The document discusses how integers are stored in computers using two's complement format. Integers and real numbers are stored differently, with integers using binary representations. Early computers stored integers in 8 bits, but now use 32 bits. Negative integers are represented by taking the two's complement of the binary representation of the positive integer of the same magnitude. This two's complement format addresses issues with representing both positive and negative zero that arose with earlier sign-magnitude representation of integers.

1. 1.15
10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers and
2’s Complement
MATH1003

2. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Goal
To be able to specify how integers are stored by the
computer in 2’s complement format.
MATH1003

3. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
MATH1003

4. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
Integers and Real Numbers are stored in different ways
inside the computer.
MATH1003

5. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
Integers and Real Numbers are stored in different ways
inside the computer.
35 and 35.0 to human beings represent the same value.
MATH1003

6. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
Integers and Real Numbers are stored in different ways
inside the computer.
35 and 35.0 to human beings represent the same value.
35 35.0
MATH1003

7. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
Integers and Real Numbers are stored in different ways
inside the computer.
35 and 35.0 to human beings represent the same value.
However, the computer “handles” these numbers with
35 35.0
different mechanisms.
MATH1003

8. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
MATH1003
35

9. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
Today’s computers store integers usually in 32 bits.
MATH1003
35

10. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
Today’s computers store integers usually in 32 bits.
For the purpose of our discussions, we will imagine that
we are working with an older computer that stores
integers in 8 bits. Concepts that we will discuss are valid
whether the integers are in 32 bits or 8 bits.
MATH1003
35

11. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
MATH1003
35

12. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
3510, as an 8 bit binary number, is 001000112
MATH1003
35

13. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
3510, as an 8 bit binary number, is 001000112
That’s how the computer will store 3510
MATH1003
35

14. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Integers
3510, as an 8 bit binary number, is 001000112
That’s how the computer will store 3510
But we know that integers include both positive and
negative whole numbers.
MATH1003
35

15. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
sign
bit
MATH1003
001000112

16. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
How does the computer store negative numbers?
(Remember: bits are only 0 or 1)
sign
bit
MATH1003
001000112

17. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
How does the computer store negative numbers?
(Remember: bits are only 0 or 1)
One option would be to use the leftmost bit as a sign bit
(0 for positive numbers and 1 for negative numbers).
sign
bit
MATH1003
001000112

18. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
How does the computer store negative numbers?
(Remember: bits are only 0 or 1)
One option would be to use the leftmost bit as a sign bit
(0 for positive numbers and 1 for negative numbers).
sign
bit
MATH1003
001000112

19. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
How does the computer store negative numbers?
(Remember: bits are only 0 or 1)
One option would be to use the leftmost bit as a sign bit
(0 for positive numbers and 1 for negative numbers).
sign
bit
MATH1003
001000112

20. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
We’ll use the other 7 bits for the value (or magnitude) of
the number.
sign
bit
MATH1003
001000112

21. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
We’ll use the other 7 bits for the value (or magnitude) of
the number.
m agnitude
MATH1003
001000112

22. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
We’ll use the other 7 bits for the value (or magnitude) of
the number.
magnitude
MATH1003
001000112

23. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Sign and Magnitude
Using this scheme, we can represent -3510 as 101000112
3510 as a 7 bit number is 01000112
The sign bit will be a 1 to represent the minus sign
sign
bit m agnitude
MATH1003
101000112

24. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
MATH1003

25. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
In this scheme, what is 000000002 and 100000002?
MATH1003

26. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
In this scheme, what is 000000002 and 100000002?
000000002 is positive 0
MATH1003

27. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
In this scheme, what is 000000002 and 100000002?
000000002 is positive 0
MATH1003
+0

28. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
In this scheme, what is 000000002 and 100000002?
000000002 is positive 0
100000002 is negative 0
MATH1003
+0

29. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
In this scheme, what is 000000002 and 100000002?
000000002 is positive 0
100000002 is negative 0
MATH1003
+0 -0

30. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
In this scheme, what is 000000002 and 100000002?
000000002 is positive 0
100000002 is negative 0
Believe or not, having two ways to represent 0 causes big
problems.
MATH1003
+0 -0

31. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
MATH1003
+0 -0

32. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
Depending on the arithmetic operation, we can get a
result of 000000002 or 100000002.
MATH1003
+0 -0

33. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
Depending on the arithmetic operation, we can get a
result of 000000002 or 100000002.
If we had this situation, there would have to be special
circuitry to test for positive 0 and negative 0 - this adds
complexity to the computer.
MATH1003
+0 -0

34. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
MATH1003

35. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
Another problem that arises has to do with the addition
of a positive and a negative number.
MATH1003

36. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
Problems
Another problem that arises has to do with the addition
of a positive and a negative number.
The computer would have to deal with the sign bit (which
does not represent value).
MATH1003

37. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
So a scheme was invented to address these issues.
It is called 2’s complement.
MATH1003

38. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
In 2’s complement, positive numbers are represented in
their normal binary equivalent.
MATH1003

39. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
MATH1003

40. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Negative number representation is found in the following
manner:
MATH1003

41. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Negative number representation is found in the following
manner:
1. calculate the magnitude of the number
MATH1003

42. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Negative number representation is found in the following
manner:
1. calculate the magnitude of the number
2. subtract the number from 111111112
MATH1003

43. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Negative number representation is found in the following
manner:
1. calculate the magnitude of the number
2. subtract the number from 111111112
3. add 1 to the result
MATH1003

44. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
1. calculate the magnitude of the number
2. subtract the number from 111111112
3. add 1 to the result
MATH1003

45. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. subtract the number from 111111112
3. add 1 to the result
MATH1003

46. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. subtract the number from 111111112
1. 10410 = 011010002 3. add 1 to the result
MATH1003

47. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. subtract the number from 111111112
1. 10410 = 011010002 3. add 1 to the result
2. 111111112 - 011010002 = 100101112
MATH1003

48. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. subtract the number from 111111112
1. 10410 = 011010002 3. add 1 to the result
2. 111111112 - 011010002 = 100101112
3. 100101112 + 12 = 100110002
MATH1003

49. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. subtract the number from 111111112
1. 10410 = 011010002 3. add 1 to the result
2. 111111112 - 011010002 = 100101112
3. 100101112 + 12 = 100110002
Therefore, -10410 = 100110002
MATH1003

50. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
MATH1003

51. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Now, let us explore a less complicated way of calculating
the 2’s complement form of a negative number:
MATH1003

52. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Now, let us explore a less complicated way of calculating
the 2’s complement form of a negative number:
1. calculate the magnitude of the number
MATH1003

53. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Now, let us explore a less complicated way of calculating
the 2’s complement form of a negative number:
1. calculate the magnitude of the number
2. starting from the rightmost bit, look for the first bit
that is a 1
MATH1003

54. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
Now, let us explore a less complicated way of calculating
the 2’s complement form of a negative number:
1. calculate the magnitude of the number
2. starting from the rightmost bit, look for the first bit
that is a 1
3. “invert” all bits to the left of the 1
MATH1003

55. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
MATH1003

56. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

57. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

58. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

59. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

60. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

61. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

62. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
MATH1003

63. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
1
MATH1003

64. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
01
MATH1003

65. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
001
MATH1003

66. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
10410 = 011010002
1 001
MATH1003

67. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -10410 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
-10410 = 100110002
MATH1003

68. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
MATH1003

69. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
MATH1003

70. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
MATH1003

71. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
MATH1003

72. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
1
MATH1003

73. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
11
MATH1003

74. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
111
MATH1003

75. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
1111
MATH1003

76. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
11111
MATH1003

77. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
111111
MATH1003

78. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 000000012
1111111
MATH1003

79. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -110 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
-110 = 111111112
MATH1003

80. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
MATH1003

81. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
4210 = 001010102
MATH1003

82. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 001010102
MATH1003

83. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 001010102
1
MATH1003

84. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 001010102
01
MATH1003

85. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 001010102
101
MATH1003

86. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 001010102
0101
MATH1003

87. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 001010102
10101
MATH1003

88. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
110 = 110101 2
00101010
MATH1003

89. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement (a better way)
Calculate the 2’s complement 1. calculate the magnitude of the number
form of -4210 2. starting from the rightmost bit, look for
the first bit that is a 1
3. “invert” all bits to the left of the 1
-4210 = 110101102
MATH1003

90. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
MATH1003

91. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
MATH1003

92. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
MATH1003

93. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
0
MATH1003

94. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
00
MATH1003

95. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
100
MATH1003

96. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
0100
MATH1003

97. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
00100
MATH1003

98. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
000100
MATH1003

99. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
0000100
MATH1003

100. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement this method is
for negative numbers
only
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112
0000100
MATH1003

101. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement this method is
for negative numbers
only
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111101112 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111101112 = -910
0000100
MATH1003

102. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
MATH1003

103. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111010002
MATH1003

104. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111010002
MATH1003

105. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111010002
1
MATH1003

106. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111010002
01
MATH1003

107. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
111010002
001
MATH1003

108. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
0001
111010002
MATH1003

109. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
2’s complement
Calculate the decimal value of (calculating the decimal equivalent)
111010002 1. starting from the rightmost bit, look for
the first bit that is a 1
2. “invert” all bits to the left of the 1
3. calculate the magnitude of the number
0001
111010002 = -2410
MATH1003

110. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
01011001
MATH1003

111. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
01011001
MATH1003

112. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210
+1710
01011001
MATH1003

113. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

114. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

115. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

116. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

117. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

118. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

119. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

120. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

121. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
01011001
MATH1003

122. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
010110012
MATH1003

123. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 7210 + 1710
Since these are both positive, this is straightforward
7210 010010002
+1710 +000100012
bit n
sig
010110012
MATH1003

124. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
MATH1003

125. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410
+ -4210
MATH1003

126. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410 011010002
+ -4210 +110101102
MATH1003

127. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410 011010002 -4210 in
2’s complement?
+ -4210 +110101102
MATH1003

128. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410 011010002
+ -4210 +110101102
MATH1003

129. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410 011010002
+ -4210 +110101102
1001111102
MATH1003

130. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410
since we are only
using 8 bit numbers,
011010002
+ -4210
the ninth bit is simply
dropped +110101102
1001111102
MATH1003

131. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410 011010002
+ -4210 +110101102
1001111102
MATH1003

132. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 10410 + (-4210)
Here we add a negative number to a positive number
10410 011010002
+ -4210 +110101102
bit n
sig
1001111102
MATH1003

133. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
MATH1003

134. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710
+ -3510
MATH1003

135. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710 000110112
+ -3510 +110111012
MATH1003

136. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710 000110112 -3510 in
2’s complement?
+ -3510 +110111012
MATH1003

137. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710 000110112
+ -3510 +110111012
MATH1003

138. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710 000110112
+ -3510 +110111012
1111110002
MATH1003

139. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710 000110112
+ -3510 +110111012
bit n
sig
1111110002
MATH1003

140. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate 2710 + (-3510)
Here we add a negative number to a positive number
2710 000110112
+ -3510 can you verify
+110111012 complement?
that this is -8 in 2’s 10
bit n
sig
1111110002
MATH1003

141. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
MATH1003

142. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210
+ -110
MATH1003

143. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210 110101102
+ -110 +111111112
MATH1003

144. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210 110101102
+ -110 +111111112
111010101
MATH1003

145. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210 110101102
+ -110 +111111112
1110101012
MATH1003

146. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210
since we are only
using 8 bit numbers,
110101102
+ -110
the ninth bit is simply
dropped +111111112
1110101012
MATH1003

147. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210 110101102
+ -110 +111111112
1110101012
MATH1003

148. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001
2’s Complement
and Addition
Calculate -4210 + (-110)
Here we add a negative number to a negative number
-4210 110101102
+ -110 +111111112
bit n
sig
1110101012
MATH1003