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**Lecture 1.2 (Chapter 1) Prepared by Dr. Lamiaa Elshenawy**

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

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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

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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

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**Complements of Numbers**

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

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**Complements of Numbers**

How can we calculate r’s complement and (r-1)’s complement ? Radix complement 𝑟 𝑛 −𝑁 Diminished radix complement 𝑟 𝑛 −1 −𝑁 Where n is number of digits

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**Complements of Numbers**

Calculate the 10’s complement and 9’s complement of Solution 10’s complement: =453300 9’s complement: =453299

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**Complements of Numbers**

Calculate the 2’s complement and 1’s complement of Hint Solution 2’s complement: 1’s complement: 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:**

Bit Positive number Bit Negative number Example (+9)10 =(01001) signed-magnitude representation (-9)10 =(11001) signed-magnitude representation (-9)10 =(10110) signed-1’s complement representation (-9)10 =(10111) signed-2’s complement representation

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**Arithmetic Addition Unsigned numbers: As ordinary arithmetic Example**

25+12=37 13+ −30 =− 30−13 =−17 Signed numbers: Positive numbers as ordinary arithmetic =+37 Negative numbers Write negative number in signed-2’s complement Discard the end carry

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Arithmetic Addition Example

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**Arithmetic Subtraction**

Unsigned numbers: As ordinary arithmetic Example 27−14=13 15− −30 = =45 Signed numbers:

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**Arithmetic Subtraction**

Example −6 − −13 =−6+13=+7 −6 10=( )2 =( ) (signed-2’s complement) +13 10=( )2 𝟏 𝟎𝟏𝟏𝟏 =+7 Discard

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**Binary Codes What is a binary code?**

An n‐bit binary code is a group of n bits that assumes up to 2 𝑛 distinct combinations of 1’s and 0’s to represent one element that is being coded. Why we use binary codes? Because digital systems understands only 1 or 0

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**Binary Codes What are the most common binary codes?**

Binary-Coded Decimal (BCD) 2421-Code Excess-3 Code 8,4,-2,-1 Code Gray Code ASCII Character Code

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**Binary-Coded Decimal Converts Decimal Numbers Binary Numbers How?**

Write each decimal digits in 4bits Example

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Binary-Coded Decimal

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**Signed-BCD Addition Write each decimal number in BCD**

Write sign in the leftmost position (0 for +, 1 for -) Add (6) 10=(0110) 2 to the binary sum If 𝑠𝑢𝑚≥ (1010) 2 Example

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**ASCII Character Code It codes 128 Characters in7-bits**

26 capital letters 26 small letters 10 decimal digits Special characters 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

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Register Transfer

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**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

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**Binary Logic What are the most common logic gates? AND: 𝑥.𝑦=𝑥 𝐴𝑁𝐷 𝑦=𝑧**

OR: 𝑥+𝑦=𝑥 𝑂𝑅 𝑦=𝑧 NOT: 𝑥 ′ = 𝑥 =𝑧

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Binary Logic

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**Thank you for your attention**

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ECE 301 – Digital Electronics Unsigned and Signed Numbers, Binary Arithmetic of Signed Numbers, and Binary Codes (Lecture #2)

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