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**The other way to represent Integers.**

Excess Notation The other way to represent Integers.

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**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: The smallest non-negative value: The largest negative value is: The smallest negative value is:

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**Excess Notation 111 110 Consider the 8 patterns in 3 bits: 101 100 011**

010 001 000

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**Excess Notation Interpreted as Natural Numbers: 111 7 110 6 101 5 100**

4 011 3 010 2 001 1 000

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**Excess Notation Interpreted as Integers in 2’s Complement: 111 -1 110**

-2 101 -3 100 -4 011 3 010 2 001 1 000

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**Excess Notation Interpreted as Integers in Excess Notation: 111 3 110**

2 101 1 100 011 -1 010 -2 001 -3 000 -4

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**Excess Notation Three different Interpretations: 111 7 -1 3 110 6 -2 2**

101 5 -3 1 100 4 -4 011 010 001 000

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**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 Show the value represented if the pattern is: an unsigned integer an integer, in 2’s Complement Notation an integer, in Excess Notation

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**Excess Notation (examples are in 8 bits to save space)**

The pattern ( ) has 2 parts:

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**Excess Notation (examples are in 8 bits to save space)**

The pattern ( ) has 2 parts: the MSB

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**Excess Notation (examples are in 8 bits to save space)**

The pattern ( ) has 2 parts: the MSB the rest

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**Excess Notation (examples are in 8 bits to save space)**

The pattern ( ) has 2 parts: the MSB the rest Let’s look at the “rest”:

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**Excess Notation (examples are in 8 bits to save space)**

The pattern ( ) has 2 parts: the MSB the rest represents the Natural number = 57

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**Excess Notation (examples are in 8 bits to save space)**

The pattern ( ) is, therefore, 57 greater than – regardless of the meaning of the MSB.

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**Excess Notation (examples are in 8 bits to save space)**

As a Natural number, is 128

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**Excess Notation (examples are in 8 bits to save space)**

As a Natural number, is 128 In 2’s Complement, is the smallest, negative value…

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**Excess Notation (examples are in 8 bits to save space)**

As a Natural number, is 128 In 2’s Complement, is the smallest, negative value… In Excess Notation, is the smallest, non-negative value…

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**Excess Notation (examples are in 8 bits to save space)**

So the pattern 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)

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Lecture 6 More Logic Functions: NAND, NOR, XOR and XNOR

Lecture 6 More Logic Functions: NAND, NOR, XOR and XNOR

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