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Representing unsigned (integer) values inside the computer in binary

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• Computers use the "binary number system" positional system to represent unsigned (integer = whole) numbers.

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• The Binary number system is similar to the decimal number system:

  Decimal number system: 10 digits are used to represent the following values: digit value -------------------------- 0 = ( ) 1 = (*) 2 = (**) 3 = (***) 4 = (****) 5 = (*****) 6 = (******) 7 = (*******) 8 = (********) 9 = (*********) A digit that appears at the n-th position to the left of the decimal point has a value equal to: value of the digit × 10n

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• Binary number system:

  Binary number system: 2 digits are used to represent the following values: digit value -------------------------- 0 = ( ) 1 = (*) A digit that appears at the n-th position to the left of the decimal point has a value equal to: value of the digit × 2n

• Example:

     Unsigned value    4 bit binary number   8 bit binary number
     --------------    -------------------   -------------------
               0             0000                 00000000
               1             0001                 00000001
               2             0010                 00000010
               3             0011                 00000011
  

• Interpreting binary numbers
  • Humans likes to work with decimal representation of integer (whole) values.

    But computers must represent integer (whole) values in the binary number system

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  • Questions:

    Suppose we have a binary number - say 10010101 What value does it represent (expressed in decimal for human "consumption) ? Suppose we have a decimal value - say 14 What is the corresponding representation of this decimal number expressed as a binary number ?

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  Converting a value between the decimal number representation and the binary number representation

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  • Given a binary number representing an unsigned value, how to determine its decimal represention ?
    • Answer:

      Sum the value of each digit multiplied by its "location factor"

    • Example:

      What unsigned value is represented by 01011001 ? Answer: Digits: 0 1 0 1 1 0 0 1 Location factor: 128 64 32 16 8 4 2 1 -------------------------------- 64 + 16 +8 + 1 = 89 Answer: 89

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  • Given an decimal representation, how to find the binary representation that represent that same value ?
    • Answer:

      Divide the value repeatedly by 2 and Collect the remainders of each division in the reverse order.

    • Example:

      What is the 8 bit binary number that represents the unsigned value 89 ? Answer: 89 /2 ------ 1 <--- remainder of 89/2 44 <--- quotient of 89/2 /2 ------ 0 <--- remainder of 44/2 22 <--- quotient of 44/2 /2 ------ 0 11 /2 ------ 1 5 /2 ------ 1 2 /2 ------ 0 1 /2 ------ 1 0 <----- Done Binary number: 1011001 (7 bits) Pad to 8 bits: 01011001 Answer: 01011001

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• Storing binary representations of unsigned values inside a computer
  • Facts:

    We can manufacture electronic switches A switch can be on or off The on state represents a 1 bit The off state represents a 0 bit A byte (memory) = 8 switches

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  • Storing the representation of unsigned values:

    Unsigned values are stored in memory using the binary number system Example: How to represent the value five (5) using 1 byte memory (= byte variable): 00000101 How to represent the value five (5) using 2 bytes memory (= short variable): 00000000 00000101 How to represent the value five (5) using 4 bytes memory (= int variable): 00000000 00000000 00000000 00000101

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  • Quiz:

    What bit pattern is stored in memory (= in the memory cells reserved for the variable) by the following Java program: int x; x = 9; What bit pattern is stored in memory (= in the memory cells reserved for the variable) by the following Java program: short x; x = 9; What bit pattern is stored in memory (= in the memory cells reserved for the variable)) by the following Java program: byte x; x = 9;

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• Test yourself if you understand binary numbers....
  • See if you understand the following jokes:

    Joke #1: Joke #2: