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**Number System and Codes**

Chapter 3 Number System and Codes

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**Decimal and Binary Numbers**

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**Decimal and Binary Numbers**

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**Converting Decimal to Binary**

Sum of powers of 2

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**Converting Decimal to Binary**

Repeated Division

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**Binary Numbers and Computers**

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Hexadecimal Numbers

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**Converting decimal to hexadecimal**

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**Converting binary to hexadecimal**

Converting hexadecimal to binary?

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Hexadecimal numbers

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Binary arithmetic Binary addition

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**Representing Integers with binary**

Some of challenges:- Integers can be positive or negative Each integer should have a unique representation The addition and subtraction should be efficient.

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**Representing a positive numbers**

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**Representing a negative numbers using Sign-Magnitude notation**

-5 = bits sign-manitude -55= bits sign-magnitude

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1’s Complement The 1’s complement representation of the positive number is the same as sign-magnitude. +84 =

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1’s Complement The 1’s complement representation of the negative number uses the following rule:- Subtract the magnitude from 2n-1 For example: -36 = ??? +36 =

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1’s Complement Example :- - 57 +57 = -57 =

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**Converting to decimal format**

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**2’s Complement For negative numbers:-**

Subtract the magnitude from 2n. Or Add 1 to the 1’s complement

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Example

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**Convert to decimal value**

Positive values:- = +89 Negative values

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**Two's Complement Arithmetic**

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**Adding Positive Integers in 2's Complement Form**

Overflow in Binary Addition

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**Overflow in Binary Addition**

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**Overflow in Binary Addition**

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**Overflow in Binary Addition**

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**Adding Positive and Negative Integers in 2's Complement Form**

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**Adding Positive and Negative Integers in 2's Complement Form**

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**Subtraction of Positive and Negative Integers**

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Digital Codes Binary Coded Decimal (BCD)

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BCD

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BCD

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4221 Code

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Gray Code In pure binary coding or 8421 BCD then counting from 7 (0111) to 8 (1000) requires 4 bits to be changed simultaneously. Gray coding avoids this since only one bit changes between subsequent numbers

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**Binary –to-Gray Code Conversion**

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**Gray –to-Binary Conversion**

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**Gray –to-Binary Conversion**

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The Excess-3- Code

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Parity The method of parity is widely used as a method of error detection. Extar bit known as parity is added to data word The new data word is then transmitted. Two systems are used: Even parity: the number of 1’s must be even. Odd parity: the number of 1’s must be odd.

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**Parity Example: Odd parity Even Parity 110010 110011 11001 111101**

111100 11110 110001 110000 11000

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