COMPUTER ARITHMETIC: Hexadecimal Number System - Hexdecimal to Decimal

• (a_n-1a_n-2 ... a₁a₀)₁₆ = (a_n-1x16^n-1+a_n-2x16^n-2+ ... +a₁x16¹+a₀x16⁰)₁₀
• Example:
  (1C7)₁₆ = 1 16² + 12 x 16¹ + 7 x 16⁰)₁₀ = (256 + 197 + 7)₁₀ = (455)₁₀

COMPUTER ARITHMETIC: Hexadecimal Number System - Decimal to Hexdecimal

• Repeated division by 16.
  (similar in principle to generating binary and octal codes).
• Example: (829)₁₀ = (what)₁₆?

  1. 829/16 = 51(quotient),13 = D(remainder) =>
       lowest octal number = D
  2. 51/16 = 3(quitient), 3(remainder) => second octal digit = 3
  3. 3/16 = 0(quotient), 3(remainder) => third octal digit = 3
  Stop, since quotient = 0
  Hence, (829)10 = (33D)16
  

COMPUTER ARITHMETIC: Hexadecimal Number System - Hexadecimal to Binary

• Expand each hexadecimal digit to 4 binary bits.
• Eaxple: (E29)₁₆ = (1110|0100|1001)₂

COMPUTER ARITHMETIC: Hexadecimal Number System - Binary to Hexadecimal

• Combine every 4 bits into one hexadecimal digit.
• Eample: (0101|1111|1010|0110)₂ = (5FA6)₁₆