Download Binary Numbers

Chapter 1: Digital Systems and Binary Numbers
• A digital system is a system that manipulates discrete
elements of information represented internally in binary
form.
• Digital computers
– general purposes
– many scientific, industrial and commercial applications
• Digital systems
–
–
–
–
telephone switching exchanges
digital camera
electronic calculators, PDA's
digital TV
Signal
• An information variable represented by physical quantity
• For digital systems, the variable takes on discrete values
– Two level, or binary values are the most prevalent values
• Binary values are represented abstractly by:
–
–
–
–
digits 0 and 1
words (symbols) False (F) and True (T)
words (symbols) Low (L) and High (H)
and words On and Off.
• Binary values are represented by values or ranges of
values of physical quantities
Binary Numbers
• Decimal number
Base or radix
aj
… a5a4a3a2a1.a1a2a3…
Decimal point
Power
 105 a5  104 a4  103 a3  102 a2  101 a1  100 a0  101 a1  102 a2  103 a3 
Example:
7,329  7 103  3 102  2 101  9 100
• General form of base-r system
an  r n  an 1  r n 1 
 a2  r 2  a1  r1  a0  a1  r 1  a2  r 2 
Coefficient: aj = 0 to r  1
 a m  r  m
Binary Numbers
Example: Base-2 number
(11010.11) 2  (26.75)10
 1  24  1  23  0  22  1  21  0  20  1  2 1  1  2 2
Example: Base-5 number
(4021.2)5
 4  53  0  52  2  51  1  50  2  51  (511.5)10
Example: Base-8 number
(127.4)8
 1  83  2  82  1  81  7  80  4  81  (87.5)10
Example: Base-16 number
(B65 F)16  11  163  6  162  5  161  15  160  (46,687)10
Binary Numbers
Example: Base-2 number
(110101)2  32  16  4  1  (53)10
Special Powers of 2
 210 (1024) is Kilo, denoted "K"
 220 (1,048,576) is Mega, denoted "M"
 230 (1,073, 741,824)is Giga, denoted "G"
Powers of two
Table 1.1
Arithmetic operation
Arithmetic operations with numbers in base r follow the same rules as decimal
numbers.
Binary Arithmetic
•
•
•
•
•
•
Single Bit Addition with Carry
Multiple Bit Addition
Single Bit Subtraction with Borrow
Multiple Bit Subtraction
Multiplication
BCD Addition
Binary Arithmetic
• Subtraction
• Addition
Augend:
Minuend:
101101
Subtrahend: 100111
Addend: +100111
The binary multiplication table is simple:
Difference:
Sum:
1010100
00=0 | 10=0 | 01=0 | 11=1
•Extending
Multiplication
multiplication to multiple digits:
Multiplicand
Multiplier
Partial Products
Product
101101
1011
 101
1011
0000 1011 - 110111
000110
Number-Base Conversions
Name
Radix
Digits
Binary
2
0,1
Octal
8
0,1,2,3,4,5,6,7
Decimal
10
0,1,2,3,4,5,6,7,8,9
Hexadecimal
16
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
 The six letters (in addition to the 10 integers) in
hexadecimal represent: 10, 11, 12, 13, 14, and 15,
respectively.
Number-Base Conversions
Example1.1
Convert decimal 41 to binary. The process is continued until the integer quotient
becomes 0.
Number-Base Conversions
 The arithmetic process can be manipulated more conveniently as follows:
Number-Base Conversions
Example 1.2
Convert decimal 153 to octal. The required base r is 8.
Example1.3
Convert (0.6875)10 to binary.
The process is continued until the fraction becomes 0 or until the number of digits has
sufficient accuracy.
Number-Base Conversions
Example1.3
 To convert a decimal fraction to a number expressed in base r, a similar
procedure is used. However, multiplication is by r instead of 2, and the
coefficients found from the integers may range in value from 0 to r  1
instead of 0 and 1.
Number-Base Conversions
Example1.4
Convert (0.513)10 to octal.
 From Examples 1.1 and 1.3:
(41.6875)10 = (101001.1011)2
 From Examples 1.2 and 1.4:
(153.513)10 = (231.406517)8
Octal and Hexadecimal Numbers
 Numbers with different bases: Table 1.2.
Octal and Hexadecimal Numbers
 Conversion from binary to octal can be done by positioning the binary number into
groups of three digits each, starting from the binary point and proceeding to the left
and to the right.
(10 110 001 101 011
2
6
1
5
3
．
111
100
000
7
4
0
110) 2 = (26153.7406)8
6
 Conversion from binary to hexadecimal is similar, except that the binary number is
divided into groups of four digits:
 Conversion from octal or hexadecimal to binary is done by reversing the preceding
procedure.
Complements
 There are two types of complements for each base-r system: the radix complement and
diminished radix complement.
the r's complement and the second as the (r  1)'s complement.
■ Diminished Radix Complement
Example:
 For binary numbers, r = 2 and r – 1 = 1, so the 1's complement of N is (2n  1) – N.
Example:
Complements
■ Radix Complement
The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0
and as 0 for N = 0. Comparing with the (r  1) 's complement, we note that the r's
complement is obtained by adding 1 to the (r  1) 's complement, since rn – N = [(rn 
1) – N] + 1.
Example: Base-10
The 10's complement of 012398 is 987602
The 10's complement of 246700 is 753300
Example: Base-2
The 2's complement of 1101100 is 0010100
The 2's complement of 0110111 is 1001001
Complements
■ Subtraction with Complements
The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:
Complements
Example 1.5
Using 10's complement, subtract 72532 – 3250.
Example 1.6
Using 10's complement, subtract 3250 – 72532
There is no end carry.
Therefore, the answer is – (10's complement of 30718) =  69282.
Complements
Example 1.7
Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a)
X – Y and (b) Y  X by using 2's complement.
There is no end carry.
Therefore, the answer is
Y – X =  (2's complement
of 1101111) =  0010001.
Complements
 Subtraction of unsigned numbers can also be done by means of the (r  1)'s
complement. Remember that the (r  1) 's complement is one less then the r's
complement.
Example 1.8
Repeat Example 1.7, but this time using 1's complement.
There is no end carry,
Therefore, the answer is
Y – X =  (1's complement
of 1101110) =  0010001.
Signed Binary Numbers
 To represent negative integers, we need a notation for negative values.
 It is customary to represent the sign with a bit placed in the leftmost position of the
number.
 The convention is to make the sign bit 0 for positive and 1 for negative.
Example:
 Table 1.3 lists all possible four-bit signed binary numbers in the three representations.
Signed Binary Numbers
Signed Binary Numbers
■ Arithmetic Addition
The addition of two numbers in the signed-magnitude system follows the rules of
ordinary arithmetic. If the signs are the same, we add the two magnitudes and give
the sum the common sign. If the signs are different, we subtract the smaller
magnitude from the larger and give the difference the sign of the larger magnitude.
 The addition of two signed binary numbers with negative numbers represented in
signed-2's-complement form is obtained from the addition of the two numbers,
including their sign bits.
 A carry out of the sign-bit position is discarded.
Example:
Binary Codes
■ BCD Code
A number with k decimal digits will
require 4k bits in BCD. Decimal 396
is represented in BCD with 12bits as
0011 1001 0110, with each group of
4 bits representing one decimal digit.
A decimal number in BCD is the
same as its equivalent binary
number only when the number is
between 0 and 9. A BCD number
greater than 10 looks different from
its equivalent binary number, even
though both contain 1's and 0's.
Moreover, the binary combinations
1010 through 1111 are not used and
have no meaning in BCD.
Signed Binary Numbers
■ Arithmetic Subtraction
 In 2’s-complement form:
1.
2.
Take the 2’s complement of the subtrahend (including the sign bit) and add it to
the minuend (including sign bit).
A carry out of sign-bit position is discarded.
(  A)  (  B )  (  A)  (  B )
(  A)  (  B )  (  A)  (  B )
Example:
( 6)  ( 13)
(11111010  11110011)
(11111010 + 00001101)
00000111 (+ 7)
Binary Codes
Example:
Consider decimal 185 and its corresponding value in BCD and binary:
■ BCD Addition
Binary Codes
Example:
Consider the addition of 184 + 576 = 760 in BCD:
■ Decimal Arithmetic
Binary Codes
■ Other Decimal Codes
Binary Codes
■ Gray Code
Binary Codes
■ ASCII Character Code
Binary Codes
■ ASCII Character Code
ASCII Character Codes
• American Standard Code for Information
Interchange (Refer to Table 1.7)
• A popular code used to represent information sent as
character-based data.
• It uses 7-bits to represent:
– 94 Graphic printing characters.
– 34 Non-printing characters
• Some non-printing characters are used for text
format (e.g. BS = Backspace, CR = carriage return)
• Other non-printing characters are used for record
marking and flow control (e.g. STX and ETX start
and end text areas).
ASCII Properties
ASCII has some interesting properties:
 Digits 0 to 9 span Hexadecimal values 3016 to 3916 .
 Upper case A - Z span 4116 to 5A16 .
 Lower case a - z span 6116 to 7A16 .
• Lower to upper case translation (and vice versa)
occurs by flipping bit 6.
 Delete (DEL) is all bits set, a carryover from when
punched paper tape was used to store messages.
 Punching all holes in a row erased a mistake!
Binary Codes
■ Error-Detecting Code
 To detect errors in data communication and processing, an eighth bit is sometimes
added to the ASCII character to indicate its parity.
 A parity bit is an extra bit included with a message to make the total number of 1's
either even or odd.
Example:
Consider the following two characters and their even and odd parity:
Binary Codes
■ Error-Detecting Code
• Redundancy (e.g. extra information), in the form of extra
bits, can be incorporated into binary code words to detect
and correct errors.
• A simple form of redundancy is parity, an extra bit
appended onto the code word to make the number of 1’s
odd or even. Parity can detect all single-bit errors and
some multiple-bit errors.
• A code word has even parity if the number of 1’s in the
code word is even.
• A code word has odd parity if the number of 1’s in the
code word is odd.
Binary Storage and Registers
■ Registers
 A binary cell is a device that possesses two stable states and is capable of storing
one of the two states.
 A register is a group of binary cells. A register with n cells can store any discrete
quantity of information that contains n bits.
n cells
2n possible states
• A binary cell
– two stable state
– store one bit of information
– examples: flip-flop circuits, ferrite cores, capacitor
• A register
– a group of binary cells
– AX in x86 CPU
• Register Transfer
– a transfer of the information stored in one register to another
– one of the major operations in digital system
– an example
Transfer of information
• The other major component of a digital system
– circuit elements to manipulate individual bits of information
Binary Logic
■ Definition of Binary Logic
 Binary logic consists of binary variables and a set of logical operations. The variables
are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each
variable having two and only two distinct possible values: 1 and 0, There are three
basic logical operations: AND, OR, and NOT.
Binary Logic
■ The truth tables for AND, OR, and NOT are given in Table 1.8.
Binary Logic
■ Logic gates
 Example of binary signals
Binary Logic
■ Logic gates
 Graphic Symbols and Input-Output Signals for Logic gates:
Fig. 1.4 Symbols for digital logic circuits
Fig. 1.5
Input-Output signals
for gates
Binary Logic
■ Logic gates
 Graphic Symbols and Input-Output Signals for Logic gates:
Fig. 1.6 Gates with multiple inputs
Number-Base Conversions
Complements
Complements
Signal Example – Physical Quantity: Voltage
OUTPUT
INPUT
5.0
HIGH
4.0
3.0
2.0
LOW
1.0
0.0
Volts
HIGH
Threshold
Region
LOW
Signal Examples Over Time
Time
Analog
Digital
Continuous in
value & time
Asynchronous
Discrete in
value &
continuous in
time
Synchronous
Discrete in
value & time
A Digital Computer Example
Memory
CPU
Inputs: Keyboard,
mouse, modem,
microphone
Control
unit
Datapath
Input/Output
Synchronous or
Asynchronous?
Outputs: CRT,
LCD, modem,
speakers
Binary Codes for Decimal Digits
 There are over 8,000 ways that you can chose 10 elements from the
16 binary numbers of 4 bits. A few are useful:
Decimal
8,4,2,1
Excess3
8,4, -2,-1
Gray
0
1
2
3
4
5
6
7
8
9
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
0000
0111
0110
0101
0100
1011
1010
1001
1000
1111
0000
0100
0101
0111
0110
0010
0011
0001
1001
1000
UNICODE
• UNICODE extends ASCII to 65,536
universal characters codes
– For encoding characters in world languages
– Available in many modern applications
– 2 byte (16-bit) code words
– See Reading Supplement – Unicode on the
Companion Website
http://www.prenhall.com/mano
Negative Numbers
• Complements
– 1's complements
(2
n
 1)  N
– 2's complements
2 N
– Subtraction = addition with the 2's complement
– Signed binary numbers
» signed-magnitude, signed 1's complement, and signed 2's
complement.
M-N
• M + the 2’s complement of N
– M + (2n - N) = M - N + 2n
• If M ≧N
– Produce an end carry, 2n, which is discarded
• If M < N
– We get 2n - (N - M), which is the 2’s complement of (N-M)
Binary Storage and Registers
• A binary cell
– two stable state
– store one bit of information
– examples: flip-flop circuits, ferrite cores, capacitor
• A register
– a group of binary cells
– AX in x86 CPU
• Register Transfer
– a transfer of the information stored in one register to another
– one of the major operations in digital system
– an example
Special Powers of 2
 210 (1024) is Kilo, denoted "K"
 220 (1,048,576) is Mega, denoted "M"
 230 (1,073, 741,824)is Giga, denoted "G"
Converting Binary to Decimal
• To convert to decimal, use decimal arithmetic to form S (digit ×
respective power of 2).
• Example:Convert 110102 to N10:
Non-numeric Binary Codes
• Given n binary digits (called bits), a binary code is a
mapping from a set of represented elements to a
subset of the 2n binary numbers.
• Example: A
Color
Binary Number
binary code
Red
000
for the seven
Orange
001
colors of the
Yellow
010
rainbow
Green
011
Blue
101
• Code 100 is
Indigo
110
not used
Violet
111
Commonly Occurring Bases
Name
Radix
Digits
Binary
2
0,1
Octal
8
0,1,2,3,4,5,6,7
Decimal
10
0,1,2,3,4,5,6,7,8,9
Hexadecimal
16
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
 The six letters (in addition to the 10
integers) in hexadecimal represent:
Binary Numbers and Binary Coding
• Information Types
– Numeric
» Must represent range of data needed
» Represent data such that simple, straightforward computation for
common arithmetic operations
» Tight relation to binary numbers
– Non-numeric
» Greater flexibility since arithmetic operations not applied.
» Not tied to binary numbers
Number of Elements Represented
• Given n digits in radix r, there are rn distinct
elements that can be represented.
• But, you can represent m elements, m < rn
• Examples:
– You can represent 4 elements in radix r = 2 with n = 2
digits: (00, 01, 10, 11).
– You can represent 4 elements in radix r = 2 with n = 4
digits: (0001, 0010, 0100, 1000).
– This second code is called a "one hot" code.
Binary Coded Decimal (BCD)
• The BCD code is the 8,4,2,1 code.
• This code is the simplest, most intuitive binary code
for decimal digits and uses the same powers of 2 as
a binary number, but only encodes the first ten
values from 0 to 9.
• Example: 1001 (9) = 1000 (8) + 0001 (1)
• How many “invalid” code words are there?
• What are the “invalid” code words?
Excess 3 Code and 8, 4, –2, –1 Code
Decimal
Excess 3
8, 4, –2, –1
0
0011
0000
1
0100
0111
2
0101
0110
3
0110
0101
4
0111
0100
5
1000
1011
6
1001
1010
7
1010
1001
8
1011
1000
9
1100
1111
• What interesting property is common to
these two codes?
Gray Code
Decimal
8,4,2,1
Gray
0
1
2
3
4
5
6
7
8
9
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
0000
0100
0101
0111
0110
0010
0011
0001
1001
1000
• What special property does the Gray code have
in relation to adjacent decimal digits?
Gray Code (Continued)
• Does this special Gray code property have any
value?
• An Example: Optical Shaft Encoder
111
000
100
000
B0
B1
110
001
B2
010
101
100
011
(a) Binary Code for Positions 0 through 7
101
111
001
G0
G1
G2
110
010
(b) Gray Code for Positions 0 through 7
011
Warning: Conversion or Coding?
• Do NOT mix up conversion of a decimal number to a binary
number with coding a decimal number with a BINARY CODE.
• 1310 = 11012 (This is conversion)
• 13  0001|0011 (This is coding)
Single Bit Binary Addition with Carry
Given two binary digits (X,Y), a carry in (Z) we get the
following sum (S) and carry (C):
Carry in (Z) of 0:
Carry in (Z) of 1:
Z
X
+Y
0
0
+0
0
0
+1
0
1
+0
0
1
+1
CS
00
01
01
10
Z
X
+Y
1
0
+0
1
0
+1
1
1
+0
1
1
+1
CS
01
10
10
11
Multiple Bit Binary Addition
• Extending this to two multiple bit examples:
Carries
Augend
Addend
Sum
0
0
01100 10110
+10001 +10111
• Note: The 0 is the default Carry-In to the least significant bit.
Binary Multiplication
The binary multiplication table is simple:
00=0 | 10=0 | 01=0 | 11=1
Extending multiplication to multiple digits:
Multiplicand
Multiplier
Partial Products
Product
1011
 101
1011
0000 1011 - 110111
BCD Arithmetic
 Given a BCD code, we use binary arithmetic to add the digits:
8
1000
Eight
+5
+0101
Plus 5
13
1101
is 13 (> 9)
 Note that the result is MORE THAN 9, so must be
represented by two digits!
 To correct the digit, subtract 10 by adding 6 modulo 16.
8
1000 Eight
+5
+0101 Plus 5
13
1101 is 13 (> 9)
+0110 so add 6
carry = 1 0011
leaving 3 + cy
0001 | 0011
Final answer (two digits)
 If the digit sum is > 9, add one to the next significant digit
BCD Addition Example
• Add 2905BCD to 1897BCD showing carries
and digit corrections.
0
0001 1000 1001 0111
+ 0010 1001 0000 0101
Error-Detection Codes
• Redundancy (e.g. extra information), in the form of extra bits,
can be incorporated into binary code words to detect and
correct errors.
• A simple form of redundancy is parity, an extra bit appended
onto the code word to make the number of 1’s odd or even.
Parity can detect all single-bit errors and some multiple-bit
errors.
• A code word has even parity if the number of 1’s in the code
word is even.
• A code word has odd parity if the number of 1’s in the code
word is odd.
4-Bit Parity Code Example
• Fill in the even and odd parity bits:
Even Parity
Odd Parity
Message - Parity Message - Parity
000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 -
• The codeword "1111" has even parity and the codeword
"1110" has odd parity. Both can be used to represent 3bit data.