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Mathematics Dictionary
Dr. K. G. Shih
Binary Numbers

  • Q01 | - Computer number systems
  • Q02 | - Hexadecimal number systems
  • Q03 | - Binary number to Hexadecimal number
  • Q04 | - Hexadecimal number to binary number
  • Q05 | - Hexadecimal number to decimal number
  • Q06 | - Octal number to hexadecimal number
  • Q07 | - Multiplication table of hexadecimal number

Q01. Computer number systems

Computer number systems
  • 1. Bianry number
    • Application - It is used as computer language
    • Digits : 0 and 1

  • 2. Octal number
    • Application - It is used for computer to communicate with computer operator
    • Digits : 0, 1, 2, 3, 4, 5, 6, 7

  • 3. Decimal number
    • Application - It is used for computer to communicate with general public
    • Digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

  • 4. Hexadecimal number
    • Application - It is used for computer to communicate with computer programmer
    • Digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Examples
  • 1 + 1 = 10 in binary
  • 1 + 7 = 10 in Octal
  • 1 + 9 = 10 in decimal
  • 1 + F = 10 in hexadecimal

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Q02. Hexadecimal number systems

Defintion
  • It has 16 digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
  • Use digits as coefficinets
  • Use position as power of base 8
  • Example : Octal 52 = 5*16^1 + 2*16^0 = 80 + 2 = 82 in decimal
Example : Binary and octal 


      0 + 1 =    1 in binary = 1 in hexadecimal
      1 + 1 =   10 in binary = 2 in hexadecimal
     10 + 1 =   11 in binary = 3 in hexadecimal
     11 + 1 =  100 in binary = 4 in hexadecimal
    100 + 1 =  101 in binary = 5 in hexadecimal
    101 + 1 =  110 in binary = 6 in hexadecimal
    110 + 1 =  111 in binary = 7 in hexadecimal
    111 + 1 = 1000 in binary = 8 in hexadecimal
   1000 + 1 = 1001 in binary = 9 in hexadecimal
   1001 + 1 = 1010 in binary = A in hexadecimal
   1010 + 1 = 1011 in binary = B in hexadecimal
   1011 + 1 = 1100 in binary = C in hexadecimal
   1100 + 1 = 1101 in binary = D in hexadecimal
   1101 + 1 = 1110 in binary = E in hexadecimal
   1110 + 1 = 1111 in binary = F in hexadecimal
   1111 + 1 =10000 in binary = 10 = 1 + F in hexadecimal

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Q03. Binary number to Hexadecimal number

Binary to hexadecimal
  • Binary number 101001001 to hexadecimal
  • Start at right
  • Each 4 binary digits as a group
  • Hence we have 0001 0100 1001 = 149 in hexadecimal

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Q04. Hexadecimal to binary number

  • Start from right
  • Express each hexadecimal digit in 4-bit binary
  • Example hexadecimal A2 in binary
  • Since 2 = 0010 and A = 1010
  • Hence A2 = 1010 0010 in binary

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Q05. Hexadecimal number to decimal number

Hexadecimal to decimal
  • Hexadecimal digits are the coefficients
  • Position of hexadecimal digits are the power of base 16
  • The right side digit is power 0
  • Second digit from right is power 1
  • Third digit from right is power 2
Example : convert A2 to decimal
  • The decimal value = 10*(16^1) + 2*(16^0)
  • = 160 + 2
  • = 162

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Q06. Octal number to hexadecimal number

Method
  • Convert octal to binary
  • Group each 4 binary digits
  • Convert each 4 digits to hexadecimal
Example : Convert octal 12 to hexadecimal
  • Octal 12 = 001 010 in binary
  • Regroup as 1010 = C in hexadecimal

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Q07. Multiplication table of octal number

            1   2   3   4   5   6   7   8   9   A   B   C   D   E   F

            ---------------------------------------------------------

      1  |  1   2   3   4   5   6   7   8   9   A   B   C   D   E   F
         |
      2  |  2   4   6   8   A   C   E  10  12  14  16  18  1A  1C  1D
         |
      3  |  3   6   9   C   F  12  15  18  1B  1E  21  24  27  2A  2D
         | 
      4  |  4   8   C  10  14  18  1C  20  24  28  2C  30  34  38  3C
         |
      5  |  5   A   F  14  19  1E  23  28  2D  32  37  3C  41  46  4B
         |
      6  |  6   C  12  18  1E  24  2A  30  36  3C  42  48  4E  54  5A
         | 
      7  |  7   E  15  1C  23  2A
         |
      8  |  8  10  18  20  28  30
         |
      9  |  9  12  1B  24  2D  36
         |
      A  |  A  14  1E  28  32  3C
         |
      B  |  B  16  21  2C  37  42
         | 
      C  |  C  18  24  30  3C  48
         |
      D  |  D  1A  27  34  41  4E
         |
      E  |  E  1C  2A  38  46  34
         |
      F  |  F  1E  2D  3C  4B  5A

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