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Chapter 3 Number System and Codes. Decimal and Binary Numbers.

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Presentation on theme: "Chapter 3 Number System and Codes. Decimal and Binary Numbers."— Presentation transcript:

1 Chapter 3 Number System and Codes

2 Decimal and Binary Numbers

3

4 Converting Decimal to Binary 1.Sum of powers of 2

5 Converting Decimal to Binary 1.Repeated Division

6 Binary Numbers and Computers

7 Hexadecimal Numbers

8 Converting decimal to hexadecimal

9 Converting binary to hexadecimal Converting hexadecimal to binary?

10 Hexadecimal numbers

11 Binary arithmetic Binary addition Binary addition

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

13 Representing a positive numbers

14 Representing a negative numbers using Sign-Magnitude notation -5 = bits sign-manitude -55= bits sign-magnitude

15 1 ’ s Complement The 1 ’ s complement representation of the positive number is the same as sign-magnitude. The 1 ’ s complement representation of the positive number is the same as sign-magnitude. +84 = =

16 1 ’ s Complement The 1 ’ s complement representation of the negative number uses the following rule:- The 1 ’ s complement representation of the negative number uses the following rule:- Subtract the magnitude from 2 n -1 Subtract the magnitude from 2 n -1 For example: For example: -36 = ??? -36 = ??? +36 = =

17 1 ’ s Complement Example :- Example : = = = =

18 Converting to decimal format

19 2 ’ s Complement  For negative numbers:-  Subtract the magnitude from 2 n. Or  Add 1 to the 1 ’ s complement

20 Example

21 Convert to decimal value Positive values:- Positive values: = = +89 Negative values Negative values

22 Two's Complement Arithmetic

23 Adding Positive Integers in 2's Complement Form Overflow in Binary Addition

24

25

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

28

29 Subtraction of Positive and Negative Integers

30 Digital Codes Binary Coded Decimal (BCD)

31 BCD

32

33 4221 Code

34 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

35 Binary – to-Gray Code Conversion

36 Gray – to-Binary Conversion

37

38 The Excess-3- Code

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

40 Parity Example: Example: Odd parity Even Parity


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