Math1003 1.15 - Integers and 2's Complement
- 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
- 149. 10110100101011010100101010111010101111011011101111011101110111101110111011110111111010110100101011110110110101111011010100111111011010100110101001 2’s Complement and Addition Calculate -4210 + (-110) Here we add a negative number to a negative number -4210 110101102 + -110 +111111112 can you verify that this is -4310 in bit n 2’s complement? sig 1110101012 MATH1003