1. Number System and Codes
Chapter 3
Number System and Codes

2. Decimal and Binary Numbers

3. Decimal and Binary Numbers

4. Converting Decimal to Binary
Sum of powers of 2

5. Converting Decimal to Binary
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

12. 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.

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.
+84 =

16. 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 =

17. 1’s Complement
Example :-
- 57
+57 =
-57 =

18. Converting to decimal format

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

20. Example

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

22. Two's Complement Arithmetic

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

24. Overflow in Binary Addition

25. Overflow in Binary Addition

26. Overflow in Binary Addition

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

28. Adding Positive and Negative Integers in 2's Complement Form

29. Subtraction of Positive and Negative Integers

30. Digital Codes
Binary Coded Decimal (BCD)

31. BCD

32. BCD

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. Gray –to-Binary Conversion

38. The Excess-3- Code

39. 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.

40. Parity Example: Odd parity Even Parity 110010 110011 11001 111101
111100
11110
110001
110000
11000