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Mathematics Dictionary
Dr. K. G. Shih
Binary Numbers

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

Defintion
  • It has only eight digits : 0, 1, 2, 3, 4, 5, 6 and 7
  • Use digits as coefficinets
  • Use position as power of base 8
  • Example : Octal 52 = 5*8^1 + 2*8^0 = 40 + 2 = 42 in decimal
Example : Binary and octal 


      0 + 1 =    1 in binary = 1 in octal
      1 + 1 =   10 in binary = 2 in octal
     10 + 1 =   11 in binary = 3 in octal
     11 + 1 =  100 in binary = 4 in octal
    100 + 1 =  101 in binary = 5 in octal
    101 + 1 =  110 in binary = 6 in octal
    110 + 1 =  111 in binary = 7 in octal
    111 + 1 = 1000 in binary = 10 in octal
   1000 + 1 = 1001 in binary = 11 in octal

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Q03. Binary number to octal number

Binary to octal
  • Binary number 101001001 to octal
  • Start at right
  • Each 3 binary digits as a group
  • Hence we have 101 001 001 = 511 in octal

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Q04. Binary number to decimal number

Binary to decimal
  • Binary digits are the coefficients
  • Position of binary digits are the power of base 2
  • The right side digit is power 0
  • Second digit from right is power 1
  • Third digit from right is power 3
Example : convert 101 to decimal
  • The decimal value = 1*(2^2) + 0*(2^1) + 1*(2^0)
  • = 4 + 0 + 1
  • = 5
Example : Convert 1111 to decimal
  • Method 1
    • 1*(2^3) + 1*(2^2) + 1*(2^1) + 1*(2^0) = 8 + 4 + 2 + 1 = 15
  • Method 2
    • Since value of 10000 = 1*(2^4)
    • Hence value of 01111 = 1*(2^4) - 1 = 16 - 1 = 15

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Q05. Octal number to binary number

Octal to binary
  • The octal digits are 0 to 7
  • Convert each digit as below
    • 0 = 000
    • 1 = 001
    • 2 = 010
    • 3 = 011
    • 4 = 100
    • 5 = 101
    • 6 = 110
    • 7 = 111
  • Example : Convert octal number 723 to binary
  • Answer : 723 = 111 010 011

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Q06. Decimal number to binary number

Decimal to binary
  • We use 2 to divide the decimal number
  • The remainder will be the coefficients
Example : Convert 8 to binary
  • 8 divide by 2 = 4 and remainder is 0
  • 4 divide by 2 = 2 and remainder is 0
  • 2 divide by 2 = 1 and remainder is 0
  • 1 divide by 2 = 0 and remainder is 1
  • Hence Decimal number of 8 is 1000 in binary or 10 in octal
Example : Convert 8 to octal
    8 divide by 8 = 1 and remainder is 0
  • 1 divide by 8 = 0 and remainder is 1
  • Hence decimal number 8 is 10 in octal number system

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Q07. Multiplication table of octal number

            1   2   3   4   5   6   7

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

      1  |  1   2   3   4   5   6   7
         |
      2  |  2   4   6  10  12  14  16
         |
      3  |  3   6  11  14  17  22  25
         | 
      4  |  4  10  14  20  24  30  34
         |
      5  |  5  12  17  24  31  36  43
         |
      6  |  6  14  22  30  36  44  52
         | 
      7  |  7  16  25  34  43  52  61

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