Digital Electronics
Principles & Applications
Seventh Edition
Roger L. Tokheim
Chapter 2
Numbers We Use in
Digital Electronics
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INTRODUCTION
• Counting in Decimal and Binary
• Place Value
• Binary to Decimal Conversion
• Decimal to Binary Conversion
• Electronic Translators
• Hexadecimal Numbers
• Octal Numbers
• Bits, Bytes and Words
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
Counting in
Decimal and Binary
• Number System Code using symbols that refer to
a number of items.
• Decimal Number System Uses ten symbols (base 10 system)
• Binary System Uses two symbols (base 2 system)
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QUIZ
1. The __________ (binary, octal)
number system uses only the symbols 0
and 1 in counting.
binary
2. The __________ (decimal,
hexadecimal) number system uses the
digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
decimal
3. The __________ (binary, decimal)
number system is also called the base 2
system because it uses only two symbols
to represent numbers.
binary
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Place Value
• Numeric value of symbols in different positions.
• Example - Place value in binary system:
Place Value
Binary
Number
8s
4s
2s
1s
Yes
Yes
No
No
1
1
0
0
RESULT: Binary 1100 = decimal 8 + 4 + 0 + 0 = decimal 12
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Binary to Decimal
Conversion
Convert Binary Number 110011
to a Decimal Number:
Binary
1
Decimal
32 + 16 + 0 + 0 + 2 + 1 = 51
1
0
0
1
1
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QUIZ
Convert the following binary
numbers into decimal numbers:
Binary 1001 =
9
Binary 1111 =
15
Binary 0010 =
2
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Decimal to Binary Conversion
Divide by 2 Process
Decimal # 13 ÷ 2 = 6
remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
Divide-by-2 Process
Stops When
1
Quotient Reaches 0
÷ 2 = 0 remainder 1
1
1 0 1
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QUIZ
Convert the following decimal
numbers into binary:
Decimal 11 =
1011
Decimal 4
0100
=
Decimal 17 = 10001
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Electronic Translators
Devices that convert from decimal to
binary numbers and from binary to
decimal numbers.
Encoders translates from decimal to binary
Decoders translates from binary to decimal
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Electronic Encoder –
Decimal to Binary
Binary output
Decimal input
7
5
3
0
0
00
1 01 1
Decimal
to
Binary
Encoder
• Encoders are available in IC form.
• This encoder translates from decimal
input to binary (BCD) output.
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Electronic Decoding –
Binary to Decimal
Binary input
Decimal output
0 10 01 01
Binary-to7-Segment
Decoder/
Driver
• Electronic decoders are available in IC form.
• This decoder translates from binary to decimal.
• Decimals are shown on an 7-segment LED display.
• This decoder also drives the 7-segment display.
©2008 The McGraw-Hill Companies, Inc. All rights reserved.
QUIZ
1. The general name for an electronic device
that translates from binary to decimal is a(n)
__________.
decoder
2. The general name for an electronic device
that translates from decimal to binary is a(n)
__________.
encoder
3. An electronic device called a(n) __________
(decoder, encoder) would probably be
located between the keypad of a calculator
and its processing circuitry to translate from
decimal to machine language (binary).
encoder
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Hexadecimal Number System
Uses 16 symbols - Base 16 System
0-9, A, B, C, D, E, F
Decimal
1
9
10
15
16
Binary
0001
1001
1010
1111
10000
Hexadecimal
1
9
A
F
10
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Hexadecimal and Binary
Conversions
• Hexadecimal to Binary Conversion
Hexadecimal
C
3
Binary
1100
0011
• Binary to Hexadecimal Conversion
Binary
1110
1010
Hexadecimal
E
A
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Decimal to Hexadecimal Conversion
Divide by 16 Process
Decimal #
47 ÷ 16 = 2 remainder 15
Divide-by-2 Process
Stops When 2
Quotient Reaches 0
÷ 16 = 0 remainder 2
2
F
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Hexadecimal to Decimal Conversion
Convert hexadecimal number
2DB to a decimal number
Place Value
256s
Hexadecimal
Decimal
16s
1s
2
D
B
(256 x 2)
(16 x 13)
(1 x 11)
512
+
208
+
11
= 731
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QUIZ
Convert Hexadecimal number A6 to Binary
A6 =
1010 0110 (Binary)
Convert Hexadecimal number 16 to Decimal
16 =
22 (Decimal)
Convert Decimal 63 to Hexadecimal
63 = 3F (Hexadecimal)
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Octal Numbers
Uses 8 symbols - Base 8 System
0, 1, 2, 3, 4, 5, 6, 7
Decimal
1
6
7
8
9
Binary
001
110
111
001 000
001 001
Octal
1
6
7
10
11
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QUIZ
1. The octal number 7 equals ______ in binary.
111
2. The octal number 11 equals ______ in binary.
001 001
3. The decimal number 23 equals ______ in binary. 010 111
4. The decimal number 23 equals ______ in octal.
27
5. The octal number 37 equals ______ in binary.
011 111
6. The octal number 37 equals ______ in decimal.
31
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Practical Suggestion on
Number System Conversions
• Use a scientific calculator
• Most scientific calculators have DEC, BIN,
OCT, and HEX modes and can either
convert between codes or perform
arithmetic in different number systems.
• Most scientific calculators also have other
functions that are valuable in digital
electronics such as AND, OR, NOT,
XOR, and XNOR logic functions.
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Groupings of Binary Digits
•Bit
1-bit (0 or 1)
•Nibble
4-bits (such as 1101)
•Byte
8-bits (such as 1100 0111)
•Word
16-bits (common definition)
•Double-word 32-bits
•Quad-word
64-bits
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QUIZ
1. A 4-bit grouping of binary digits is called a
_____ (byte, nibble).
nibble
2. An byte refers to an a(n) _____ (8, 64)-bit
group of binary digits.
8
3. A single binary digit (such as a 0 or 1) is called a
_____ (bit, nibble).
bit
4. A common definition for a word in computer
jargon is a _____ (1, 16)-bit group of binary
digits.
16
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REVIEW
• Counting in Decimal and Binary
• Place Value
• Binary to Decimal Conversion
• Decimal to Binary Conversion
• Electronic Translators
• Hexadecimal Numbers
• Octal Numbers
• Bits, Bytes and Words
©2008 The McGraw-Hill Companies, Inc. All rights reserved.