1. The document discusses data representation and computer arithmetic in various number systems. 2. It provides examples of addition in binary, decimal, and other number systems. Binary addition is demonstrated through examples of adding positive and negative numbers represented in 1's and 2's complement forms. 3. Additional examples show addition in decimal and other number bases such as octal and hexadecimal. Proper carrying and representation of results are emphasized.

1. Chapter 1
DATA REPRESENTATION

2. Complements
Examples Examples
 10’s C  1’s C
• 512790 = 487210 • 011100 = 100011
• 14672.3 = 85327.7
 2’s C
 7’s C • 010101 = 101011
• 65172 = 12605
 16’s C
• 59A1D = A65E3

3. Systems Used to Represent Negative Numbers
Signed-Magnitude Representation
 A number consists of a magnitude and a symbol
indicating whether the magnitude is positive or
negative.
Examples:
+85 = 010101012 -85 = 110101012
+127 = 011111112 -127 = 111111112

4. Systems Used to Represent Negative Numbers
Signed-Complement System
• This system negates a number by taking its
complement as defined by the system.
Examples:
+85 = 010101012 -85 = 101010102 (1s)
+127 = 011111112 -127 = 100000012 (2s)

5. Chapter 2
COMPUTER ARITHMETIC

6. Arithmetic in various Number Systems
• Addition of numbers in any number system
– Add numbers starting at the least significant
digit.
– Perform addition on numbers of the same
number base.
• Subtraction of numbers
– Must use complements

7. Binary Addition
• To add binary numbers: (X + Y)
– Get the SCR of the negative numbers
– Add the two numbers
– If the SCR used is:
• 2's C: Discard end carry
• 1's C: Add the end carry to the sum

8. Example: Binary Addition
• Add the following numbers. Use 8 bits to
represent each number.
– 6 + 13
– 6 + (-13)
– (-6) + 13
– (-6) + (-13)

9. Example: Binary Addition
• 6 + 13 = 19

10. Example: Binary Addition
• 6 + 13 = 19
0 0000110

11. Example: Binary Addition
• 6 + 13 = 19
0 0000110
+ 0 0001101

12. Example: Binary Addition
• 6 + 13 = 19
0 0000110
+ 0 0001101
= 0 0010011

13. Example: Binary Addition
• 6 + 13 = 19 • 6 + (-13) = -7
0 0000110 0 0000110
+ 0 0001101
= 0 0010011

14. Example: Binary Addition
• 6 + 13 = 19 • 6 + (-13) = -7
0 0000110 0 0000110
+ 0 0001101 (1's)
= 0 0010011

15. Example: Binary Addition
• 6 + 13 = 19 • 6 + (-13) = -7
0 0000110 0 0000110
+ 0 0001101 + 1 1110010 (1's)
= 0 0010011

16. Example: Binary Addition
• 6 + 13 = 19 • 6 + (-13) = -7
0 0000110 0 0000110
+ 0 0001101 + 1 1110010 (1's)
= 0 0010011 = 1 1111000

17. Example: Binary Addition
• (-6) + 13 = 7
(2's)
6 = 0 0000110

18. Example: Binary Addition
• (-6) + 13 = 7
1 1111010 (2's)
6 = 0 0000110

19. Example: Binary Addition
• (-6) + 13 = 7
1 1111010 (2's)
+ 0 0001101
6 = 0 0000110

20. Example: Binary Addition
• (-6) + 13 = 7
1 1111010 (2's)
+ 0 0001101
=10 0000111
6 = 0 0000110

21. Example: Binary Addition
• (-6) + 13 = 7
1 1111010 (2's)
+ 0 0001101
=10 0000111
6 = 0 0000110

22. Example: Binary Addition
• (-6) + 13 = 7 • (-6) + (-13) = -19
1 1111010 (2's) 1 1111010 (2's)
+ 0 0001101
=10 0000111
6 = 0 0000110

23. Example: Binary Addition
• (-6) + 13 = 7 • (-6) + (-13) = -19
1 1111010 (2's) 1 1111010 (2's)
+ 0 0001101 + 1 1110011 (2's)
=10 0000111
6 = 0 0000110

24. Example: Binary Addition
• (-6) + 13 = 7 • (-6) + (-13) = -19
1 1111010 (2's) 1 1111010 (2's)
+ 0 0001101 + 1 1110011 (2's)
=10 0000111 = 11 1101101
6 = 0 0000110

25. Example: Binary Addition
• (-6) + 13 = 7 • (-6) + (-13) = -19
1 1111010 (2's) 1 1111010 (2's)
+ 0 0001101 + 1 1110011 (2's)
=10 0000111 = 11 1101101
6 = 0 0000110

26. Examples: Addition
• (999.5 + 281.6)10
1 1 1
999.5
+ 281.6
1 281.1

27. Examples: Addition
• (999.5 + 281.6)10 • (110.11 + 101010.11)2
1 1 1 1 1 1 1 1
999.5 110.11
+ 281.6 + 101010.11
1 281.1 1100 01.10

28. Examples: Addition
• (355.45 + 240.664)8
355.45
+ 240.664
=

29. Examples: Addition
• (355.45 + 240.664)8
1
355.45
+ 240.664
= 4

30. Examples: Addition
• (355.45 + 240.664)8
1
355.45
+ 240.664
= 34

31. Examples: Addition
• (355.45 + 240.664)8
1 1
355.45
+ 240.664
= 6.3 34

32. Examples: Addition
• (355.45 + 240.664)8
1 1 1
355.45
+ 240.664
= 6 16.3 3 4

33. Examples: Addition
• (355.45 + 240.664)8 • (A0C.D + E72.9)16
1 1 1
355.45 A0C.D
+ 240.664 + E72.9
= 6 16.3 3 4 =

34. Examples: Addition
• (355.45 + 240.664)8 • (A0C.D + E72.9)16
1 1 1 1
355.45 A0C.D
+ 240.664 + E72.9
= 6 16.3 3 4 = .6

35. Examples: Addition
• (355.45 + 240.664)8 • (A0C.D + E72.9)16
1 1 1 1
355.45 A0C.D
+ 240.664 + E72.9
= 6 16.3 3 4 = 1 8 7 F.6