Transcript lec02-numrep2

Data Representation
Lecture 21
5.3.2001.
13/2/2001
Sudeshna sarkar, CSE, IIT
Kharagpur
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1001011101101
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 1001001100101
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12ed
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Kharagpur
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Signed Integer Representation
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Many of the numerical data items that
are used in a program are signed
(positive or negative).
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Question:: How to represent sign?
Three possible approaches:
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Sign-magnitude representation
One’s complement representation
Two’s complement representation
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Sign-magnitude Representation
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For an n-bit number representation
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The most significant bit (MSB) indicates sign
 0  positive
 1  negative
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The remaining n-1 bits represent magnitude.
b1 b0
bn-1bn-2
Sign
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Magnitude
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Signed Magnitude scheme
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Range of numbers that can be
represented:
Maximum :: + (2n-1 – 1)
Minimum ::  (2n-1 – 1)
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A problem:
Two different representations of zero.
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+0  0 000….0
-0  1 000….0
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One’s Complement Representation
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Basic idea:
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Positive numbers are represented exactly as in signmagnitude form.
Negative numbers are represented in 1’s complement
form.
How to compute the 1’s complement of a number?
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Complement every bit of the number (10 and 01).
MSB will indicate the sign of the number.
 0  positive
 1  negative
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n=4
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0000
0001
0010
0011
0100
0101
0110
0111

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
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+0
+1
+2
+3
+4
+5
+6
+7
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1000
1001
1010
1011
1100
1101
1110
1111
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-7
-6
-5
-4
-3
-2
-1
-0
To find the representation of, say, -4, first note that
+4 = 0100
-4 = 1’s complement of 0100 = 1011
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Range of numbers that can be represented:
Maximum :: + (2n-1 – 1)
Minimum ::  (2n-1 – 1)
A problem:
Two different representations of zero.
 +0
 0 000….0
 -0
 1 111….1
Advantage of 1’s complement representation
 Subtraction can be done using addition.
 Leads to substantial saving in circuitry.
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Kharagpur
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Two’s Complement Representation
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Basic idea:
 Positive numbers are represented exactly as in
sign-magnitude form.
 Negative numbers are represented in 2’s
complement form.
How to compute the 2’s complement of a number?
 Complement every bit of the number (10 and
01), and then add one to the resulting number.
 MSB will indicate the sign of the number.
 0  positive
 1  negative
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In C
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short int
 16 bits
 + (215-1) to -215
int
 32 bits
 + (231-1) to -231
long int
 64 bits
 + (263-1) to -263
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Two’s complement
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Consider 8 bit numbers. In this system
if the value of the MSB is:
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zero : then the rest of the seven bits
represent normal binary numbers (0 to
127).
one : then this represents -128 which is
added to the total of the other seven bits
that represent a normal binary number.
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Example : n=8
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00000001
00000101
10000001
10000101
11011111
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+1
+5
-127
-123
-33
To find the representation of, say, -4, first note that
+4 = 00000100
-4 = 2’s complement of 00000100
= 11111011+1 = 11111100
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Range of numbers that can be represented:
Maximum :: + (2n-1 – 1)
Minimum ::  2n-1
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Advantage:
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Unique representation of zero.
Subtraction can be done using addition.
Leads to substantial saving in circuitry.
Most computers today use the 2’s complement
representation for storing negative numbers.
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Two’s complement arithmetic
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Two’s complement is very important
when we wish to perform a subtraction
between two binary numbers.
Adding one binary number to the 2’s
complement of another binary number
is equivalent to a subtraction operation.
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Negative numbers in 2’s complement
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1. Flip all the binary digits within the
corresponding positive number (1’s
complement).
2. Add 1 to the result.
 13: 00001101
 1’s complement : 11110010
 2’s complement : 11110011
This represents -13
(-128+64+32+16+0+0+2+1)
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2’s complement arithmetic
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Example 1: 18-11 = ?
18 is represented as 00010010
11 is represented as 00001011
 1’s complement of 11 is 11110100
 2’s complement of 11 is 11110101
Add 18 to 2’s complement of 11
00010010
+ 11110101
00000111 is binary
---------------for
7.
00000111 (with a carry of 1
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which
is
ignored)
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2’s complement arithmetic
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Example 2: 7 - 9 = ?
7 is represented as 00000111
9 is represented as 00001001
 1’s complement of 9 is 11110110
 2’s complement of 9 is 11110111
 Add 7 to 2’s complement of 9
00000111
+ 11110111
00000111 is binary
---------------for
-2.
11111110 (with a carry of 0
Sudeshna sarkar, CSE, IIT
which
is
ignored)
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Kharagpur
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Overflow
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Overflow condition : carry in and carry
out from MSB are different.
(64) 01000000
( 4) 00000100
-------------(68) 01000100
carry (out)(in)
0 0
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(64) 01000000
(96) 01100000
-------------(-96) 10100000
carry out in
0 1
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differ:
overflow
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Real Numbers:
floating point representation
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1.5, 36.415 : how to represent these
numbers ?
In floating point representation, a
number is represented in two parts :
1. mantissa (M)
F = M x BE
 2. exponent (E)
B: exponent base
1903810 = 1.9038 x 104
0.0032510 = 3.25 x 10-3
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Floating point : binary numbers
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All of the mantissa appears to the right
of the decimal (binary?) point.
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1101.00112 becomes 0.110100112 x 24
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0.0001012 becomes 0.1012 x 2-3
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Storage of floating point numbers
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Two pieces of information need to be stored :
 1. mantissa : more bits for mantissa means
more precision
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2. exponent : more bits in the exponent
provides a greater range of numbers.
1. M represents a 2’s
complement fraction
M
E
1 > M > -1
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2. E represents the
exponent in 2’s
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
In C:
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4 bytes for single precision float
8 bytes for double
Different computer systems use
different data representations for
floating point numbers.
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Example
Mantissa Decimal Exponent Decimal
eqv of M
eqv of E
Decimal
value
01000000 ½+0+0+0
+0+0+0
0000
(=1/2)
0001
21=2
01000000 ½+0+0+0
+0+0+0
0000
(=1/2)
1111
-8+4+2+1 ½ x 2-1
= -1
=
01101001 ½+1/4+0 0101
0000
+1/16+0
+0+1/128
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1/4
25 = 32
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Range of values
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4 bytes are used to represent a floating point
number.
Mantissa (3 bytes) and
 exponent (1 byte)
are in 2’s complement form.
What would be the smallest, greatest, and the smallest positive
fraction that could be handled by this system ?
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1. The largest single precision number is
01111111 11111111 11111111 01111111
2. The smallest number (most -ve) would be
10000000 00000000 00000000 01111111
3. The smallest fraction would be:
00000000 00000000 00000001 10000000
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Representation of Characters
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Many applications have to deal with non-numerical
data.
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Characters and strings.
There must be a standard mechanism to represent
alphanumeric and other characters in memory.
Two standards in use:
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Extended Binary Coded Decimal Interchange Code
(EBCDIC) -- Used in older IBM machines
American Standard Code for Information Interchange
(ASCII)
 Most widely used today
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ASCII code
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Each character (letters, numerals,
punctuation marks, control characters and
other symbols) is assigned a numeric code (7
or 8 bits).
The binary encoding of the characters follow
a regular ordering.
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Digits are ordered consecutively in their proper
numerical sequence (0 to 9).
Letters (uppercase and lowercase) are arranged
consecutively in their proper alphabetic order.
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Some Common ASCII Codes
‘A’ :: 41 (H) 65 (D)
‘B’ :: 42 (H) 66 (D)
………..
‘Z’ :: 5A (H) 90 (D)
‘a’ :: 61 (H) 97 (D)
‘b’ :: 62 (H) 98 (D)
………..
‘z’ :: 7A (H) 122 (D)
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‘0’ :: 30 (H) 48 (D)
‘1’ :: 31 (H) 49 (D)
………..
‘9’ :: 39 (H) 57 (D)
‘(‘ ::
‘+’ ::
‘?’ ::
‘\n’ ::
‘\0’ ::
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28 (H)
2B (H)
3F (H)
0A (H)
00 (H)
40 (D)
43 (D)
63 (D)
10 (D)
00 (D)
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Character Strings
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Two ways of representing a sequence of characters in
memory.
 The first location contains the number of characters in
the string, followed by the actual characters.
 The characters follow one another, and is terminated
by a special delimiter.
In C, the second approach is used.
 The ‘\0’ character is used as the string delimiter.
Example: “Hello”
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
H
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e l
l
o “0”
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