1. Chapter 4
Operations on
Bits

2. Figure 4-1
Operations on bits

3. 4.1
ARITHMETIC
OPERATIONS

4. Arithmetic Operation on Integer
All arithmetic operation (addition, subtraction, multiplication and division) can be applied to integers. However we focus on addition and subtraction.
Because multiplication is repetitive addition and division is repetitive subtraction.
Addition and subtraction can be applied to all integer representation but we interested in two’s complement representation.

5. Adding bits Bits ------------ 0+0 0+1 1 1+0 1+1 1 1+1+1=? 1+1+0=?
Result 1
Carry 1
1+1+1=?
1+1+0=?
1+0+0+?

6. Example 1
Add two numbers in two’s complement representation: (+17) + (+22)  (+39)
Solution
Carry
Result  39

7. Example 2
Add two numbers in two’s complement representation: (+24) + (-17)  (+7)
Solution
Carry
Result  +7

8. Example 3
Add two numbers in two’s complement representation: (-35) + (+20)  (-15)
Solution
Carry
Result  -15

9. Example 4
Add two numbers in two’s complement representation: (+127) + (+3)  (+130)
Solution
Carry
Result  -126 (Error) An overflow has occurred.

10. Overflow
Overflow in binary addition occurs when the result of addition can not be stored within the range defined by the allocation.
For two’s complement representation the addition result should be in the range between -2N-1 and 2N-1-1 where N is number of bits allocation

11. Why did we get wrong answer?
Figure 4-2
Why did we get wrong answer?
Number Visualization

12. Subtraction in two’s complement
In two’s complement representation there is no difference between addition and subtraction.
To subtract “negate” (using two’s complement) the subtracted number and add.
Would overflow occurs in subtraction? How?

13. Example 5
Subtract 62 from 101 in two’s complement: (+101) - (+62)  (+101) + (-62)
Solution
Carry
Result  39 The leftmost carry is discarded.

14. Arithmetic Operation on Floating-Point Number
Check sign
If the sign is positive, add numbers.
If the sign is negative, compare the absolute values, subtract the smaller from the larger, attach the sign of the larger.
Move the decimal point to make the exponent the same.
Add or subtract
Normalize the result

15. After normalization +26 x 1.000001, which is stored as:
Example 6
Add two floats:
The exponents are 5 and 3. The numbers are: x and
+23 x Make the exponents the same.
+25 x
+25 x
+25 x
After normalization +26 x , which is stored as: +

16. 4.2
LOGICAL
OPERATIONS

17. Logical Operation
In Computer, you can interpret 0 as logical value false and 1 as logical value true.
Logical operation can accept 1 or 2 bits to create only 1 bits.
If operation is applied to only one input, it is a unary operation.
If operation is applied to 2 inputs, it is a binary operation.

18. Unary and binary operations
Figure 4-3
Unary and binary operations

19. Figure 4-4
Logical operations

20. Figure 4-5
Truth tables

21. NOT Operator NOT Operator has one input.
NOT Operator inverts the bits.
NOT Symbol
Input x
Output
NOT x
NOT

22. Figure 4-6
NOT operator

23. Example 7
Use the NOT operator on the bit pattern
Solution
Target NOT Result

24. AND Operator
AND operator is a binary operator, it takes two inputs and create one output
AND Symbol x
AND
AND
Output y

25. Figure 4-7
AND operator

26. Example 8
Use the AND operator on bit patterns and
Solution
Target AND Result

27. Inherent rule of the AND operator
Figure 4-8
Inherent rule of the AND operator

28. OR Operator
OR operator is a binary operator, it takes two inputs and create one output
OR Symbol x OR
Output y

29. Figure 4-9
OR operator

30. Example 9
Use the OR operator on bit patterns and
Solution
Target OR Result

31. Inherent rule of the OR operator
Figure 4-10
Inherent rule of the OR operator

32. XOR Operator
XOR operator is a binary operator, it takes two inputs and create one output
OR Symbol x
XOR
Output y

33. Figure 4-11
XOR operator

34. Example 10
Use the XOR operator on bit patterns and
Solution
Target XOR Result

35. Inherent rule of the XOR operator
Figure 4-12
Inherent rule of the XOR operator

36. Applications of Binary Operation
Logical binary operations can be used to modify a bit pattern.
They could be used to set, unset or invert specific bits.
The Mask is the a bit pattern used to modify another bit pattern.
A bit pattern can be ANDed, ORed or XORed with the Mask

37. Figure 4-13
Mask

38. Unsetting Specific Bits
AND operator can be used to unset (force to 0) specific bits in a bit pattern.
The rule for constructing an unsetting mask are:
To unset a bit in the target, use 0 for the corresponding bit in the mask.
To leave a bit in the target unchanged, use 1 for the corresponding bit in the mask

39. Example of unsetting specific bits
Figure 4-14
Example of unsetting specific bits

40. Example 11
Use a mask to unset (clear) the 5 leftmost bits of a pattern. Test the mask with the pattern
Solution
The mask is
Target AND Mask Result

41. Solution on the next slide.
Example 12
Imagine a power plant that pumps water to a city using eight pumps. The state of the pumps (on or off) can be represented by an 8-bit pattern. For example, the pattern shows that pumps 1 to 3 (from the right), 7 and 8 are on while pumps 4, 5, and 6 are off. Now assume pump 7 shuts down. How can a mask show this situation?
Solution on the next slide.

42. Solution
Use the mask to AND with the target pattern. The only 0 bit (bit 7) in the mask turns off the seventh bit in the target.
Target AND Mask Result

43. Setting Specific Bits
OR operator can be used to set (force to 1) specific bits in a bit pattern.
The rule for constructing an unsetting mask are:
To set a bit in the target, use 1 for the corresponding bit in the mask.
To leave a bit in the target unchanged, use 0 for the corresponding bit in the mask

44. Example of setting specific bits
Figure 4-15
Example of setting specific bits

45. Example 13
Use a mask to set the 5 leftmost bits of a pattern. Test the mask with the pattern
Solution
The mask is
Target OR Mask Result

46. Example 14
Using the power plant example, how can you use a mask to show that pump 6 is now turned on?
Solution
Use the mask
Target OR Mask Result

47. Flipping (inverting) Specific Bits
XOR operator can be used to flipping specific bits (changing 0 to 1 and 1 to 0) in a bit pattern.
The rule for constructing an unsetting mask are:
flip a bit in the target, use 1 for the corresponding bit in the mask.
To leave a bit in the target unchanged, use 0 for the corresponding bit in the mask

48. Example of flipping specific bits
Figure 4-16
Example of flipping specific bits

49. Example 15
Use a mask to flip the 5 leftmost bits of a pattern. Test the mask with the pattern
Solution
Target XOR Mask Result

50. 4.3
SHIFT
OPERATIONS

51. Shift Operation A bit pattern can be shifted to the left or the right.
The right-shift operation discards the rightmost bit every shift and insert 0 to the leftmost bit.
The left-shift operation discards the leftmost bit every shift and insert 0 to the rightmost bit.
Note: Shifting may change the sign of the number.

52. Figure 4-17
Shift operations

53. Example 16
Show how you can divide or multiply a number by 2 using shift operations.
Solution
If a bit pattern represents an unsigned number, a right-shift operation divides the number by two. The pattern represents 59. When you shift the number to the right, you get , which is 29. If you shift the original number to the left, you get , which is 118.

54. Example 17
Use a combination of logical and shift operations to find the value (0 or 1) of the fourth bit (from the right).
Solution
Use the mask to AND with the target to keep the fourth bit and clear the rest of the bits.
Continued on the next slide

55. Solution (continued)
Target a b c d e f g h AND Mask Result e
Shift the new pattern three times to the right e000  00000e00  e0  e Now it is easy to test the value of the new pattern as an unsigned integer. If the value is 1, the original bit was 1; otherwise the original bit was 0.