The other way to represent Integers.

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Presentation transcript:

The other way to represent Integers. 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 largest non-negative value: 11111111 The smallest non-negative value: 10000000 The largest negative value is: 01111111 The smallest negative value is: 00000000

Excess Notation 111 110 Consider the 8 patterns in 3 bits: 101 100 011 010 001 000

Excess Notation Interpreted as Natural Numbers: 111 7 110 6 101 5 100 4 011 3 010 2 001 1 000

Excess Notation Interpreted as Integers in 2’s Complement: 111 -1 110 -2 101 -3 100 -4 011 3 010 2 001 1 000

Excess Notation Interpreted as Integers in Excess Notation: 111 3 110 2 101 1 100 011 -1 010 -2 001 -3 000 -4

Excess Notation Three different Interpretations: 111 7 -1 3 110 6 -2 2 101 5 -3 1 100 4 -4 011 010 001 000

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: the MSB

Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: the MSB the rest

Excess Notation (examples are in 8 bits to save space) The pattern (10111001) has 2 parts: the MSB 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: the MSB 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 128 In 2’s Complement, 10000000 is the smallest, negative value… -128

Excess Notation (examples are in 8 bits to save space) As a Natural number, 10000000 is 128 In 2’s Complement, 10000000 is the smallest, negative value… -128 In Excess Notation, 10000000 is the smallest, non-negative value… 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)