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Excess Notation The other way to represent Integers. Excess Notation (examples are in 8 bits to save space) Fixed length notation system. Uses 0 to represent negative values. The The The The largest non-negative value: smallest non-negative value: largest negative value is: smallest negative value is: 11111111 10000000 01111111 00000000 Excess Notation Consider the 8 patterns in 3 bits: 111 110 101 100 011 010 001 000 Excess Notation Interpreted as Natural Numbers: 111 110 101 100 011 010 001 000 7 6 5 4 3 2 1 0 Excess Notation Interpreted as Integers in 2’s Complement: 111 110 101 100 011 010 001 000 -1 -2 -3 -4 3 2 1 0 Excess Notation Interpreted as Integers in Excess Notation: 111 110 101 100 011 010 001 000 3 2 1 0 -1 -2 -3 -4 Excess Notation Three different Interpretations: 111 110 101 100 011 010 001 000 7 6 5 4 3 2 1 0 -1 -2 -3 -4 3 2 1 0 3 2 1 0 -1 -2 -3 -4 Excess Notation (examples are in 8 bits to save space) To better understand how binary patterns unpack under the 3 notations, let’s look at an example. Consider the pattern 10111001 Show the value represented if the pattern is: an unsigned integer an integer, in 2’s Complement Notation an integer, in Excess Notation Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: 1. the MSB Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: 1. the MSB 2. the rest Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: 1. the MSB 2. the rest Let’s look at the “rest”: 0111001 Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: 1. the MSB 2. the rest 0111001 represents the Natural number 32 + 16 + 8 + 1 = 57 Excess Notation (examples are in 8 bits to save space) The pattern (10111001) is, therefore, 57 greater than 10000000 – regardless of the meaning of the MSB. Excess Notation (examples are in 8 bits to save space) As a Natural number, 10000000 is 128 Excess Notation (examples are in 8 bits to save space) As a Natural number, 10000000 is In 2’s Complement, 10000000 is the smallest, negative value… 128 -128 Excess Notation (examples are in 8 bits to save space) As a Natural number, 10000000 is In 2’s Complement, 10000000 is the smallest, negative value… In Excess Notation, 10000000 is the smallest, non-negative value… 128 -128 0 Excess Notation (examples are in 8 bits to save space) So the pattern 10111001 is 57 greater than: 128 if it’s natural (57+128=185) -128 if it’s 2’s Complement (57-128=-71) 0 if it’s Excess (57+ 0= 57)