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TRANSCRIPT
Data Representation in
Computer Systems
Chapter 2
2
Chapter 2 Objectives
• เพอใหเขาใจการแทนตวเลข และการจดการตวเลขใน digital
Computer
• มทกษะในการเปลยนระหวางระบบเลขฐาน (radix system)
• เขาใจ ขอผดพลาดทเกดขน
– overflow
– Truncation / underflow
3
Chapter 2 Objectives
• เขาใจความคดพนฐานการแทนเลขทศนยม (floating point)
• การใชรหสตวอกขระทนยม
• เขาใจความคดเกยวกบการตรวจจบขอผดพลาด และการแกไขรหส
• A bit is the most basic unit of information in a
computer.
• A byte is a group of eight bits.
• A word is a contiguous group of bytes.
• A group of four bits is called a nibble (or nybble). : 4 บตสง 4 บตต า
4
2.1 Introduction
5
2.2 Positional Numbering Systems
• The decimal number 947 in powers of 10 is:
• The decimal number 5836.47 in powers of 10 is:
5 10 3 + 8 10 2 + 3 10 1 + 6 10 0
+ 4 10 -1 + 7 10 -2
9 10 2 + 4 10 1 + 7 10 0
6
2.2 Positional Numbering Systems
• The binary number 11001 in powers of 2 is:
• When the radix of a number is something other
than 10, the base is denoted by a subscript.
– Sometimes, the subscript 10 is added for emphasis:
110012 = 2510
1 2 4 + 1 2 3 + 0 2 2 + 0 2 1 + 1 2 0
= 16 + 8 + 0 + 0 + 1 = 25
7
2.3 Decimal to Binary Conversions
• ในการเปลยนฐานเลข สามารถท าได 2 แบบ
–การลบ (subtraction)
–การหา เอาเศษ
(ไมนยมใช การลบ)
8
• Suppose we want to convert
the decimal number 190 to
base 3.
– We know that 3 5 = 243 so
our result will be less than six
digits wide. The largest
power of 3 that we need is
therefore 3 4 = 81, and
81 2 = 162.
– Write down the 2 and subtract
162 from 190, giving 28.
2.3 Decimal to Binary Conversions
9
• Converting 190 to base 3...
– The next power of 3 is
3 3 = 27. We’ll need one of
these, so we subtract 27 and
write down the numeral 1 in
our result.
– The next power of 3, 3 2 = 9,
is too large, but we have to
assign a placeholder of zero
and carry down the 1.
2.3 Decimal to Binary Conversions
10
2.3 Decimal to Binary Conversions
• Converting 190 to base 3...
– 3 1 = 3 is again too large, so
we assign a zero placeholder.
– The last power of 3, 3 0 = 1,
is our last choice, and it gives
us a difference of zero.
– Our result, reading from top
to bottom is:
19010 = 210013
สรปใชวธการลบ
11
เปลยน190 จากฐาน 10 เปน ฐาน 3 : (190)10 = (???)3 5 4 3 2 1 0
2 1 0 0 1
243 81 27 9 3 1
162 = 81 x 2
28
27 = 27 x 1
1 0 = 9 x 0
0
0 = 3 x 0
0 1 = 1 x 1
1
190
12
2.3 Decimal to Binary Conversions
• Another method of converting integers from
decimal to some other radix uses division.
• This method is mechanical and easy.
• It employs the idea that successive division by a
base is equivalent to successive subtraction by
powers of the base.
• Let’s use the division remainder method to again
convert 190 in decimal to base 3.
13
• Converting 190 to base 3...
– First we take the number
that we wish to convert and
divide it by the radix in
which we want to express
our result.
– In this case, 3 divides 190
63 times, with a remainder
of 1.
– Record the quotient and the
remainder.
2.3 Decimal to Binary Conversions
14
• Converting 190 to base 3...
– 63 is evenly divisible by 3.
– Our remainder is zero, and
the quotient is 21.
2.3 Decimal to Binary Conversions
15
• Converting 190 to base 3...
– Continue in this way until
the quotient is zero.
– In the final calculation, we
note that 3 divides 2 zero
times with a remainder of 2.
– Our result, reading from
bottom to top is:
19010 = 210013
2.3 Decimal to Binary Conversions
16
2.3 Decimal to Binary Conversions
การเปลยนเลขฐาน 10 เปนฐาน 2 โดยการหารเอาเศษ
• (37)10 = (100101)2
• (103)10 = (1100111)2
• (317)10 = (100111101)2
17
2.3 Decimal to Binary Conversions
การเปลยนเลขฐาน 10 เปนฐาน 2 โดยการหารเอาเศษ
• (120)10 = (xxx)2
• (517)10 = (xxx)2
• (1023)10 = (xxx)2
18
2.3 Decimal to Binary Conversions
การเปลยนเลขฐาน 2 เปนฐาน 10 โดยน า คาต าแหนงนน คณดวยฐานยกก าลงในต าแหนงทอย
• (1001)2 = (9)10
• (110100)2 = (52)10
• (10010010)2 = (146)10
19
2.3 Decimal to Binary Conversions
การเปลยนเลขฐาน 2 เปนฐาน 10 โดยน า คาต าแหนงนน คณดวยฐานยกก าลงในต าแหนงทอย
• (100101)2 = (x)10
• (110111)2 = (x)10
• (10110110)2 = (x)10
20
2.3 Decimal to Binary Conversions
• สวนทเปน ทศนยม หรอ เศษสวน ฐาน 10
709.52 = 7x102 + 0x101 + 9x 100
+ 5x10-1 + 2x10-2
• สวนทเปนเศษสวน ฐาน 2 ของ 101.012
101.01 = 1x22 + 0x21 + 1x 20 +
0x2-1 + 1x2-2
• เปลยนเศษสวน 5.25 ฐาน 10 เปน ฐาน 2
(5.25)10 = (101.01)2
101.01
21
2.3 Decimal to Binary Conversions
.25 x 2 = 0.5
.5 x 2 = 1.0
หยด
• เปลยน 100.62510 เปน ฐาน 2
(100.625)10 = (1100100.101)2
1100100.101
22
2.3 Decimal to Binary Conversions
.625 x 2 = 1.250
.25 x 2 = 0.5
.5 x 2 = 1.0
หยด
• เปลยน 17.810 เปน ฐาน 2
(17.8)10 = (10001.11001)2
23
2.3 Decimal to Binary Conversions
.8 x 2 = 1.6 .6 x 2 = 1.2 .2 x 2 = 0.4 .4 x 2 = 0.8 .8 x 2 = 1.6 .6 x 2 = 1.2
หยด ถาตองการ 5ต าแหนง
24
2.3 Decimal to Binary Conversions
• เปลยนเลขทศนยมฐาน 10 ตอไปนเปน ฐาน 2
(โดยใหมทศนยม 5 ต าแหนง)
51.12510 = (xxx)2
101.910 = (xxx)2
131.562510 = (xxx)2
199.4210 = (xxx)2
25
2.3 Decimal to Binary Conversions
• แปลง 1011.0112 เปน ฐาน 10
1011.0112 = 11.37510
0 + 1 + 1
2 4 8
0 + 0.25 + 0.125
26
2.3 Decimal to Binary Conversions
• เปลยนเศษสวน ฐาน 2 เปน ฐาน 10
10.11012 = xxx10
1011.1012 = xxx10
11.11002 = xxx10
27
2.3 Decimal to Binary Conversions
• เปลยนเลข ฐาน 2 เปน ฐาน 8 (นบ 3 จาก LSB : บตทมนยส าคญนอยสด ทเหลอไมครบ 3 เตม 0 ดานหนา)
(10 100 101 001)2 = (2451)8
010
28
2.3 Decimal to Binary Conversions
• เปลยนเลข ฐาน 2 เปน ฐาน 16 (นบ 4 จาก LSB : บตทมนยส าคญนอยสด ทเหลอไมครบ 4 เตม 0 ดานหนา)
(101 0010 1001)2 = (529)16
0101
29
2.3 Decimal to Binary Conversions
• เปลยนจ านวนจรง ฐาน 2 เปน ฐาน 8 (นบ 3)
– หนาจด นบจาก LSB ขาดเตม 0 ดานหนา
– หลงจด นบจาก MSB ขาดเตม 0 ดานหลง
(10011.011)2 = ?8
(010 011.011)2 = (23.3)8
(1111.11)2 = ?8
(001 111.110)2 = (17.6)8
30
2.3 Decimal to Binary Conversions
• เปลยนจ านวนจรง ฐาน 2 เปน ฐาน 8
(11000110.1)2 = (xxxxx)8
(1.11010)2 = (xxxxx)8
31
2.3 Decimal to Binary Conversions
• การแปลงเลขจ านวนจรงฐาน 2 เปน ฐาน 16
– หนาจด นบจาก LSB ขาดเตม 0 ดานหนา
– หลงจด นบจาก MSB ขาดเตม 0 ดานหลง
(10011.011)2 = ?16
(0001 0011.0110)2 = (13.6)16
(1111.11)2 = ?16
(1111.1100)2 = (F.C)16
32
2.3 Decimal to Binary Conversions
• การแปลงเลขจ านวนจรงฐาน 2 เปน ฐาน 16
(11000110.1)2 = (xxxxx)16
(1.11010)2 = (xxxxx)16
33
2.3 Decimal to Binary Conversions
• เปลยนเศษสวน ฐาน 10 เปน ฐาน 8
• ท างานเหมอนกบเปนเปลยนเศษสวน ฐาน 10 เปน ฐาน 2 คอหารดวยฐานไปเรอยๆ
(12.5)10 = ?8
[(12)/8.(58)]10 = (14.4)8
4.0 181+480
34
2.3 Decimal to Binary Conversions
• เปลยนเศษสวน ฐาน 10 เปน ฐาน 16
• ท างานเหมอนกบเปนเปลยนเศษสวน ฐาน 10 เปน ฐาน 2 คอหารดวยฐานไปเรอยๆ
(12.5)10 = ?16
[(12/16).(516)]10 = (C.8)16
8.0 C160
35
2.4 Signed Integer Representation
• The conversions we have so far presented
have involved only positive numbers.
• To represent negative values, computer
systems allocate the high-order bit to indicate
the sign of a value.
– The high-order bit is the leftmost bit in a byte. It
is also called the most significant bit.
• The remaining bits contain the value of the
number.
36
2.4 Signed Integer Representation
Signed Integer Representation
การแสดงเลขจ านวนเตม แบบมเครองหมาย
37
2.4 Signed Integer Representation
• There are three ways in which signed binary
numbers may be expressed:
– Signed magnitude,
– One’s complement and
– Two’s complement.
• In an 8-bit word, signed magnitude
representation places the absolute value of
the number in the 7 bits to the right of the
sign bit.
38
2.4 Signed Integer Representation
การแทนขอมลขนาด 8 บต
LSB MSB
7 6 0
Sign Bit = 1 มคาเปนลบ
= 0 มคาเปนบวก
Sign bit : เครองหมาย Magnitude : ขนาด
39
2.4 Signed Integer Representation
1 111 1111
:
1 000 0001
1 000 0000
0 000 0000
0 000 0001
:
0 111 1111 -127 … -1 -0 +0 +1 … +127
• Data range
Binary Decimal
0 0001001
1 0110101
1 0100100
0 1111111
+9
-53
-40
+127
• For 8-bit in memory
40
2.4 Signed Integer Representation
1 111 1111
:
1 000 0000
0 000 0000
0 000 0001
:
0 111 1111
-127 … -1 -0 +0 +1 … +127
Binary Decimal
0 0001001
1 1110110
1 1011000
0 1111111
0 1111111
+9
-9
-40
-127
+127
• For 8-bit in memory for 1’s Complement
• Data range
41
2.4 Signed Integer Representation
• 2’s Complement
–Invert and +1 1 000 0000
:
1 111 1111
0 000 0000
0 000 0001
:
0 111 1111
-128 … -1 +0 +1 … +127
Binary Decimal
0 0001001
1 1110111
1 1011100
1 0000000
0 1111111
+9
-9
-40
-128
+127
• Data range
42
2.4 Signed Integer Representation
43
2.4 Signed Integer Representation
• For example, in 8-bit signed magnitude, positive 3 is: 00000011
• Negative 3 is: 10000011
• Computers perform arithmetic operations on
signed magnitude numbers in much the same
way as humans carry out pencil and paper
arithmetic.
– Humans often ignore the signs of the operands
while performing a calculation, applying the
appropriate sign after the calculation is complete.
44
2.4 Signed Integer Representation
• Binary addition is as easy as it gets. You need
to know only four rules: 0 + 0 = 0 0 + 1 = 1
1 + 0 = 1 1 + 1 = 10
• The simplicity of this system makes it possible
for digital circuits to carry out arithmetic
operations.
0 + 0 = 00 0 + 1 = 01
1 + 0 = 01 1 + 1 = 10
SUM Carry
2.4 Signed Integer Representation
• Positive numbers is the same in one’s
complement and two’s complement (as
well as signed-magnitude), the process of
adding two positive binary numbers is the
same.
45
2.4 Signed Integer Representation
• How to perform addition using two’s
complement notation (and hence
subtraction, because we subtract a number
by adding its opposite).
• Ex: Add 910 to –2310 using two’s
complement arithmetic.
46
Two’s complement arithmetic
operation
• Ex: Find the sum of 2310 and –910 in binary
using two’s complement arithmetic.
47
• Ex: Find the sum of 111010012 (–23) and
111101112 (–9) using two’s complement addition.
Overflow A Simple Rule for Detecting an Overflow
Condition in Signed Numbers:
• If the carry into the sign bit equals the carry out of
the bit, no overflow has occurred.
• If the carry into the sign bit is different from the
carry out of the sign bit, overflow (and thus an
error) has occurred. See Ex.
48
Overflow
Consider 4-bit unsigned and signed numbers.
• If we add the two unsigned values 0111 (7)
and 0001 (1), we get 1000 (8).
– There is no carry (out), and thus no error.
• If we add the two unsigned values 0111 (7)
and 1011 (11), we get 0010 with a carry,
indicating that there is an error (indeed, 7 + 11
is not 2).
49
Overflow in signed numbers
• Adding the two’s complement integers 0101
(+5) and 0011 (+3) equal to 1000 (–8),
which is clearly incorrect.
– The problem is that we have a carry in to the
sign bit, but no carry out, which indicates that
we have an overflow.
• However, add 0111 (+7) and 1011 (–5), we
get the correct result: 0010 (+2).
– We have both a carry in to and a carry out of
the leftmost bit, so there is no error. 50
Overflow summation
• The rule of thumb used to determine when
carry indicates an error depends on
whether we are using signed or unsigned
numbers.
– Unsigned numbers: a carry (out of the leftmost
bit) indicates the total number of bits was not
large enough to hold the resulting value, and
overflow has occurred.
– Signed numbers: if the carry in to the sign bit
and the carry (out of the sign bit) differ, then
overflow has occurred. 51
Examples of Carry and Overflow
in Signed Numbers
52
2.4.4 Unsigned Versus Signed Numbers
• Unsigned numbers represent values that are guaranteed
not to be negative.
• A computer programmer must be able to manage both
signed and unsigned numbers.
– For instance, the C language has int and unsigned int for
integer variables, defining signed and unsigned integers,
respectively.
– Different type declarations, many languages have different
arithmetic operations with signed and unsigned numbers.
• Good programmers understand this condition and
make appropriate plans to deal with the situation
before it occurs. 53
• วงจรเกทสรางการบวก บทแรก (LSB)
54
2.4 Signed Integer Representation
A B Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
XOR AND
2.4.7 Binary Multiplication and
Division Using Shifting
55
56
2.4 Signed Integer Representation
signed magnitude
• มบตบอกเครองหมาย
0 : บวก 1 : ลบ
• บตบอกขนาด 57
2.4 Signed Integer Representation
58
2.4 Signed Integer Representation
• Example:
– Using signed magnitude
binary arithmetic, find the
sum of 75 and 46.
• First, convert 75 and 46 to
binary, and arrange as a sum,
but separate the (positive)
sign bits from the magnitude
bits.
59
2.4 Signed Integer Representation
• Example:
– Using signed magnitude
binary arithmetic, find the
sum of 75 and 46.
• Just as in decimal arithmetic,
we find the sum starting with
the rightmost bit and work left.
60
2.4 Signed Integer Representation
• Example:
– Using signed magnitude
binary arithmetic, find the
sum of 75 and 46.
• In the second bit, we have a
carry, so we note it above the
third bit.
61
2.4 Signed Integer Representation
• Example:
– Using signed magnitude
binary arithmetic, find the
sum of 75 and 46.
• The third and fourth bits also
give us carries.
62
2.4 Signed Integer Representation
• Example:
– Using signed magnitude binary
arithmetic, find the sum of 75
and 46.
• Once we have worked our way
through all eight bits, we are
done.
In this example, we were careful careful to pick two
values whose sum would fit into seven bits. If that is not
the case, we have a problem.
63
2.4 Signed Integer Representation
• Example:
– Using signed magnitude binary
arithmetic, find the sum of 107
and 46.
• We see that the carry from the
seventh bit overflows and is
discarded, giving us the
erroneous result: 107 + 46 = 25.
64
2.4 Signed Integer Representation
• The signs in signed
magnitude representation
work just like the signs in
pencil and paper arithmetic.
– Example: Using signed
magnitude binary arithmetic,
find the sum of - 46 and - 25.
• Because the signs are the same, all we do is
add the numbers and supply the negative sign
when we are done.
65
2.4 Signed Integer Representation
• Mixed sign addition (or
subtraction) is done the
same way.
– Example: Using signed
magnitude binary arithmetic,
find the sum of 46 and - 25.
• The sign of the result gets the sign of the number
that is larger.
– Note the “borrows” from the second and sixth bits.
66
2.4 Signed Integer Representation
• Signed magnitude representation is easy for
people to understand, but it requires complicated computer hardware.
• Another disadvantage of signed magnitude is
that it allows two different representations for
zero: positive zero and negative zero.
67
2.4 Signed Integer Representation
• In complement systems, negative values are
represented by some difference between a number and its base.
• In diminished radix complement systems, a
negative value is given by the difference between
the absolute value of a number and one less than
its base.
• In the binary system, this gives us one’s
complement. It amounts to little more than flipping
the bits of a binary number.
one’s complement.
•คาลบเกบแบบสลบบต
ตองการเกบในหนวยความจ าขนาด 8 บต (5)10 = (0000 0101)2
(-5)10 = (1111 1010)2
68
2.4 Signed Integer Representation
69
2.4 Signed Integer Representation
• one’s complement.
• For example, in 8-bit one’s complement, positive 3 is: 00000011
• Negative 3 is: 11111100
• In one’s complement, as with signed
magnitude, negative values are indicated by a
1 in the high order bit.
70
2.4 Signed Integer Representation
• With one’s complement
addition, the carry bit is
“carried around” and added
to the sum.
– Example: Using one’s
complement binary arithmetic,
find the sum of 48 and - 19
We note that 19 in one’s complement is 00010011,
so -19 in one’s complement is: 11101100.
71
2.4 Signed Integer Representation
• Although the “end carry around” adds some
complexity, one’s complement is simpler to implement than signed magnitude.
• But it still has the disadvantage of having two
different representations for zero: positive
zero and negative zero.
• Two’s complement solves this problem.
• Two’s complement is the radix complement of
the binary numbering system.
Two’s complement.
•คาลบเกบแบบสลบบต +1
ตองการเกบในหนวยความจ าขนาด 8 บต (5)10 = (0000 0101)2
(-5)10 = (1111 1011)2
72
2.4 Signed Integer Representation
73
2.4 Signed Integer Representation
• To express a value in two’s complement:
– If the number is positive, just convert it to binary and
you’re done.
– If the number is negative, find the one’s complement of
the number and then add 1.
• Example:
– In 8-bit one’s complement, positive 3 is: 00000011
– Negative 3 in one’s complement is: 11111100
– Adding 1 gives us -3 in two’s complement form: 11111101.
74
2.4 Signed Integer Representation
• With two’s complement arithmetic, all we do is add
our two binary numbers. Just discard any carries
emitting from the high order bit.
We note that 19 in one’s complement is: 00010011,
so -19 in one’s complement is: 11101100,
and -19 in two’s complement is: 11101101.
– Example: Using one’s
complement binary
arithmetic, find the sum of
48 and - 19.
75
2.4 Signed Integer Representation
• When we use any finite number of bits to
represent a number, we always run the risk of
the result of our calculations becoming too large
to be stored in the computer.
• While we can’t always prevent overflow, we can
always detect overflow.
• In complement arithmetic, an overflow condition
is easy to detect.
76
2.4 Signed Integer Representation
• Example:
– Using two’s complement binary
arithmetic, find the sum of 107
and 46.
• We see that the nonzero carry
from the seventh bit overflows into
the sign bit, giving us the
erroneous result: 107 + 46 = -103.
Rule for detecting signed two’s complement overflow: When
the “carry in” and the “carry out” of the sign bit differ,
overflow has occurred.
77
2.4 Signed Integer Representation
0110
78
2.4 Signed Integer Representation
• Signed and unsigned numbers are both useful.
– For example, memory addresses are always unsigned.
• Using the same number of bits, unsigned integers
can express twice as many values as signed
numbers.
• Trouble arises if an unsigned value “wraps around.”
– In four bits: 1111 + 1 = 0000.
• Good programmers stay alert for this kind of
problem.
79
2.4 Signed Integer Representation
80
2.4 Signed Integer Representation
81
2.4 Signed Integer Representation
82
2.4 Signed Integer Representation
83
2.4 Signed Integer Representation
84
2.4 Signed Integer Representation
85
2.4 Signed Integer Representation
86
2.4 Signed Integer Representation
87
2.4 Signed Integer Representation
88
2.4 Signed Integer Representation
89
2.5 Floating-Point Representation
Floating-Point Representation
การแสดง Floating-Point
90
2.5 Floating-Point Representation
• Floating-point numbers allow an arbitrary
number of decimal places to the right of the
decimal point.
– For example: 0.5 0.25 = 0.125
• They are often expressed in scientific notation.
– For example:
0.125 = 1.25 10-1
5,000,000 = 5.0 106
91
2.5 Floating-Point Representation
• Computers use a form of scientific notation for
floating-point representation
• Numbers written in scientific notation have three
components:
92
• Computer representation of a floating-point
number consists of three fixed-size fields:
• This is the standard arrangement of these fields.
2.5 Floating-Point Representation
93
• The one-bit sign field is the sign of the stored value.
• The size of the exponent field, determines the
range of values that can be represented.
• The size of the significand determines the precision
of the representation.
2.5 Floating-Point Representation
94
• The IEEE-754 single precision floating point
standard uses an 8-bit exponent and a 23-bit
significand.
• The IEEE-754 double precision standard uses an
11-bit exponent and a 52-bit significand.
2.5 Floating-Point Representation
For illustrative purposes, we will use a 14-bit model
with a 5-bit exponent and an 8-bit significand.
Example IEEE-754 Single-Precision
Floating-Point Numbers
95
Range of IEEE-754
Double-Precision Numbers
96
2127 × 1.1111111111 11111111111112 (let’s call this value MAX)
2–127 × .000000000000000000000012 (let’s call this value MIN).
Single precision floating-point numbers cannot represent:
- negative numbers less than –MAX (negative overflow);
- negative numbers greater than –MIN (negative underflow);
- positive numbers less than +MIN (positive underflow); and
- positive numbers greater than +MAX (positive overflow).
98
• Example:
– Express 3210 in the simplified 14-bit floating-point
model.
• We know that 32 is 25. So in (binary) scientific
notation 32 = 1.0 x 25 = 0.1 x 26.
• Using this information, we put 110 (= 610) in the
exponent field and 1 in the significand as shown.
2.5 Floating-Point Representation
99
• Another problem with our system is that we have
made no allowances for negative exponents. We
have no way to express 0.5 (=2 -1)! (Notice that
there is no sign in the exponent field!)
2.5 Floating-Point Representation
All of these problems can be fixed with no
changes to our basic model.
100
• To provide for negative exponents, we will use a
biased exponent.
• 5 bits แทนไดถง 32 เพอเกบคาลบดวย ถงแบงครง 16 เรยกวา excess-
16
– In our case, we have a 5-bit exponent. We will use 16 for
our bias. This is called excess-16 representation.
2.5 Floating-Point Representation
16 31 0 - +
101
• Example:
– Express 3210 in the revised 14-bit floating-point model.
• We know that 32 = 1.0 x 25 = 0.1 x 26.
• To use our excess 16 biased exponent, we add 16 to
6, giving 2210 (=101102).
• Graphically:
2.5 Floating-Point Representation
102
• Example:
– Express 0.062510 in the revised 14-bit floating-point
model.
• We know that 0.0625 is 2-4. So in (binary) scientific
notation 0.0625 = 1.0 x 2-4 = 0.1 x 2 -3.
• To use our excess 16 biased exponent, we add 16 to
-3, giving 1310 (=011012).
2.5 Floating-Point Representation
103
• Example:
– Express -26.62510 in the revised 14-bit floating-point
model.
• We find 26.62510 = 11010.1012. Normalizing, we
have: 26.62510 = 0.11010101 x 2 5.
• To use our excess 16 biased exponent, we add 16 to
5, giving 2110 (=101012). We also need a 1 in the sign
bit.
2.5 Floating-Point Representation
104
• Calculations aren’t useful until their results can
be displayed in a manner that is meaningful to
people.
• We also need to store the results of calculations,
and provide a means for data input.
• Thus, human-understandable characters must be
converted to computer-understandable bit
patterns using some sort of character encoding
scheme.
2.6 Character Codes
รหสแทนขอมลภายในเครองคอมพวเตอร
• รหสแทนขอมลทเปนตวเลข – Sign and Magnitude
– 1’s Complement
– 2’s Complement
• รหสแทนขอมลทเปนตวอกษร – BCD : Binary Code Decimal
– EBCDIC : Extended Binary Code Decimal Interchange
Code
– ASCII : American Standard Code Interchange International
จ านวนแบบตวอกขระในระบบคอมพวเตอร นอยสด
• ตวเลข 10 ตว ( 0, 1 ,2 … 9)
• ตวอกษรภาษาองกฤษ 26 ตว ( A,B,…, Z)
• เครองหมายพเศษ 16 ตว ( + - * / > < ; ,!@#$%&() )
รวมทงหมด 52 ตว สามารถใชกลมเลขฐานสองเพยง 6 bits
ในระบบ BCD เพม Parity bit อก 1 bit เพอการตรวจสอบความถกตอง
รหสแทนขอมลทเปนตวอกษร แบบ BCD
• ใน 1 Byte ม 7 Bits
BA Numeric แทน 00 0 - 9 0 - 9 11 1 - 9 A - I 10 1 - 9 J - R 01 2 - 9 S - Z
• อกขระพเศษแตละบรษทก าหนดเอง
C B A 8 4 2 1
Numeric Zone
Parity
Binary-Coded Decimal
108
Digit BCD
0
1
2
3
4
5
6
7
8
9
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
Zones
1111
1100
1101
Unsigned
Positive
Negative
BCD
• For example, to encode 146, the decimal
digits are replaced by 0001, 0100, and 0110,
respectively.
• Computers use bytes as the smallest unit of
access, most values are stored in 8 bits, not
4. Using this approach, padding with zeros,
146 would be stored as 00000001 00000100
00000110.
109
packed BCD
110
รหสแทนขอมลทเปนตวอกษร แบบ EBCDIC
• 1 Byte ใช 8 bits
• แทนอกขระได 256 ตว 8 4 2 1 8 4 2 1
Zone Numeric
EBCDIC Code
112
American Standard Code
for Information Interchange (ASCII)
• ออกโดย สมาคมรกษามาตรฐานของ สหรฐอเมรกา • เพอใหการตดตอสอสารระหวางเครองคอมพวเตอร มความสะดวก
และเปนมาตรฐานเดยวกน
• 1 Byte ใช 8 bit
• มใชในเครองไมโครคอมพวเตอร ทดสอบการแทนรหสไดโดยการกด Alt + Numeric Key Pad แทนรหส ASCII
114
รหสแทนขอมลทเปนตวอกษร แบบ ASCII
Parity bit
ใชในการตรวจสอบความผดพลาด โดยกระท าภายใตขอสมมตฐาน 2 ขอ • คาของความผดพลาดทจะเกดม Percent นอยมาก • ความผดพลาดทเกดขน เกดไดเพยง bit เดยวเทานนใน 1 byte
Check Parity Bit ม 2 แบบ
• Odd Parity จ านวน bit ค • Even Parity จ านวน bit ค
116
Parity bit
รหสแทนขอมลทเปนตวอกษร แบบ ASCII
117
Unicode
• In 1991, to establish a new international
information exchange code called Unicode.
• Unicode is a 16-bit alphabet that is downward
compatible with ASCII and the Latin-1
character set.
• Conformant with the ISO/IEC 10646-1
international alphabet.
• Capacity to encode the majority of characters
used in every language of the world. 118
Unicode Codespace
119
120
End of Chapter 2