©2008 The McGraw-Hill Companies, Inc. All rights reserved. Digital Electronics Principles & Applications Seventh Edition Chapter 2 Numbers We Use in Digital Electronics Roger L. Tokheim ©2008 The McGraw-Hill Companies, Inc. All rights reserved.

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)

©2008 The McGraw-Hill Companies, Inc. All rights reserved. QUIZ 1. The __________ (binary, octal) number system uses only the symbols 0 and 1 in counting. 2. The __________ (decimal, hexadecimal) number system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and The __________ (binary, decimal) number system is also called the base 2 system because it uses only two symbols to represent numbers. binary decimal binary

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Place Value Numeric value of symbols in different positions. Example - Place value in binary system: Binary 8s4s 2s 1s Number Place Value Yes No 1001 RESULT: Binary 1100 = decimal = decimal 12

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Binary to Decimal Conversion Convert Binary Number to a Decimal Number: = Decimal Binary

©2008 The McGraw-Hill Companies, Inc. All rights reserved. QUIZ Convert the following binary numbers into decimal numbers: Binary 1001 = 9 Binary 1111 = Binary 0010 = 15 2

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Decimal to Binary Conversion Divide by 2 Process Decimal # 13 ÷ 2 = 6 remainder 1 6 ÷ 2 = ÷ 2 = ÷ 2 = Divide-by-2 Process Stops When Quotient Reaches 0

©2008 The McGraw-Hill Companies, Inc. All rights reserved. QUIZ Convert the following decimal numbers into binary: Decimal 11 = Decimal 4 = Decimal 17 =

©2008 The McGraw-Hill Companies, Inc. All rights reserved. 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

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Electronic Encoder – Decimal to Binary 0 Decimal to Binary Encoder Binary output Decimal input Encoders are available in IC form. This encoder translates from decimal input to binary (BCD) output.

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Binary-to- 7-Segment Decoder/ Driver Electronic Decoding – Binary to Decimal Binary input 0 0 Decimal output 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) __________. 2. The general name for an electronic device that translates from decimal to binary is a(n) __________. decoder 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

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Uses 16 symbols - Base 16 System 0-9, A, B, C, D, E, F Decimal Binary Hexadecimal 1 9 A F 10 Hexadecimal Number System

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Hexadecimal to Binary Conversion HexadecimalC 3 Binary Binary Hexadecimal E A Binary to Hexadecimal Conversion Hexadecimal and Binary Conversions

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Decimal to Hexadecimal Conversion Divide by 16 Process Decimal #47 ÷ 16 = 2 remainder 15 2 ÷ 16 = 0 remainder 2 F2 Divide-by-2 Process Stops When Quotient Reaches 0

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Hexadecimal to Decimal Conversion Convert hexadecimal number 2DB to a decimal number = D B Hexadecimal Decimal Place Value 256s16s1s (256 x 2)(16 x 13)(1 x 11)

©2008 The McGraw-Hill Companies, Inc. All rights reserved. QUIZ Convert Hexadecimal number A6 to Binary Convert Hexadecimal number 16 to Decimal Convert Decimal 63 to Hexadecimal 63 = 16 = A6 = (Binary) 22 (Decimal) 3F (Hexadecimal)

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Octal Numbers Uses 8 symbols - Base 8 System 0, 1, 2, 3, 4, 5, 6, 7 Decimal Octal Binary

©2008 The McGraw-Hill Companies, Inc. All rights reserved. QUIZ 1.The octal number 7 equals ______ in binary. 2. The octal number 11 equals ______ in binary. 3. The decimal number 23 equals ______ in binary. 4. The decimal number 23 equals ______ in octal. 6. The octal number 37 equals ______ in decimal The octal number 37 equals ______ in binary

©2008 The McGraw-Hill Companies, Inc. All rights reserved. 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.

©2008 The McGraw-Hill Companies, Inc. All rights reserved. Groupings of Binary Digits Bit1-bit (0 or 1) Nibble4-bits (such as 1101) Byte8-bits (such as ) Word16-bits (common definition) Double-word32-bits Quad-word64-bits

©2008 The McGraw-Hill Companies, Inc. All rights reserved. QUIZ 1.A 4-bit grouping of binary digits is called a _____ (byte, nibble). 2. An byte refers to an a(n) _____ (8, 64)-bit group of binary digits. 3. A single binary digit (such as a 0 or 1) is called a _____ (bit, nibble). 4. A common definition for a word in computer jargon is a _____ (1, 16)-bit group of binary digits. nibble 8 bit 16

©2008 The McGraw-Hill Companies, Inc. All rights reserved. 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