SD & D Negative Numbers
- 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
- 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.
- 4. 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
- 5. 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: Binary Decimal 11111101 -3 11111110 -2 11111111 -1 00000000 0 00000001 1 00000010 2 00000011 3 1. the set of integers show symmetry about zero
- 6. Two’s Complement Representation Two’s complement is designed so that: 00000010 + 1 00000011 2. adding 1 to any number produces the next number (ignoring carry bits)
- 7. Two’s Complement Representation To find the Two’s Complement of a number (its opposite sign): 1. Change all the 1’s to 0 and 0’s to 1. 2. Add 1.
- 8. Two’s Complement Representation For example, how would -5 be represented using Two’s Complement? 5 = 000000101 1. Change all the 1’s to 0 and 0’s to 1. 11111010
- 9. Two’s Complement Representation For example, how would -5 be represented using Two’s Complement? 5 = 000000101 1. Change all the 1’s to 0 and 0’s to 1. 11111010 2. Add 1. 11111010 +1 11111011 So -5 as Two’s Complement = 11111011
- 10. Two’s Complement Representation Example 2 - find the Two’s Complement of 88 88 = 01011000 1. Change all the 1’s to 0 and 0’s to 1. 10100111
- 11. Two’s Complement Representation Example 2 - find the Two’s Complement of 88 88 = 01011000 1. Change all the 1’s to 0 and 0’s to 1. 10100111 2. Add 1. 10100111 +1 10101000 So -88 as Two’s Complement = 10101000
- 12. Range The number of integers which could be stored in one byte (8 bits) is 28 = 256
- 13. Range The number of integers which could be stored in one byte (8 bits) is 28 = 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.
- 14. Range What range of numbers could be stored in two bytes using twos complement? 216 = 65536
- 15. Range What range of numbers could be stored in two bytes using twos complement? 216 = 65536 The range of integers which could be stored in two bytes (16 bits) is -32768 to +32767
- 16. Range What range of numbers could be stored in two bytes using twos complement? 216 = 65536 The range of integers which could be stored in two bytes (16 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.
- 17. 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