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Unit 3
Binary Representation
ANALOG VS. DIGITAL
Analog vs. Digital
The analog world is based oncontinuous events.Observations can take on(____) any value.
The digital world is based ondiscrete events. Observationscan only take on a _______number of discrete values
Analog vs. Digital
Q. Which is better?
A. Depends on what you are trying to do.
Some tasks are better handled with analogdata, others with digital data.
Analog means continuous/real valued signals withan infinite number of possible values
Digital signals are discrete [i.e. 1 of n values]
Analog vs. Digital
How much money is in my checking account?^ –
Analog: Oh, some, but not too much.
Digital: $243.
Analog vs. Digital
How much do you love me?
Analog: I love you with all my heart!!!!
Digital: 3.2 x 10
3
MegaHearts
The Real (Analog) World
The real world is inherently analog.
To interface with it, our digital systems needto:^ –
Convert analog signals to digital values (numbers)at the input.
Convert digital values to analog signals at theoutput.
Analog signals can come in many forms^ –
Voltage, current, light, color, magnetic fields,pressure, temperature, acceleration, orientation
Digital is About Numbers
In a digital world, numbers are used to represent allthe possible discrete events (i.e. 1-of-n possibilities)
Numerical values (5.7, 1923.8, …)
Computer instructions (ADD, SUB, BLE, …)
Characters ('a', 'b', 'c', …)
Conditions (on, off, ready, paper jam, …)
Numbers allow for easy manipulation
Add, multiply, compare, store, …
Results are repeatable
Each time we add the same two number we getthe same result
4 Skills
We will teach you 4 skills that you shouldknow and be able to apply with confidence^ –
Convert a number in any base (base r) to decimal(base 10)
Convert a decimal number (base 10) to binary
Use the shortcut for conversion between binary(base 2) and hexadecimal (base 16)
Understand the finite number of combinationsthat can be made with n bits (binary digits) and itsimplication for codes including ASCII and Unicode
Number Systems
Number systems consist of
1. A base (radix) r2. ___ coefficients ___________
Human System:
Decimal (Base 10):
Computer System:
Binary (Base 2): 0,
Human systems for working with computer systems(shorthand for human to read/write binary)
Octal (Base 8): 0,1,2,3,4,5,6,
Hexadecimal (Base 16): 0-9,A,B,C,D,E,F
(A thru F = 10 thru 15)
Skill 1: Converting to Decimal
-^
Number systems use implied place values (weights) that arepowers of the base( ____
____ ___. ____ )
10
( ___ ___ ___ ___. ____ ____ )
2
10
2 =100 10
1 =10 10
0 =1 10
=0.
(^32) =8 2
2 =
(^12) =2 2
0 =1 2
=0.5 2
=0.
-^
Converting from
another base
to
decimal
requires summing the
product of each digit with its implied place value
(^
.^
)^2
(_________)
10
__
__
__
__
Conversion Examples 1
__
__
__
(1A5)
___
+ ___*
___
___
Skill 2: Converting from Decimal
-^
Number systems use implied place values (weights) that arepowers of the base( ____
____ ___. ____ )
10
( ___ ___ ___ ___. ____ ____ )
2
10
2 =100 10
1 =10 10
0 =1 10
=0.
(^32) =8 2
2 =
(^12) =2 2
0 =1 2
=0.5 2
=0.
-^
Converting from
decimal
to
another base
requires finding the
digits that, when multiplied times each place value, will sum to thedesired value (i.e. making change).
10
( ___ ___ ___ ___ ___. ___ ___ )
2
16
8
4
2
1
Conversion Examples 2
)^2
)^16
Shortcuts for Converting Binary (r=2), Hexadecimal (r=16) and Octal (r=8) SHORTHAND FOR BINARY
Binary, Octal, and Hexadecimal
Octal (base 8 = 2
3
1 Octal digit (? )
8
can
represent: _______
3 bits of binary (?? ?)
2
can represent:000-111 = _______
Conclusion…1 Octal digit = ________
Hex (base 16=
4
1 Hex digit (? )
16
can
represent: _________
4 bits of binary(??? ?)
2
can represent:
Conclusion…1 Hex digit = _______
Binary Representation Systems
-^
Integer Systems^ –
Unsigned
-^
Unsigned (Normal) binary
Signed
-^
Signed Magnitude
-^
2’s complement
-^
1’s complement*
-^
Excess-N*
-^
Floating Point*^ –
For very large and small(fractional) numbers
-^
Codes
Text
-^
ASCII / Unicode
Decimal Codes
•^
BCD (Binary Coded Decimal)/ (8421 Code)
- = Not covered in this class
Skill 4: Unique Combinations
Given
n
digits of base
r
, how many unique numbers can be
formed?
__
What is the range? [
___________
]
Use the examples below to generalize the relationship
Main Point: Given n digits of base r,
___
unique numbers
can be made with the range [_________]
2-digit, decimal numbers (r=10, n=2)3-digit, decimal numbers (r=10, n=3)4-bit, binary numbers (r=2, n=4)6-bit, binary numbers
(r=2, n=6)
0-
0-
100 combinations:
00-
0-
0-
0-
0-
1000 combinations:
000- 16 combinations:
0000- 64 combinations: 000000-
Approximating Large Powers of 2
-^
Often need to find decimalapproximation of a large powers of 2like 2
16
32
, etc.
-^
Use following approximations:^ –
10
20
30
40
-^
For other powers of 2, decomposeinto product of 2
10
or 2
20
or 2
30
and a
power of 2 that is less than 2
10
16-bit half word: ___ numbers
32-bit word: ____ numbers
64-bit dword: 16 million trillion numbers
(^162)
= (^242)
= (^282)
= (^322)
=
Binary Codes
Using binary we can represent any kind ofinformation by coming up with a code
Using
n
bits we can represent 2
n
distinct items
Colors of the rainbow:•Red = 000•Orange = 001•Yellow = 010•Green = 100•Blue = 101•Purple = 111
Letters:•‘A’ = 00000•‘B’ = 00001•‘C’ = 00010
...
•‘Z’ = 11001
ASCII Code
Used for representing text characters
Originally 7-bits but usually stored as ___________________ in modern computer systems
Example:
"Hello\n"
Each character is converted to ASCII equivalent
•^
‘H’ = 0x48, ‘e’ = 0x65, …
-^
\n = newline character is represented by either one or two ASCIIcharacter
LF (0x0A) = line feed (moves cursor down a line) - CR (0x0D) = carriage return character (moves cursor to start of currentline) - Newline for Unix / Mac = LF only - Newline for Windows = CR + LF
ASCII Table
LSD/MSD
0
1
2
3
4
5
6
7
0
NULL
DLW
SPACE
0
@
P^
`^
p
1
SOH
DC
!^
1
A^
Q^
a^
q
2
STX
DC
“^
2
B^
R^
b^
r
3
ETX
DC
#^
3
C^
S^
c^
s
4
EOT
DC
$^
4
D^
T^
d^
t
5
ENQ
NAK
%
5
E^
U^
e^
u
6
ACK
SYN
&
6
F^
V^
f^
v
7
BEL
ETB
‘^
7
G^
W
g^
w
8
BS
CAN
(^
8
H^
X^
h^
x
9
TAB
EM
)^
9
I^
Y^
i^
y
A^
LF
SUB
*^
:^
J^
Z^
j^
z
B^
VT
ESC
+^
;^
K^
[^
k^
{
C^
FF
FS
,^
<^
L^
^
l^
|
D^
CR
GS
-^
=^
M
]^
m
}
E^
SO
RS
.^
^
N^
^^
n^
~
F^
SI
US
/^
?^
O^
_^
o^
DEL
'M' = 0x4D = 0100 1101
UniCode
•^
ASCII can represent only the Englishalphabet, decimal digits, andpunctuation^ –
7-bit code => 2
7 = 128 characters
It would be nice to have one codethat represented morealphabets/characters for commonlanguages used around the world
•^
Unicode^ –
Up to 32-bit Code => up to 4 billioncombinations
137,000 character defined for manylanguages
Used by Java as standard charactercode
Unicode hex value
(i.e. FB52 => 1111101101010010)
Review: Chars and Ints
-^
The C/C++ language supports types
char
and
int
. Let's see
what you know about these:
True/False
Question
____
In C/C++: '1' and 1 are the same.- '1' is replaced with its equiv. _______________________- Thus, storing '1' or ______________ to a char variable are allequivalent (the hardware does not ___________ how the bits willbe interpretated…that is up to the code)
____
chars and ints have different ranges of the values they canrepresent.-^
Range of char: _________________
-^
______________________________
____
chars are only for ASCII characters(e.g.
char c =
and not
char c =
Using positional weights/place values BASE R TO BASE 10
Skill 1: Converting Base r to Decimal
•^
10
10
10
2 =
10
1 =
10
0 =
10
=0.
•^
2
10
2 3 =
2 2 =
2 1 =
(^02)
-1 2
=0.
•^
3B.
16
B
10
16
1 =
16
0 =
16
=0.
Main Point
: To convert any base to decimal (base 10), apply the
implicit place values (weights) which are just the powers of the
base and sum each digit times its place value.
General Conversion From Base r to Decimal •
A number in base r has place values/weightsthat are the powers of the base
Denote the coefficients as: a
i
Left-most digit =Most Significant
Digit (MSD)
Right-most digit =Least Significant
Digit (LSD)
N
r^
(ai
*ri
i) => D
(a
a 3
a 2
a 1
.a 0
-
a
-
)r
a
*r 3
3
+ a
*r 2
2
+ a
*r 1
1
+ a
*r 0
0
+ a
-
*r
-
a
-
*r
-
Number in base r
Decimal Equivalent
(^1
)r
3
2
1
0
-
-
Generalized approach: Example:
Examples
(1A5)
(AD2)
Binary Examples
. 1 2 4 8
16 32
128
1
Powers of 2
512
256
128
64
32
16
8
4
2
1
1024
It helps to memorize the first 11 powers of 2^2
0
4
5
6
7
8
9
10
"Making change" BASE 10 TO BASE 2 OR BASE 16
Skill: Decimal to Base r
•^
To convert a decimal number,
x,
to binary:
Only coefficients of 1 or 0. So simply find place valuesthat add up to the desired values, starting with largerplace values and proceeding to smaller values and placea 1 in those place values and 0 in all others
Similar to how one would
make change
16
8
4
2
1
25
10
=
1
1
1
32
For 25
10
the place value 32 is too large to include so we include
16. Including 16 means we have to make 9 left over. Include 8
and 1.
0
0
0
