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Logic Design (CE1111 ) Lecture 1.2 (Chapter 1) Prepared by Dr. Lamiaa Elshenawy 1

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Outlines Digital Systems Binary Numbers Number-Base Conversions Octal and Hexadecimal Numbers Complements of Numbers Signed Binary Numbers Binary Codes Binary Storage and Registers Binary Logic 2

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Complements of Numbers Complements are used in digital computers to simplify the subtraction operation and for logical manipulation Simplifying operations leads to simpler, less expensive circuits to implement the operations 3

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Complements of Numbers Types of complements 1. Radix complement (r’s complement) 2. Diminished radix complement ((r-1)’s complement) Decimal Numbers O 10’s complement and 9’s complement Binary Numbers O 2’s complement and 1’s complement 4

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Complements of Numbers 5

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Calculate the 10’s complement and 9’s complement of 546700. Solution 10’s complement: 1000000 - 546700=453300 9’s complement: 999999 - 546700=453299 6

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Complements of Numbers Calculate the 2’s complement and 1’s complement of 1011000. Hint Solution 2’s complement: 1011000 0101000 1’s complement: 1011000 0100111 7 2’s complement: leave all least significant 0’s and first 1 unchanged, change from 0 to 1 or from 1 to 0 for all other higher significant digits 1’s complement: change from 0 to 1 or from 1 to 0

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Signed Binary Numbers Leftmost position of the number used for sign: O Bit 0 Positive number O Bit 1 Negative number Example (+9) 10 =(01001) signed-magnitude representation (-9) 10 =(11001) 2 signed-magnitude representation (-9) 10 =(10110) 2 signed-1’s complement representation (-9) 10 =(10111) 2 signed-2’s complement representation 8

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Arithmetic Addition 10

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Arithmetic Addition Example 11

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Arithmetic Subtraction 12

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Arithmetic Subtraction 13 Discard

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Binary Codes 14

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Binary Codes What are the most common binary codes? 1. Binary-Coded Decimal (BCD) 2. 2421-Code 3. Excess-3 Code 4. 8,4,-2,-1 Code 5. Gray Code 6. ASCII Character Code 15

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Binary-Coded Decimal Converts Decimal Numbers Binary Numbers How? Write each decimal digits in 4bits Example 16

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Binary-Coded Decimal 17

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Signed-BCD Addition 18

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ASCII Character Code It codes 128 Characters in7-bits 1. 26 capital letters 2. 26 small letters 3. 10 decimal digits 4. Special characters 21 ASCII: American Standard Code for Information Interchange

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Registers What is a register? A register is a group of binary cells. A register with n cells can store any discrete quantity of information that contains n bits 24

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25 Register Transfer

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26 Information Processing

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Binary Logic What is a binary logic? Binary logic consists of binary variables and a set of logical operations 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 27

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Binary Logic 28

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Binary Logic 29

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