1. The other way to represent Integers.
Excess Notation
The other way to represent Integers.

2. 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:

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

4. Excess Notation Interpreted as Natural Numbers: 111 7 110 6 101 5 1004
011 3
010 2
001 1
000

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

6. Excess Notation Interpreted as Integers in Excess Notation: 111 3 1102
101 1
100
011 -1
010 -2
001 -3
000 -4

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

8. 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

9. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:

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

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

12. 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”:

13. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) has 2 parts:
the MSB
the rest
represents the Natural number = 57

14. Excess Notation (examples are in 8 bits to save space)
The pattern ( ) is, therefore, 57 greater than – regardless of the meaning of the MSB.

15. Excess Notation (examples are in 8 bits to save space)
As a Natural number, is 128

16. 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…

17. 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…

18. 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)