Representation of Negative Numbers

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Representation of Negative Numbers

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Representation of Negative Numbers

  1. 2. Real Numbers & Integers All numbers belong to the set or group called real numbers . Inside the set of real numbers is a set of all positive and negative whole numbers. This set is called integers . Real numbers -4.6 2.46 13.7 Integers 0 -7 5 221 -31 100.052 -52.414 0.0014
  2. 3. Signed Bit Representation The simplest way of representing a negative number in binary is to use the first bit of the number to represent whether the number is positive or negative: 011 = 3 111 = -3 This is known as signed bit representation. The problem with signed bit representation is that there are 2 values for zero: 000 = 0 100 = -0
  3. 4. Two’s Complement Representation A better way of representing negative numbers in binary is by using Two’s Complement . Two’s Complement is designed so that: <ul><li>the set of integers show symmetry about zero </li></ul>Binary Decimal 11111101 -3 11111110 -2 11111111 -1 00000000 0 00000001 1 00000010 2 00000011 3
  4. 5. Two’s Complement Representation Two’s complement is designed so that: <ul><li>adding 1 to any number produces the next number (ignoring carry bits) </li></ul>00000010 + 1 00000011
  5. 6. Two’s Complement Representation To find the Two’s Complement of a number (it’s opposite sign ): <ul><li>Change all the 1 ’s to 0 and 0 ’s to 1 . </li></ul><ul><li>Add 1. </li></ul>
  6. 7. Two’s Complement Representation For example, how would -5 be represented using Two’s Complement? 5 = 000000101 <ul><li>Change all the 1 ’s to 0 and 0 ’s to 1 . 11111010 </li></ul><ul><li>Add 1. 11111010 +1 11111011 So -5 as Two’s Complement = 11111011 </li></ul>
  7. 8. Two’s Complement Representation Example 2 - find the Two’s Complement of how would 88 88 = 01011000 <ul><li>Change all the 1 ’s to 0 and 0 ’s to 1 . 10100111 </li></ul><ul><li>Add 1. 10100111 +1 10101000 So -88 as Two’s Complement = 10101000 </li></ul>
  8. 9. Range The number of integers which could be stored in one byte (8 bits) is 2 8 = 256 The range of integers which could be stored in one byte (8 bits) using Two’s Complement is -128 to +127 Why does there seem to be one less positive number? There are 255 numbers plus the value 0 . So there are 256 numbers in all.
  9. 10. Range What range of numbers could be stored in two bytes using twos complement? 2 16 = 65536 The range of integers which could be stored in one byte (8 bits) is -32768 to +32767 This method of representing large numbers is unsuitable because of the increased memory needed to store the large number of bits needed. A solution to this is to use Floating Point Representation .
  10. 11. Credits Higher Computing – Data Representation – Representation of Negative Numbers Produced by P. Greene and adapted by R. G. Simpson for the City of Edinburgh Council 2004 Adapted by M. Cunningham 2010

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