程序代写代做代考 Bits, Data Types, and Operations

Bits, Data Types, and Operations

Chapter 2
Bits, Data Types,
and Operations

ECE 206 – Fall 2000 – G. Byrd

How do we represent data in a computer?
At the lowest level, a computer is an electronic machine.
works by controlling the flow of electrons

Easy to recognize two conditions:
presence of a voltage – we’ll call this state “1”
absence of a voltage – we’ll call this state “0”

Could base state on value of voltage,
but control and detection circuits more complex.
compare turning on a light switch to
measuring or regulating voltage

ECE 206 – Fall 2000 – G. Byrd

Computer is a binary digital system.
Basic unit of information is the binary digit, or bit.
Values with more than two states require multiple bits.
A collection of two bits has four possible states:
00, 01, 10, 11
A collection of three bits has eight possible states:
000, 001, 010, 011, 100, 101, 110, 111
A collection of n bits has 2n possible states.

Binary (base two) system:
has two states: 0 and 1

Digital system:
finite number of symbols

ECE 206 – Fall 2000 – G. Byrd

What kinds of data do we need to represent?

Numbers – signed, unsigned, integers, floating point,
complex, rational, irrational, …
Text – characters, strings, …
Images – pixels, colors, shapes, …
Sound
Logical – true, false
Instructions
…

Data type:
representation and operations within the computer

We’ll start with numbers…

ECE 206 – Fall 2000 – G. Byrd

Unsigned Integers
Non-positional notation
could represent a number (“5”) with a string of ones (“11111”)
problems?

Weighted positional notation
like decimal numbers: “329”
“3” is worth 300, because of its position, while “9” is only worth 9

3×100 + 2×10 + 9×1 = 329
1×4 + 0x2 + 1×1 = 5
most
significant
least
significant
329
102
101
100

101
22
21
20

ECE 206 – Fall 2000 – G. Byrd

Unsigned Integers (cont.)
An n-bit unsigned integer represents 2n values:
from 0 to 2n-1.
22 21 20
0 0 0 0
0 0 1 1
0 1 0 2
0 1 1 3
1 0 0 4
1 0 1 5
1 1 0 6
1 1 1 7

ECE 206 – Fall 2000 – G. Byrd

Unsigned Binary Arithmetic
Base-2 addition – just like base-10!
add from right to left, propagating carry

10010 10010 1111
+ 1001 + 1011 + 1
11011 11101 10000

10111
+ 111
carry
Subtraction, multiplication, division,…

ECE 206 – Fall 2000 – G. Byrd

Signed Integers
With n bits, we have 2n distinct values.
assign about half to positive integers (1 through 2n-1)
and about half to negative (- 2n-1 through -1)
that leaves two values: one for 0, and one extra

Positive integers
just like unsigned – zero in most significant (MS) bit
00101 = 5

Negative integers
sign-magnitude – set MS bit to show negative,
other bits are the same as unsigned
10101 = -5
one’s complement – flip every bit to represent negative
11010 = -5
in either case, MS bit indicates sign: 0=positive, 1=negative

ECE 206 – Fall 2000 – G. Byrd

Two’s Complement
Problems with sign-magnitude and 1’s complement
two representations of zero (+0 and –0)
arithmetic circuits are complex

How to add two sign-magnitude numbers?
e.g., try 2 + (-3)
How to add to one’s complement numbers?
e.g., try 4 + (-3)
Two’s complement representation developed to make
circuits easy for arithmetic.
for each positive number (X), assign value to its negative (-X),
such that X + (-X) = 0 with “normal” addition, ignoring carry out

00101 (5) 01001 (9)
+ 11011 (-5) + (-9)
00000 (0) 00000 (0)

ECE 206 – Fall 2000 – G. Byrd

Two’s Complement Representation
If number is positive or zero,
normal binary representation, zeroes in upper bit(s)

If number is negative,
start with positive number
flip every bit (i.e., take the one’s complement)
then add one

00101 (5) 01001 (9)
11010 (1’s comp) (1’s comp)
+ 1 + 1
11011 (-5) (-9)

ECE 206 – Fall 2000 – G. Byrd

Two’s Complement Shortcut
To take the two’s complement of a number:
copy bits from right to left until (and including) the first “1”
flip remaining bits to the left

011010000 011010000
100101111 (1’s comp)
+ 1
100110000 100110000
(copy)
(flip)

ECE 206 – Fall 2000 – G. Byrd

Two’s Complement Signed Integers
MS bit is sign bit – it has weight –2n-1.
Range of an n-bit number: -2n-1 through 2n-1 – 1.
The most negative number (-2n-1) has no positive counterpart.

-23 22 21 20
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
0 1 1 0 6
0 1 1 1 7

-23 22 21 20
1 0 0 0 -8
1 0 0 1 -7
1 0 1 0 -6
1 0 1 1 -5
1 1 0 0 -4
1 1 0 1 -3
1 1 1 0 -2
1 1 1 1 -1

ECE 206 – Fall 2000 – G. Byrd

Converting Binary (2’s C) to Decimal
If leading bit is one, take two’s complement to get a positive number.
Add powers of 2 that have “1” in the
corresponding bit positions.
If original number was negative,
add a minus sign.

X = 01101000two
= 26+25+23 = 64+32+8
= 104ten
Assuming 8-bit 2’s complement numbers.
n 2n
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024

ECE 206 – Fall 2000 – G. Byrd

More Examples

Assuming 8-bit 2’s complement numbers.
X = 00100111two
= 25+22+21+20 = 32+4+2+1
= 39ten
X = 11100110two
-X = 00011010
= 24+23+21 = 16+8+2
= 26ten
X = -26ten
n 2n
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024

ECE 206 – Fall 2000 – G. Byrd

Converting Decimal to Binary (2’s C)
First Method: Division
Find magnitude of decimal number. (Always positive.)
Divide by two – remainder is least significant bit.
Keep dividing by two until answer is zero,
writing remainders from right to left.
Append a zero as the MS bit;
if original number was negative, take two’s complement.

X = 104ten 104/2 = 52 r0 bit 0
52/2 = 26 r0 bit 1
26/2 = 13 r0 bit 2
13/2 = 6 r1 bit 3
6/2 = 3 r0 bit 4
3/2 = 1 r1 bit 5
X = 01101000two 1/2 = 0 r1 bit 6

ECE 206 – Fall 2000 – G. Byrd

Converting Decimal to Binary (2’s C)
Second Method: Subtract Powers of Two
Find magnitude of decimal number.
Subtract largest power of two
less than or equal to number.
Put a one in the corresponding bit position.
Keep subtracting until result is zero.
Append a zero as MS bit;
if original was negative, take two’s complement.

X = 104ten 104 – 64 = 40 bit 6
40 – 32 = 8 bit 5
8 – 8 = 0 bit 3
X = 01101000two
n 2n
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024

ECE 206 – Fall 2000 – G. Byrd

Operations: Arithmetic and Logical
Recall:
a data type includes representation and operations.
We now have a good representation for signed integers,
so let’s look at some arithmetic operations:
Addition
Subtraction
Sign Extension

We’ll also look at overflow conditions for addition.
Multiplication, division, etc., can be built from these
basic operations.
Logical operations are also useful:
AND
OR
NOT

ECE 206 – Fall 2000 – G. Byrd

Addition
As we’ve discussed, 2’s comp. addition is just
binary addition.
assume all integers have the same number of bits
ignore carry out
for now, assume that sum fits in n-bit 2’s comp. representation

01101000 (104) 11110110 (-10)
+ 11110000 (-16) + (-9)
01011000 (98) (-19)
Assuming 8-bit 2’s complement numbers.

ECE 206 – Fall 2000 – G. Byrd

Subtraction
Negate subtrahend (2nd no.) and add.
assume all integers have the same number of bits
ignore carry out
for now, assume that difference fits in n-bit 2’s comp. representation

01101000 (104) 11110110 (-10)
– 00010000 (16) – (-9)
01101000 (104) 11110110 (-10)
+ 11110000 (-16) + (9)
01011000 (88) (-1)
Assuming 8-bit 2’s complement numbers.

ECE 206 – Fall 2000 – G. Byrd

Sign Extension
To add two numbers, we must represent them
with the same number of bits.
If we just pad with zeroes on the left:

Instead, replicate the MS bit — the sign bit:
4-bit 8-bit
0100 (4) 00000100 (still 4)
1100 (-4) 00001100 (12, not -4)
4-bit 8-bit
0100 (4) 00000100 (still 4)
1100 (-4) 11111100 (still -4)

ECE 206 – Fall 2000 – G. Byrd

Overflow
If operands are too big, then sum cannot be represented as an n-bit 2’s comp number.

We have overflow if:
signs of both operands are the same, and
sign of sum is different.

Another test — easy for hardware:
carry into MS bit does not equal carry out

01000 (8) 11000 (-8)
+ 01001 (9) + 10111 (-9)
10001 (-15) 01111 (+15)

ECE 206 – Fall 2000 – G. Byrd

Logical Operations
Operations on logical TRUE or FALSE
two states — takes one bit to represent: TRUE=1, FALSE=0

View n-bit number as a collection of n logical values
operation applied to each bit independently

A B A AND B
0 0 0
0 1 0
1 0 0
1 1 1

A B A OR B
0 0 0
0 1 1
1 0 1
1 1 1

A NOT A
0 1
1 0

ECE 206 – Fall 2000 – G. Byrd

Examples of Logical Operations
AND
useful for clearing bits

AND with zero = 0
AND with one = no change

OR
useful for setting bits

OR with zero = no change
OR with one = 1

NOT
unary operation — one argument
flips every bit

11000101
AND 00001111
00000101
11000101
OR 00001111
11001111
NOT 11000101
00111010

ECE 206 – Fall 2000 – G. Byrd

Hexadecimal Notation
It is often convenient to write binary (base-2) numbers
as hexadecimal (base-16) numbers instead.
fewer digits — four bits per hex digit
less error prone — easy to corrupt long string of 1’s and 0’s

Binary Hex Decimal
0000 0 0
0001 1 1
0010 2 2
0011 3 3
0100 4 4
0101 5 5
0110 6 6
0111 7 7

Binary Hex Decimal
1000 8 8
1001 9 9
1010 A 10
1011 B 11
1100 C 12
1101 D 13
1110 E 14
1111 F 15

ECE 206 – Fall 2000 – G. Byrd

Converting from Binary to Hexadecimal
Every four bits is a hex digit.
start grouping from right-hand side

011101010001111010011010111
7
D
4
F
8
A
3
This is not a new machine representation,
just a convenient way to write the number.

ECE 206 – Fall 2000 – G. Byrd

Fractions: Fixed-Point
How can we represent fractions?
Use a “binary point” to separate positive
from negative powers of two — just like “decimal point.”
2’s comp addition and subtraction still work.

if binary points are aligned
No new operations — same as integer arithmetic.
00101000.101 (40.625)
+ 11111110.110 (-1.25)
00100111.011 (39.375)
2-1 = 0.5
2-2 = 0.25
2-3 = 0.125

ECE 206 – Fall 2000 – G. Byrd

Very Large and Very Small: Floating-Point
Large values: 6.023 x 1023 — requires 79 bits
Small values: 6.626 x 10-34 — requires >110 bits

Use equivalent of “scientific notation”: F x 2E
Need to represent F (fraction), E (exponent), and sign.
IEEE 754 Floating-Point Standard (32-bits):
S
Exponent
Fraction
1b
8b
23b

ECE 206 – Fall 2000 – G. Byrd
Exponent = 255 used for special values:
If Fraction is non-zero, NaN (not a number).
If Fraction is zero and sign is 0, positive infinity.
If Fraction is zero and sign is 1, negative infinity.

Floating Point Example
Single-precision IEEE floating point number:
10111111010000000000000000000000

Sign is 1 – number is negative.
Exponent field is 01111110 = 126 (decimal).
Fraction is 0.100000000000… = 0.5 (decimal).

Value = -1.5 x 2(126-127) = -1.5 x 2-1 = -0.75.
sign
exponent
fraction

ECE 206 – Fall 2000 – G. Byrd

Floating-Point Operations
Will regular 2’s complement arithmetic work for
Floating Point numbers?
(Hint: In decimal, how do we compute 3.07 x 1012 + 9.11 x 108?)

ECE 206 – Fall 2000 – G. Byrd

Text: ASCII Characters
ASCII: Maps 128 characters to 7-bit code.
both printable and non-printable (ESC, DEL, …) characters

00 nul 10 dle 20 sp 30 0 40 @ 50 P 60 ` 70 p
01 soh 11 dc1 21 ! 31 1 41 A 51 Q 61 a 71 q
02 stx 12 dc2 22 ” 32 2 42 B 52 R 62 b 72 r
03 etx 13 dc3 23 # 33 3 43 C 53 S 63 c 73 s
04 eot 14 dc4 24 $ 34 4 44 D 54 T 64 d 74 t
05 enq 15 nak 25 % 35 5 45 E 55 U 65 e 75 u
06 ack 16 syn 26 & 36 6 46 F 56 V 66 f 76 v
07 bel 17 etb 27 ‘ 37 7 47 G 57 W 67 g 77 w
08 bs 18 can 28 ( 38 8 48 H 58 X 68 h 78 x
09 ht 19 em 29 ) 39 9 49 I 59 Y 69 i 79 y
0a nl 1a sub 2a * 3a : 4a J 5a Z 6a j 7a z
0b vt 1b esc 2b + 3b ; 4b K 5b [ 6b k 7b {
0c np 1c fs 2c , 3c < 4c L 5c \ 6c l 7c | 0d cr 1d gs 2d - 3d = 4d M 5d ] 6d m 7d } 0e so 1e rs 2e . 3e > 4e N 5e ^ 6e n 7e ~
0f si 1f us 2f / 3f ? 4f O 5f _ 6f o 7f del

ECE 206 – Fall 2000 – G. Byrd

Interesting Properties of ASCII Code
What is relationship between a decimal digit (‘0’, ‘1’, …)
and its ASCII code?

What is the difference between an upper-case letter
(‘A’, ‘B’, …) and its lower-case equivalent (‘a’, ‘b’, …)?

Given two ASCII characters, how do we tell which comes first in alphabetical order?

Are 128 characters enough?
(http://www.unicode.org/)
No new operations — integer arithmetic and logic.

ECE 206 – Fall 2000 – G. Byrd

Other Data Types
Text strings
sequence of characters, terminated with NULL (0)
typically, no hardware support

Image
array of pixels

monochrome: one bit (1/0 = black/white)
color: red, green, blue (RGB) components (e.g., 8 bits each)
other properties: transparency
hardware support:

typically none, in general-purpose processors
MMX — multiple 8-bit operations on 32-bit word
Sound
sequence of fixed-point numbers

ECE 206 – Fall 2000 – G. Byrd

LC-3 Data Types
Some data types are supported directly by the
instruction set architecture.

For LC-3, there is only one hardware-supported data type:
16-bit 2’s complement signed integer
Operations: ADD, AND, NOT

Other data types are supported by interpreting
16-bit values as logical, text, fixed-point, etc.,
in the software that we write.

ECE 206 – Fall 2000 – G. Byrd

0
exponent
,
2
fraction
.
0
)
1
(
254
exponent
1
,
2
fraction
.
1
)
1
(
126
127
exponent
=
´
´
–
=
£
£
´
´
–
=
–
–
S
S
N
N

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