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Transcript of Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B...
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Chapter 1:Chapter 1:
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Digital SystemsDigital Systems
Discrete Data
● Examples:
♦ 26 letters of the alphabet (A, B … etc)
♦ 10 decimal digits (0, 1, 2 … etc)
● Combine together
♦ Words are made of letters (University … etc)
♦ Numbers are made of digits (4241 … etc)
Binary System
● Only ‘0’ and ‘1’ digits
● Can be easily implemented in electronic circuits
Analog Discrete
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Decimal Number SystemDecimal Number System
Base (also called radix) = 10
● 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
Digit Position
● Integer & fraction
Digit Weight
● Weight = (Base) Position
Magnitude
● Sum of “Digit x Weight”
Formal Notation
1 0 -12 -2
5 1 2 7 4
10 1 0.1100 0.01
500 10 2 0.7 0.04
d2*B2+d1*B
1+d0*B0+d-1*B
-1+d-2*B-2
(512.74)10
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Octal Number SystemOctal Number System
Base = 8
● 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }
Weights
● Weight = (Base) Position
Magnitude
● Sum of “Digit x Weight”
Formal Notation
1 0 -12 -2
8 1 1/864 1/64
5 1 2 7 4
5 *82+1 *81+2 *80+7 *8-1+4 *8-2
=(330.9375)10
(512.74)8
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Binary Number SystemBinary Number System
Base = 2
● 2 digits { 0, 1 }, called binary digits or “bits”
Weights
● Weight = (Base) Position
Magnitude
● Sum of “Bit x Weight”
Formal Notation
Groups of bits 4 bits = Nibble
8 bits = Byte
1 0 -12 -2
2 1 1/24 1/4
1 0 1 0 1
1 *22+0 *21+1 *20+0 *2-1+1 *2-2
=(5.25)10
(101.01)2
1 0 1 1
1 1 0 0 0 1 0 1
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Hexadecimal Number SystemHexadecimal Number System
Base = 16
● 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }
Weights
● Weight = (Base) Position
Magnitude
● Sum of “Digit x Weight”
Formal Notation
1 0 -12 -2
16 1 1/16256 1/256
1 E 5 7 A
1 *162+14 *161+5 *160+7 *16-1+10 *16-2
=(485.4765625)10
(1E5.7A)16
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The Power of 2The Power of 2
n 2n
0 20=1
1 21=2
2 22=4
3 23=8
4 24=16
5 25=32
6 26=64
7 27=128
n 2n
8 28=256
9 29=512
10 210=1024
11 211=2048
12 212=4096
20 220=1M
30 230=1G
40 240=1T
Mega
Giga
Tera
Kilo
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AdditionAddition
Decimal Addition
5 5
55+
011
= Ten ≥ Base
Subtract a Base
11 Carry
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Binary AdditionBinary Addition
Column Addition
1 0 1111
1111 0+
0000 1 11
≥ (2)10
111111
= 61
= 23
= 84
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Binary SubtractionBinary Subtraction
Borrow a “Base” when needed
0 0 1110
1111 0−
0101 1 10
= (10)2
2
2
2 2
1
000
1
= 77
= 23
= 54
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Binary MultiplicationBinary Multiplication
Bit by bit
01 1 1 1
01 1 0
00 0 0 0
01 1 1 1
01 1 1 1
0 0 000
0110111 0
x
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Number Base ConversionsNumber Base Conversions
Decimal(Base 10)
Octal(Base 8)
Binary(Base 2)
Hexadecimal(Base 16)
Evaluate Magnitude
Evaluate Magnitude
Evaluate Magnitude
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Decimal (Decimal (IntegerInteger) to Binary Conversion) to Binary Conversion
Divide the number by the ‘Base’ (=2)
Take the remainder (either 0 or 1) as a coefficient
Take the quotient and repeat the division
Example: (13)10
Quotient Remainder Coefficient
Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2
MSB LSBMSB LSB
13/ 2 = 6 1 a0 = 1 6 / 2 = 3 0 a1 = 0 3 / 2 = 1 1 a2 = 1 1 / 2 = 0 1 a3 = 1
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Decimal (Decimal (FractionFraction) to Binary Conversion) to Binary Conversion
Multiply the number by the ‘Base’ (=2)
Take the integer (either 0 or 1) as a coefficient
Take the resultant fraction and repeat the division
Example: (0.625)10
Integer Fraction Coefficient
Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2
MSB LSBMSB LSB
0.625 * 2 = 1 . 250.25 * 2 = 0 . 5 a-2 = 00.5 * 2 = 1 . 0 a-3 = 1
a-1 = 1
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Decimal to Octal ConversionDecimal to Octal Conversion
Example: (175)10
Quotient Remainder Coefficient
Answer: (175)10 = (a2 a1 a0)8 = (257)8
175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2
Example: (0.3125)10
Integer Fraction Coefficient
Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8
0.3125 * 8 = 2 . 50.5 * 8 = 4 . 0 a-2 = 4
a-1 = 2
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Binary Binary −− Octal Conversion Octal Conversion
8 = 23
Each group of 3 bits represents an octal digit
Octal Binary
0 0 0 0
1 0 0 1
2 0 1 0
3 0 1 1
4 1 0 0
5 1 0 1
6 1 1 0
7 1 1 1
Example:
( 1 0 1 1 0 . 0 1 )2
( 2 6 . 2 )8
Assume Zeros
Works both ways (Binary to Octal & Octal to Binary)
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Binary Binary −− Hexadecimal Conversion Hexadecimal Conversion
16 = 24
Each group of 4 bits represents a hexadecimal digit
Hex Binary0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1A 1 0 1 0B 1 0 1 1C 1 1 0 0D 1 1 0 1E 1 1 1 0F 1 1 1 1
Example:
( 1 0 1 1 0 . 0 1 )2
( 1 6 . 4 )16
Assume Zeros
Works both ways (Binary to Hex & Hex to Binary)
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Octal Octal −− Hexadecimal Conversion Hexadecimal Conversion
Convert to Binary as an intermediate step
Example:
( 0 1 0 1 1 0 . 0 1 0 )2
( 1 6 . 4 )16
Assume Zeros
Works both ways (Octal to Hex & Hex to Octal)
( 2 6 . 2 )8
Assume Zeros
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Decimal, Binary, Octal and HexadecimalDecimal, Binary, Octal and Hexadecimal
Decimal Binary Octal Hex00 0000 00 001 0001 01 102 0010 02 203 0011 03 304 0100 04 405 0101 05 506 0110 06 607 0111 07 708 1000 10 809 1001 11 910 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F
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ComplementsComplements
1’s Complement (Diminished Radix Complement)
● All ‘0’s become ‘1’s
● All ‘1’s become ‘0’s
Example (10110000)2
(01001111)2
If you add a number and its 1’s complement …
1 0 1 1 0 0 0 0
+ 0 1 0 0 1 1 1 1
1 1 1 1 1 1 1 1
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ComplementsComplements
2’s Complement (Radix Complement)
● Take 1’s complement then add 1
● Toggle all bits to the left of the first ‘1’ from the right
Example:
Number:
1’s Comp.:
0 1 0 1 0 0 0 0
1 0 1 1 0 0 0 0
0 1 0 0 1 1 1 1
+ 1
OR
1 0 1 1 0 0 0 0
00001010
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Negative NumbersNegative Numbers
Computers Represent Information in ‘0’s and ‘1’s
● ‘+’ and ‘−’ signs have to be represented in ‘0’s and ‘1’s
3 Systems
● Signed Magnitude
● 1’s Complement
● 2’s Complement
All three use the left-most bit to represent the sign:
♦ ‘0’ positive
♦ ‘1’ negative
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Signed Magnitude RepresentationSigned Magnitude Representation
Magnitude is magnitude, does not change with sign
(+3)10 ( 0 0 1 1 )2
(−3)10 ( 1 0 1 1 )2
Can’t include the sign bit in ‘Addition’0 0 1 1 (+3)10
+ 1 0 1 1 (−3)10
1 1 1 0 (−6)10
Sign Magnitude
S Magnitude (Binary)
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1’s Complement Representation1’s Complement Representation
Positive numbers are represented in “Binary”
Negative numbers are represented in “1’s Comp.”
(+3)10 (0 011)2
(−3)10 (1 100)2
There are 2 representations for ‘0’
(+0)10 (0 000)2
(−0)10 (1 111)2
0 Magnitude (Binary)
1 Code (1’s Comp.)
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1’s Complement Range1’s Complement Range
4-Bit Representation
24 = 16 Combinations
− 7 ≤ Number ≤ + 7
−23+1 ≤ Number ≤ +23 − 1
n-Bit Representation
−2n−1+1 ≤ Number ≤ +2n−1 − 1
Decimal 1’s Comp.+ 7 0 1 1 1+ 6 0 1 1 0+ 5 0 1 0 1+ 4 0 1 0 0+ 3 0 0 1 1+ 2 0 0 1 0+ 1 0 0 0 1+ 0 0 0 0 0− 0 1 1 1 1− 1 1 1 1 0− 2 1 1 0 1− 3 1 1 0 0− 4 1 0 1 1− 5 1 0 1 0− 6 1 0 0 1− 7 1 0 0 0
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2’s Complement Representation2’s Complement Representation
Positive numbers are represented in “Binary”
Negative numbers are represented in “2’s Comp.”
(+3)10 (0 011)2
(−3)10 (1 101)2
There is 1 representation for ‘0’
(+0)10 (0 000)2
(−0)10 (0 000)2
0 Magnitude (Binary)
1 Code (2’s Comp.)
1’s Comp. 1 1 1 1
+ 1
1 0 0 0 0
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2’s Complement Range2’s Complement Range
4-Bit Representation
24 = 16 Combinations
− 8 ≤ Number ≤ + 7
−23 ≤ Number ≤ + 23 − 1
n-Bit Representation
−2n−1 ≤ Number ≤ + 2n−1 − 1
Decimal 2’s Comp.+ 7 0 1 1 1+ 6 0 1 1 0+ 5 0 1 0 1+ 4 0 1 0 0+ 3 0 0 1 1+ 2 0 0 1 0+ 1 0 0 0 1+ 0 0 0 0 0− 1 1 1 1 1− 2 1 1 1 0− 3 1 1 0 1− 4 1 1 0 0− 5 1 0 1 1− 6 1 0 1 0− 7 1 0 0 1− 8 1 0 0 0
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Number RepresentationsNumber Representations
4-Bit Example
Unsigned Binary
Signed Magnitude
1’s Comp. 2’s Comp.
Range 0 ≤ N ≤ 15 -7 ≤ N ≤ +7 -7 ≤ N ≤ +7 -8 ≤ N ≤ +7
PositiveBinary Binary Binary Binary
Negative XBinary 1’s Comp. 2’s Comp.
0 0 0
1 1 1
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Binary Subtraction Using 1’s Comp. AdditionBinary Subtraction Using 1’s Comp. Addition
Change “Subtraction” to “Addition”
If “Carry” = 1then add it to theLSB, and the resultis positive(in Binary)
If “Carry” = 0then the resultis negative(in 1’s Comp.)
0 1 0 1
+ 1 1 1 0
(5)10 – (1)10
(+5)10 + (-1)10
0 0 1 1+
0 1 0 0
0 1 0 1
+ 1 0 0 1
(5)10 – (6)10
(+5)10 + (-6)10
0 1 1 1 0
1 1 1 0
+ 4 − 1
1
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Binary Subtraction Using 2’s Comp. AdditionBinary Subtraction Using 2’s Comp. Addition
Change “Subtraction” to “Addition”
If “Carry” = 1ignore it, and the result is positive(in Binary)
If “Carry” = 0then the resultis negative(in 2’s Comp.)
0 1 0 1
+ 1 1 1 1
(5)10 – (1)10
(+5)10 + (-1)10
1 0 1 0 0
0 1 0 1
+ 1 0 1 0
(5)10 – (6)10
(+5)10 + (-6)10
0 1 1 1 1
+ 4 − 1
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Binary CodesBinary Codes
Group of n bits
● Up to 2n combinations
● Each combination represents an element of information
Binary Coded Decimal (BCD)
● Each Decimal Digit is represented by 4 bits
● (0 – 9) Valid combinations
● (10 – 15) Invalid combinations
Decimal BCD0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1
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BCD AdditionBCD Addition
One decimal digit + one decimal digit
● If the result is 1 decimal digit ( ≤ 9 ), then it is a simple binary addition
Example:
● If the result is two decimal digits ( ≥ 10 ), then binary addition gives invalid combinations
Example:
5
+ 3
8
0 1 0 1
+ 0 0 1 1
1 0 0 0
5
+ 5
1 0
0 1 0 1
+ 0 1 0 1
1 0 1 00 0 0 1 0 0 0 0
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BCD AdditionBCD Addition
If the binary resultis greater than 9,correct the result byadding 6
5
+ 5
1 0
0 1 0 1
+ 0 1 0 1
1 0 1 0
+ 0 1 1 0
0 0 0 1 0 0 0 0
Two Decimal Digits
Multiple Decimal Digits
3 5 1
0 0 1 1 0 1 0 1 0 0 0 1
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Gray CodeGray Code
One bit changes fromone code to the nextcode
Different than Binary
Decimal Gray00 000001 000102 001103 001004 011005 011106 010107 010008 110009 110110 111111 111012 101013 101114 100115 1000
Binary0000000100100011010001010110011110001001101010111100110111101111
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ASCII CodeASCII Code
American Standard Code for Information Interchange
Info 7-bit CodeA 1000001B 1000010...
.
.
.Z 1011010
a 1100001b 1100010...
.
.
.z 1111010
@ 1000000? 0111111+ 0101011
![Page 37: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/37.jpg)
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Error Detecting CodesError Detecting Codes
Parity
One bit added to a group of bits to make the total number of ‘1’s (including the parity bit) even or odd
● Even
● Odd
Good for checking single-bit errors
1 0 1 1 1 1 0 0 0 0 0 10
0 0 1 1 1 1 0 0 0 0 0 11
4-bit Example 7-bit Example
![Page 38: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/38.jpg)
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Binary LogicBinary Logic
Operators
● NOT
If ‘x’ = 0 then NOT ‘x’ = 1
If ‘x’ = 1 then NOT ‘x’ = 0
● AND
If ‘x’ = 1 AND ‘y’ = 1then ‘z’ = 1
Otherwise ‘z’ = 0
● OR
If ‘x’ = 1 OR ‘y’ = 1 then ‘z’ = 1
Otherwise ‘z’ = 0
![Page 39: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/39.jpg)
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Binary LogicBinary Logic
Truth Tables, Boolean Expressions, and Logic Gates
x y z
0 0 0
0 1 0
1 0 0
1 1 1
x y z
0 0 0
0 1 1
1 0 1
1 1 1
x z
0 1
1 0
AND OR NOT
xy z x
y z
z = x • y = x y z = x + y z = x = x’
x z
![Page 40: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/40.jpg)
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Logic SignalsLogic Signals
Binary ‘0’ is representedby a “low” voltage(range of voltages)
Binary ‘1’ is representedby a “high” voltage(range of voltages)
The “voltage ranges” guardagainst noise
of binary signalsExample
![Page 41: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/41.jpg)
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Switching CircuitsSwitching Circuits
AND OR
![Page 42: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/42.jpg)
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HomeworkHomework
Mano
● Chapter 1
♦ 1-2
♦ 1-7
♦ 1-9
♦ 1-10
♦ 1-11
♦ 1-16
♦ 1-18
♦ 1-20
♦ 1-24(a)
♦ 1-29
Write your family name in ASCII with odd parity
Decode the following ASCII string (with MSB = parity):
11000011 01101111 11101101 11110000 11000000 01010000 01010011 01010101 11010100
Is the parity even or odd?
![Page 43: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/43.jpg)
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HomeworkHomework
Mano1-2 What is the exact number of bytes in a system that
contains (a) 32K byte, (b) 64M byte, and (c) 6.4G byte?
1-7 Express the following numbers in decimal: (10110.0101)2, (16.5)16, and (26.24)8.
1-9 Convert the hexadecimal number 68BE to binary and then from binary convert it to octal.
1-10 Convert the decimal number 345 to binary in two ways: (a) convert directly to binary, (b) convert first to hexadecimal, then from hexadecimal to binary. Which method is faster?
![Page 44: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/44.jpg)
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HomeworkHomework
1-11 Do the following conversion problems:
(a) Convert decimal 34.4375 to binary.
(b) Calculate the binary equivalent of 1/3 out to 8 places. Then convert from binary to decimal. How close is the result to 1/3?
(c) Convert the binary result in (b) into hexadecimal. Then convert the result to decimal. Is the answer the same?
1-16 Obtain the 1’s and 2’s complements of the following binary numbers:
(a) 11101010 (b) 01111110 (c) 00000001 (d) 10000000 (e) 00000000
![Page 45: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/45.jpg)
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HomeworkHomework
1-18 Perform subtraction on the following unsigned binary numbers using the 2’s-complement of the subtrahend. Where the result should be negative, 2’s complement it and affix a minus sign.
(a) 11011 – 11001 (b) 110100 – 10101 (c) 1011 – 110000 (d) 101010 – 101011
1-20 Convert decimal +61 and +27 to binary using the signed-2’s complement representation and enough digits to accommodate the numbers. Then perform the binary equivalent of (+27) + (– 61), (–27) + (+61) and(–27) + (–61). Convert the answers back to decimal and verify that they are correct.
![Page 46: Chapter 1: 2 / 45 Digital Systems Discrete Data ●Examples: ♦ 26 letters of the alphabet (A, B … etc) ♦ 10 decimal digits (0, 1, 2 … etc) ●Combine.](https://reader036.fdocuments.us/reader036/viewer/2022062312/551bba9d550346b9588b4662/html5/thumbnails/46.jpg)
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HomeworkHomework
1-24 Represent decimal number 6027 in (a) BCD
1-29 The following is a string of ASCII characters whose bit patterns have been converted into hexadecimal for compactness: 4A EF 68 6E 20 C4 EF E5. Of the 8 bits in each pair digit, the leftmost is a parity bit. The remaining bits are the ASCII code.
(a) Convert to bit form and decode the ASCII
(b) Determine the parity used: odd or even.