Representation of signed numbers

• an unsigned n-bit word can be used to represent 2ⁿ integers in the range [0, ..., 2ⁿ-1]
  • for example, with an 8-bit word we can represent 2⁸ integers in the range [0, ..., 255]
    
    
• we look at two ways of representing signed numbers (i.e. positive and negative)
• we always begin by stating n, the number of bits we are using to represent numbers, the number of significant digits
  
  
• sign and magnitude representation
  • the most significant bit is reserved as the sign bit
  • a 0 in the sign bit indicates a positive number
  • a 1 in the sign bit indicates a negative number
    
    
    
  • in an n-bit word, this leaves us only n-1 bits to represent the magnitude
  • we can represent integers in the range [-2^n-1+1, ..., 2^n-1-1]
  • for example, with an 8-bit word we can represent the integers in the range [-127, ..., 127]
    
    
  • disadvantages
    • there are two representations for zero
      
    • the sign bit requires special treatment when performing binary arithmetic
    • specialized hardware would be required for performing subtraction
      
      
  • this representation is not typically used for representation of signed integers
  • it is used in the representation of real numbers
    
    
• complementary representations
  • negative numbers are represented as the complement of the corresponding positive number
    
    
  • the sign of the number is inherent in the representation
  • half of the values representable in n bits are used for positive numbers, half for negative numbers
    
    
  • we demonstrate this concept first in decimal by looking at ten's complement
    • for an n-digit decimal number N, its 10's complement is given by 10ⁿ - N
    • suppose we have a 2-digit (decimal) word size, then we take the complement by subtracting from 100
    • for example, the ten's complement of 15 is 85

      100 - 15 ------ 85

• another approach to calculating the complement is to subtract from 10ⁿ - 1 and then add 1
• using this approach, the ten's complement of 15 is still 85

  99 - 15 ------ 84 + 1 ------ 85

• note that taking the complement of the complement gives you the original number
• example, the ten's complement of 85 is 15

  100 - 85 ------ 15

• adding a number to it's complement gives zero (in the significant digits)

  85 + 15 ------ 100

• in this example:
  • 85 is the ten's complement representation for -15
  • 85 is the ten's complement of 15
  • 15 is the ten's complement of 85
    
    
• in binary we use two's complement to represent signed integers
  • for an n-bit binary number N, its two's complement is given by 2ⁿ - N
    
    
  • alternatively, we can calculate the two's complement by subtracting from 2ⁿ-1 and then adding 1
    • 2ⁿ-1 will always consist of n one bits
    • subtracting N from this has the effect of inverting each bit in the number N
    • adding 1 to this result gives the two's complement of N
    • so, to take the two's complement of a binary number, simply flip the bits and add one
      
      
  • example using n = 5 (n is the number of bits or significant digits)
    
    
    
  • properties of two's complement representation
    • if you add a number and its complement you get zero: X + (-X) = 0
    • there is only one representation for zero
    • the most significant bit still indicates the sign
    • the range of integers that can be represented with n bits is [-2^n-1, ..., 2^n-1-1]
      • for example, with 5 bits you can represent [-16, ..., 15]
      • this is 2⁵=32 unique numbers: 16 negative integers, zero, and 15 positive integers
    • there is a very simple (and easily implemented in hardware) algorithm for taking a two's complement
    • both addition and subtraction can be performed in hardware using addition (subtraction is implemented by taking the two's complement of the subtrahend and then adding)

References:

• "68000 Family Assembly Language", by Alan Clements, PWS Publishing Company,1994.