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Binary Representation of Integers

  • In all number system both positive and negative numbers are possible.
  • A conventional method of denoting these values is to use a (+) plus or (-) minus sign preceding the number.
  • This representation is called signed number.
  • Generally an additional bit, known as the sign bit, is placed at the most significant end to represent the sign 0 and 1 are used to indicate positive and negative respectively.
  • They are of three types, namely
    • Sign and Magnitude Representation
    • One's complement
    • Two's complement

Sign and Magnitude Representation

  • The simplest signed number system is the sing and magnitude notation often referred to as signed binary
  • An n-bit number employs the MSB to represent the sign of the number ad the remaining n-1 bits to express the magnitude in binary.
  • As an example, consider -1310 expressed using five bits.
  • We have
    • -1310 = 1.1101

      (Sign bit of -)
  • In this representation, a comma has been used to separate the sign bit from the magnitude bits for clarity.
  • A positive number is represented as
    +2710 = 0,110112

One's Complement

  • One's complement represents positive numbers and negative numbers.
  • Hence the value of +2 and -2 will be represented as 0010 in binary form.
  • For example convert the given number using one's complement
-3 (convert number in the positive format)
+3 = 0000 0011 (8 bit form)
In 1's complement +3= 11111100
  • Hence the result of -3 is 111111002
Note: The simple method to convert the given value into one's complement is change all the given 1 as 0(zero) and 0 as 1 in the given value.

Two's Complement

  • In a binary computer, one practical choice of the complementation constant R is as a power of 2.
  • Since the radix of the system is 2 and R should be less than or equal to the radix, the obvious choices of complementation constants become 1 and 2.
  • To convert the value in to two's complement first make the given 8bit digit into one's complement and add 1 to the LSB.
  • For example convert -4 using two's complement.

4 = 0000 0100

Change to one's complement = 1111 1011

Add one to LSB 1

  • Hence the result of -4 is 111111002

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