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Mathematics Dictionary
Dr. K. G. Shih
Binary Numbers

  • Q01 | - Value of 2 to power n
  • Q02 | - Binary 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

Q01. Value of 2 to power n 



    n    2^n                 n     2^n
    0    1                   0     1.0
    1    2                  -1     0.5 
    2    4                  -2     0.25
    3    8                  -3     0.125 
    4    16                 -4     0.0625
    5    32                 -5     0.03125
    6    64                 -6     0.015625
    7    128                -7     0.0078125
    8    256                -8     0.00390625
    9    5012               -9     0.001953125
   10    1024 (K)          -10     0.0009765125
   11    2048
   12    4096
   13    8192
   14    16384    
   15    32768
   16    65536
   17    131072
   18    262144
   19    524288
   20    1048576 (M)

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Q02. Binary number systems

Defintion
  • It has only two digits : 0 and 1
  • Use 0 and 1 as coefficinets
  • Use position as power of base 2
Example 


      1 + 1 =   10 = 2
     10 + 1 =   11 = 3
     11 + 1 =  100 = 4
    100 + 1 =  101 = 5
    101 + 1 =  110 = 6
    110 + 1 =  111 = 7
    111 + 1 = 1000 = 8
   1000 + 1 = 1001 = 9

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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 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 binary number of 8 is 1000

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