Computer Representation of
Information
Outline
 Representation
of Characters
 Mathematical operations on
(addition)
 Signed integer representation
numbers
Sign-magnitude notation
 Two’s complement notation
 Binary Notations Exercises
 Real Numbers Representation

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Representation of Characters





A unique binary pattern/value is used to represent each of the
printable characters on your keyboard (e.g. A, a, 4, *, [,’, etc.
…), as well as the special control/unprintable characters (e.g.
carriage return, line feed, tab, space, etc. …).
A byte (8 bits) gives you the opportunity to have 28 (256)
unique representations/patterns/values.
There are standard character coding schemes to ease
interchange of information, e.g. ASCII.
The code is designed in such a way not only to preserve
uniqueness, but also to keep the order which is often needed in
manipulating characters/text, e.g. sorting.
The value/pattern used to represent “A” is less by one than the
value used to represent “B”, … etc.
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Right-hand
Left-hand digit
digit
3
0
0
1
1
A
Q
a
q
2
2
B
R
b
r
3
3
C
S
c
s
4
4
D
T
d
t
5
5
E
U
e
u
6
6
F
V
f
v
7
7
G
W
g
w
8
8
H
X
h
x
9
9
I
Y
i
y
A
J
Z
j
z
B
K
k
C
L
l
D
M
m
E
N
n
F
O
o
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5
6
P
7
p
4
Mathematical Operations on
Numbers (Addition)
Most of the mathematical operations on numbers from
different number systems are conceptually identical to the
decimal arithmetic you are used to.
 To add two numbers you do the following:
Start at the rightmost digit.
While there are more digits:
Add the current digit of each operand.
If the sum is less than the base/radix
then
record that sum,
otherwise record the difference between the sum
and the base/radix,
and add one to the next digit of operand 1.

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Mathematical Operations on
Numbers (Addition) (cont’d)
Decimal numbers:
1610 +
1510
-----3110
Start by adding 6 + 5. The sum (11) is not less than the base (10), so (11-10)
record 1, and carry 1 by adding it to the first digit of 16, giving 2, then add the
next digit from each operand (2+1) to give 3.
 Octal numbers:
168 +
158
-----338
Start by adding 6 + 5. The sum (1110) is not less than the base (8), so (11-8)
record 3, and carry 1 by adding it to the first digit of 16, giving 2, then add the
next digit from each operand (2+1) to give 3.
The above discussions can be generalized to any base.


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Signed Integers




So far, we were representing unsigned integers in
binary.
We need to have a way for representing the sign of the
number (positive or negative).
We need to have a sign bit, or figure out some other
way, so as to be able to represent negative values as
well as positive ones.
We will study two notations for representing signed
integers:
 sign-magnitude notation, and
 two’s complement notation.
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Sign-Magnitude Notation




It leaves the left most bit for the sign (sign bit), and uses
the rest of the bits (m - 1) to represent the integer.
0 in the sign bit is used to represent positive values, and
1 is used to represent negative values.
Therefore, the range of integer values i which can be
represented with m bits is:
-(2m-1 – 1) <= i <= +(2m-1 – 1)
Using the sign–magnitude notation (in a byte):
 +2910 is represented as 0001 1101
 -2910 is represented as 1001 1101
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Problems with Sign-Magnitude
Notation
 You
might find this technique simple, natural,
and straight forward, as it closely resembles the
way you are used to writing numbers.
 However this notation creates mainly two
problems for the computers:
1. 0 representation (00000000 is not 10000000).
2. Addition of mixed sign numbers (10000101 (–510)
and 00000001 doesn’t result in –4).
 Thus
the sign-magnitude notation is not used in
computers.
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Two’s Complement Notation
Two’s complement of an m-bit number N =
(Bitwise/one’s complement of N) + 1
 Examples of 8-bit numbers and their 2’s
complement representation:

+1 = 000000012
 –1 = 11111110 + 1 = 11111111
 +29 = 000111012
 –29 = 11100010 + 1 = 11100011
(2’s complement of 111000112 = 000111002 + 1 =
000111012 (29))

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Two’s Complement Notation (cont’d)
an m-bit signed integer in two’s complement
notation, the range of integer values is from
–2m–1 to 2m–1 – 1.
 For
When m = 8, the range is from –128 to 127.
 When m = 16, the range is from –32768 to 32767.

 Two’s
complement notation overcomes the
problems of sign-magnitude notation.
We have only one representation for the 0.
 Addition of mixed sign integers give correct results
(with no need for extra logic).

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Binary Notations Exercises
Representations of integers in a byte (8 bits) using
different binary notations:
Integer Unsigned
34
-23
-128
310
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001000102
Not Possible
Not Possible
Not Possible
Sign-magnitude 2's-complement
001000102
100101112
Not Possible
Not Possible
001000102
111010012
100000002
Not Possible
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Exercises (cont’d)
Add (-37) to (25) using 8-bit two’s
complement representation.
11 11  Carry
Ans:
000110012
2510
11011011
2
-37
+
+
10
-----------------111101002
-1210

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Exercises (cont’d)
Subtract (39) from (-17) using 8-bit two’s
complement representation.
1111111  Carry
Ans:
111011112
-1710
11011001
2
39
+
10
-----------------110010002
-5610

Final carry overflows 8 bits.
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Real Numbers Representation

Real or Floating-Point Numbers, e.g. 22.625
101 100 10-1 10-2 10-3
2 2 . 6
2
5

In binary
decimal point
24 23 22 21 20 2-1 2-2 2-3
1 0 1 1 0 . 1 0 1
binary point
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Real Numbers Representation
(cont’d)

Scientific notation: 2.2625 x 101
exponent
mantissa

Binary scientific notation: 1.0110101 x 24
exponent
mantissa

Hence it consists of a sign bit, a mantissa field, and an
exponent field.
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Next lecture will be about
Problem Solving Methods
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