Unformatted text preview: Chapter 4: Data Representation
EEC 70 Fall 2010
Professor Kent Wilken 1 Data Representation
We need a specific way of representing integers,
floating point, and characters in binary
floating
Memory is organized as a array of bytes (a
bytes (a
groups of 8 bits), so integers, reals, and chars
are an integer number of bytes
are
7 0 2 Integer Representation
Integer size usually is processor’s “bit size”.
Integer
32bit processor like MIPS has 32bit (4 byte)
32
bit
integers
integers
Goals of integer representations:
Goals
• Allow an all positive representation (0 to 2321)
• Allow positive and negative representation (roughly
Allow
231 to +231)
• Allow fast arithmetic
• Provide a unique representation for each value
makes comparison easier
makes • Make it easy for humans to read ??
can automatically (via I/O programs) convert to base 10 as needed
can 3 Different Integer Representations
Different
Unsigned
Unsigned
Sign Magnitude
Sign
One’s Complement
One
Two’s Complement
Two
Biased
Biased
Binary coded decimal
Binary
Most modern computers use two’s
Most
complement and unsigned
complement
• Hardware is simple and fast
4 Unsigned
Weighted representation we have already
seen
seen
• Range: 0 to 2n1, for n bits. For 3 bits:
Decimal
0
1
2
3
4
5
6
7 Unsigned
000
001
010
011
100
101
110
111
5 Sign Magnitude
Use leftmost bit as the sign bit:
Use
most
sign
• sign bit of 0 indicates positive, 1 negative
Remaining bits represent number’s unsigned
Remaining
unsigned
magnitude
magnitude
n1 n1 Range: 2n1+1 to 2n11 for nbit number
Seems “natural” but problematic because
• two representations for 0
two
⇒ more complex hardware
• different hardware for
different
addition and subtraction Decimal
Decimal
0
1
2
3
0
1
2
3 Sign Mag.
000
001
010
011
100
101
110
111 6 One’s Complement
Used in early machines, leads to method in
modern machines: two’s complement
modern
Positive numbers use same representation as
unsigned.
unsigned.
Negation (producing the additive inverse) is
additive
is
done by complementing (inverting) each bit:
done
1 → 0, 0 → 1
0,
Two representations
Two
for zero (not so good)
Negative numbers
Negative
have leading “1”
have Decimal
0
1
2
3
0
1
2
3 One’s Comp.
000
001
010
011
111
110
101
100
7 Two’s Complement
Improvement on one’s complement that has a single
Improvement
complement
representation for zero, allows addition and subtraction
to be done uniformly
to
All bits except the most significant bit (MSB) are given
most
(MSB)
the same weight as unsigned representation.
the
For an nbit number, MSB has weight 2n1
For
bit
Allows range from 2n1 to +2n1 1
to
The additive inverse (negation) of a two’s complement
The
complement
number is its one’s complement plus 1:
number
Example: 3ten = 011 ; 3ten = 100 +001 = 101
011
011 + 101 = 000 (carry out ignored, see next slide)
(3) (3)
8 Two’s Complement: Additive Inverse
B + (B + 1) = (B + B) + 1
= [111 … 111] + 1
= [2n1 + ( 2n2 + 2n3 + … + 21 + 20 )] + 1
= 2n1 + [(2n2 + 2n3 + … + 21 + 20 ) + 1]
= 2n1 + 2n1
=0
Decimal Two’s Comp.
0
1
2
3
1
2
3
4 000
001
010
011
111
110
101
100
9 Two’s Complement (cont.)
Subtraction can be done by producing the
additive inverse and then adding:
additive
A  B = A + (B)
Example: 00101  11001 = 00101 + 00111 = 01100
Example:
(5)
(7)
(5)
(7)
(12)
The complement and increment (+1) operations are
easily done in hardware, ⇒ subtraction can be
subtraction
done using an simple modification of the hardware
used for addition (see EEC180A, EEC170).
used
Note: negative numbers have a “1” in the leftmost
most
bit position, the sign bit.
sign
10 Two’s Complement (cont.)
How to represent an integer in two’s
How
complement:
complement:
• Positive numbers: binary value with a zero sign bit
• Negative numbers: take positive value
Negative
take the one’s complement
complement
add 1 00101 (+5)
00101
11010
11010
+ 00001
=====
11011 (5) Negate any number (positive or negative) by:
complement, increment
complement,
11 Biased Representation
Representation which has consecutive
bit patterns in the same order as
unsigned (0000 ... 1111), but allows
positive and negative numbers
positive
• For 4 bits we can bias by +4: the value
For
represented is 4 less than the unsigned value
of the same bit pattern
of
• Could bias by a different value
for a specific purpose
• Biased notation used for
floating point exponent
to make >, < comparisons
easier (details in Chapter 6) Decimal
4
3
2
1
0
1
2
3 Biased +4
000
001
010
011
100
101
110
111
12 Binary Coded Decimal (BCD)
Encoding that makes each decimal digit an even
number of bits (4), used in very early computers
number
Easier for humans to work with base 10
numbers stored in the computer??
numbers
Inefficient use of memory: 4 bits represents 10
values rather than 16
values
Decimal
0
1
2
3
4
5
6
7 BCD
0000
0001
0001
0010
0010
0011
0011
0100
0100
0101
0101
0110
0110
0111
0111 Decimal
8
9
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A
N/A BCD
1000
1001
1001
1010
1011
1100
1101
1110
1111
13 Sign Extension
How to change a two’s complement integer
How
complement
represented using with smaller number of bits
(e.g., 8) into the same integer represented using
a larger number of bits (e.g., 32)?
larger
• Replicate the sign bit to fill the extra bits:
0xxxxxxx becomes 00000000 0xxxxxxx
for positive number, this clearly adds zero to
for
the number
1xxxxxxx becomes 11111111 1xxxxxxx
1xxxxxxx
for negative number, we are moving the 2n1
for
term to a higher bit position, but we are
adding additional positive terms. The net
result is no change in the value.
14 Overflow
Sometimes an arithmetic result cannot be
represented with the number of bits used.
represented
For example, for 3bit unsigned representation:
011 (3)
+110 (6)
001 (1?!) This condition is called overflow
This
overflow
It may be necessary to detect overflow and take
appropriate action
appropriate
For unsigned addition, overflow occurs when
the result is smaller than either of the operand
(see above example)
(see
15 Overflow
For two’s complement addition, overflow
For
complement
occurs when:
occurs
• adding two positive integers and the result is
adding
negative
negative
011
+010
101 (3)
(2)
(3?!) • adding two negative integers and the result is
adding
positive
positive
101 (3)
+110 (2)
011 (3?!)
16 Character Representation
The traditional character representation is
ASCII: American Standard for Computer
merican tandard
omputer
Information Interchange
nformation
ASCII uses 8bits to represent each character
ASCII
bits
(see Table 4.4, page 102). Examples:
(see
Binary Hex Character 0100 1000 48 ‘H’ 0111 0101
0111 75 ‘u’ 0110 1000 68 ‘h’ 0011 1111 3f ‘?’ Note: lowercase/uppercase have different codes
17 Terminal Input/Output
Terminal I/O is done using ASCII
characters
characters
• On input, terminal keystroke is mapped to
On
ASCII code, passed to processor
ASCII
• On output, ASCII code from processor is
On
mapped to dot matrix pattern for display 18 ASCII Table 19 Parity
ASCII table shows characters for values 0127ten.
ASCII
Only requires 7 bits. Why 8 bits?
Only
Eighth bit (MSB) is called a parity bit can be
parity
can
used used to detect errors
used
• Parity bit can be set so the total number of 1s is odd
Parity
(called odd parity):
odd
1100 1000
‘H’
0111 0101 ‘u’ 0110 1000 ‘h’ 1011 1111 ‘?’ 1000 0000 nul • Parity allows any single bit error (single bit flip) to be
Parity
detected
detected
20 Unicode
What about nonEnglish alphabets?
What
Unicode (Universal Code) uses 2 bytes per character,
Unicode (Universal
allows 65,536 different characters. Provides a unique
code for characters in most all languages, including:
code
• Latin
• Greek
• Cyrillic
• Hebrew
• Arabic
• Persian
• Armenian
• Korean
• Chinese
• Japanese
• nine major Indian scripts
• various Southeast Asian scripts
• various African scripts
• Scandinavian
• Turkish
• etc.
21 String Declaration
String declarations put corresponding ASCII
codes in consecutive memory locations
codes
.data
error_ reply: .ascii null_char: .byte .
.
. “Huh?”
0 01001000
01110101
01101000
00111111
00000000 .text
puts error_reply 22 Memory is Typeless
Memory has not notion of data types, only
binary values
binary
.data
error_ reply: .word 0x4875683f null_char: .byte .
.
. 0
01001000
01110101
01101000
00111111
00000000 .text
puts error_reply Type is determined by the instructions that
operate on that memory
operate
23 Characters as Integers
Characters can be treated as 8bit integers (memory is
Characters
bit
typeless)
typeless)
• Can do addition, subtraction, etc.:
char1: .byte ‘8’ # ASCII representation is 0x38
char2: .byte ‘9’ # ASCII representation is 0x39
ASCII
digit: .word
digit:
...
sub digit, char1, 0x30 # subtraction produces digit value
• Can compare character values:
blt char_input, 0x30, not_digit #first digit character is 0x30
bgt char_input, 0x39, not_digit #last digit character is 0x39
#input character is a digit
...
not_digit:
not_digit: #input character is nondigit
#input
24 Converting ASCII Strings to Integers
For input ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, we want integer value
we
12345ten.
12345
• Must subtract 0x30 bias out of each character to yield true digit
Must
bias out
value
value
• Must give each digit proper weight. Because digits come left to
right, we don’t know digit weight right when it is read in. char string → integer algorithm outline:
integer = 0
while there are more digit characters
get character
digit = character  0x30
# remove the bias to get true digit
digit
integer = integer x 10 + digit # prior part of integer must be
# increased in weight by 10x 25 Converting Integers to ASCII Strings
For value 12345ten must produce character sequence
For
must
‘1’, ‘2’, ‘3’, ‘4’, ‘5’
• Must determine digit sequence, from MSD to LSD
• Must add 0x30 bias to digit for proper character value integer → char string algorithm outline:
weight = 1
while (weight x 10) <= integer
weight = weight x 10
while weight >= 1
digit = integer / weight
character = digit + 0x30
putc character
integer = integer mod weight
weight = weight / 10 # determine the max weight # get the next digit
# create proper character
# output character
# get remainder
# get next digit weight
26 Floating Point Representation
MIPS and most other processors use IEEE
IEEE
standard floating point representation.
standard
Representation consists of three parts:
Representation
• Sign Bit: 1 bit # sign bit for mantissa • Exponent: 8 bits # biased notation allows pos/neg exp.
biased • Fraction: 23 bits
31 30 S 0 23 22 Exp Fraction The value represented is:
The
(1)S x f x 2e
where
e = Exp  bias
Exp
f = Fraction/223 + 1
27 Floating Point Representation (cont.)
For single precision numbers (32bits)
For
bias = 127 Exp is biased to allow easier magnitude
comparison
comparison
• larger unsigned value is a larger exponent
00000000
00000001
00000001
00000010
00000010
00000011
00000011
...
11111101
11111110
11111111 reserved
126
125
124
...
126
127
reserved
28 Floating Point Representation (cont.)
The floating point numbers are normalized:
The
1.fraction x 2e
Because the integer part of the mantissa is
mantissa is
always* 1, there is no need to store it. The 1 is
implied. Called the hidden bit.
implied Called
hidden
31 30 S 0 23 22 Exp . Fraction 1 *Except for 0, which has a 0 for the integer part.
0 is treated specially. The all zeros pattern is
reserved, used to represent 0.
• Allows comparison with zero to be the same for integer
Allows
and floating point.
and
29 Floating Point Representation:
Floating
Example
Example
Let’s put 64.2 in floating point representation
Let
First convert to binary presentation:
64 is 1000000two
0.2 is .0011two (see Chapter 3 slides)
thus 64.2 = 1000000.0011two Next normalize the number:
Next
1000000.0011two = 1.0000000011two x 26
The FP fraction is the first 23 digits to the right of the
The
binary point (ignoring rounding, See Chapter 6):
rounding
0000 0000 1100 1100 1100 110 The exponent must be biased:
The
6ten + 127ten = 133ten = 10000101two 30 Example (cont.)
The resulting FP representation is:
The
0 10000101 00000000110011001100110
S Exp Fraction In Hex:
In
0100 0010 1000 0000 0110 0110 0110 0110
0x
0x 4 2 8 0 6 6 6 6 When we look in memory and see 0x42806666,
how do we know it is a floating point number
and not an integer or 4 characters?
and 31 Special Floating Point
Special
Representations
Representations
Zero: 0x 0000 0000
Zero:
+infinity (result is too large to represent):
+infinity
0 11111111 0000...0000
infinity (result is too small to represent):
1 11111111 0000...0000
Not A Number (NaN), e.g., result of
division by 0:
division
s 11111111 fraction
where s is either 0 or 1, and fraction is
where
nonzero
non
More on floating point in Chapter 6
More
32 Review Questions
A variation on Problem 2.13 from the text:
Write a SAL program that counts the
Write
number of punctuation characters in a
line of text entered by the user, where a
punctuation character is any nonpunctuation
numeral, nonalpha and nonspace
space
character.
character. 33 Review Questions What number does 1000 0001 represent?
What
Is it possible to tell? If not, what extra
information is needed?
information 34 Review Questions
In a normal computer each byte or word consists
In
of a collection of memory cells that are the same
size (e.g., each memory cell is one bit that holds
two values). Consider a weird computer where a
weird word consists of a collection of different
weird
consists
size memory cells. Specifically, a weird word
includes one cell of each of the following sizes:
2value, 3value, 4value, 5value, 6value, 7value, and 8value. To the nearest bit, a weird
value.
word provides storage which is equivalent to a
normal word with how many bits? 35 Review Questions
Give the single precision floating point
Give
representation for the number 83.7ten 36 ...
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Full Document
 Fall '05
 Wilken
 Addition, Negative and nonnegative numbers, ASCII

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