Transcript Excess Notation

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)