ComputerArithmetic
Number System                             Used by Used inSystem    Base Symbols       humans? computers?Decimal    10   0,...
Binary?– Uses only two digits, 0 and 1– It is base or radix of 2     In State 0               In state 1
Binary?• Each digit has a value depending on its  position:   102 = (1x21)+(0x20) = 210   112 = (1x21)+(1x20) = 310   1...
Why Binary ?• digital on" and "off“          digits – 0 and 1• binary use more storage than decimal• Easier to handle 2-di...
Why Binary?• Recall: we can use numbers to represent  marital status information:  • 0 = single  • 1 = married  • 2 = divo...
Binary Addition RulesRules:  0+0        =0  0+1        =1  1+0        =1          (just like in decimal)     1+1     =...
Decimal Addition Example                    1) Add 8 + 7 = 15Add 3758 to 4657:         Write down 5, carry 1              ...
Decimal Addition Explanation                                  What just happened?      111                              1 ...
Binary Addition Example 1                         Col 1) Add 1 + 0 = 1 Write 1Example 1: Addbinary 110111 to 11100   Col 2...
Binary Addition ExplanationWhat is actually       In the first two columns,happened when we            there were no carri...
Binary Addition VerificationYou can always check your       Verificationanswer by converting the         1101112      551...
Binary Addition Example 2Example 2:            VerificationAdd 1111 to 111010.    1110102      5810                      ...
Binary subtraction By compliment method
1’S Complement                   01010011 Invert All Bits   10101100                              15
2’S Complement                   01010011 Invert All Bits   10101100                         +1  Add One          10101101...
Add/Sub : 4 Combinations                         9                            (-9)Positive / Positive            Negative ...
Positive / Positive Combination                          9      00001001  Positive / Positive   Positive Answer      + 5  ...
Positive / Negative Combination                          9      00001001Positive / Negative Positive Answer               ...
Negative / Positive Combination                       (-9)         11110111Positive / Negative Negative Answer            ...
Negative / Negative Combination                                                               2’s Complement              ...
2’S Complement Quick Method    Example:        111011001) Start at the LSB and write down all zeros movingto the left.2) W...
Binary SubtractionBy borrow method
Binary Subtraction               Explanation In binary, the base unit is 2 So when you cannot subtract, you borrow from ...
Binary Subtraction                 Example 1                           Col 1) Subtract 1 – 0 = 1Example 1: Subtract       ...
Binary Subtraction                Verification                           Verification                           1100112  ...
Binary Subtraction              Example 2                            VerificationExample 2: Subtract         1010012  411...
Binary Multiplication
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Computer arthtmetic,,,

  1. 1. ComputerArithmetic
  2. 2. Number System Used by Used inSystem Base Symbols humans? computers?Decimal 10 0, 1, … 9 Yes NoBinary 2 0, 1 No YesOctal 8 0, 1, … 7 No NoHexa- 16 0, 1, … 9, No Nodecimal A, B, … F
  3. 3. Binary?– Uses only two digits, 0 and 1– It is base or radix of 2 In State 0 In state 1
  4. 4. Binary?• Each digit has a value depending on its position:  102 = (1x21)+(0x20) = 210  112 = (1x21)+(1x20) = 310  1002 = (1x22)+ (0x21)+(0x20) = 410
  5. 5. Why Binary ?• digital on" and "off“ digits – 0 and 1• binary use more storage than decimal• Easier to handle 2-digits for circuits, transistors i.e (1,0) rather then more
  6. 6. Why Binary?• Recall: we can use numbers to represent marital status information: • 0 = single • 1 = married • 2 = divorced • 3 = widowed
  7. 7. Binary Addition RulesRules: 0+0 =0 0+1 =1 1+0 =1 (just like in decimal)  1+1 = 210 = 102 = 0 with 1 to carry  1+1+1 = 310 = 112 = 1 with 1 to carry
  8. 8. Decimal Addition Example 1) Add 8 + 7 = 15Add 3758 to 4657: Write down 5, carry 1 2) Add 5 + 5 + 1 = 11 111 Write down 1, carry 1 3758 3) Add 7 + 6 + 1 = 14 + 4657 Write down 4, carry 1 8 415 4) Add 3 + 4 + 1 = 8 Write down 8
  9. 9. Decimal Addition Explanation What just happened? 111 1 1 1 (carry) 3758 3 7 5 8 +4 6 5 7 + 4657 - 8 14 11 15 (sum) 10 10 10 (subtract the base) 8 4 1 5 8415So when the sum of a column is equal to or greater than the base, wesubtract the base from the sum, record the difference, and carry one to thenext column to the left.
  10. 10. Binary Addition Example 1 Col 1) Add 1 + 0 = 1 Write 1Example 1: Addbinary 110111 to 11100 Col 2) Add 1 + 0 = Write 1 Col 3) Add 1 + 1 = 2 (10 in binary) Write 0, carry 1 Col 4) Add 1+ 0 + 1 = 2 Write 0, carry 1 1 1 1 1 1 1 0 1 1 1 Col 5) Add 1 + 1 + 1 = 3 (11 in binary) Write 1, carry 1 + 0 1 1 1 0 0 Col 6) Add 1 + 1 + 0 = 2 10 1 00 1 1 Write 0, carry 1 Col 7) Bring down the carried 1 Write 1
  11. 11. Binary Addition ExplanationWhat is actually In the first two columns,happened when we there were no carries.carried in binary? In column 3, we add 1 + 1 = 2 Since 2 is equal to the base, subtract the base from the sum and carry 1. In column 4, we also subtract 1 1 1 1 the base from the sum and carry 1. 1 1 01 1 1 In column 5, we also subtract the base from the sum and carry 1. + 0 1 11 0 0 In column 6, we also subtract 2 3 22 the base from the sum and carry 1. - 2 2 22 . In column 7, we just bring down the carried 1 1 0 1 0 0 1 1
  12. 12. Binary Addition VerificationYou can always check your Verificationanswer by converting the 1101112  5510figures to decimal, doing the +0111002 + 2810addition, and comparing the 8310answers. 64 32 16 8 4 2 1 1 0 1 0 0 1 1 1 1 0 1 1 1 = 64 + 16 + 2 +1 + 0 1 1 1 0 0 = 8310 1 0 1 0 0 1 1
  13. 13. Binary Addition Example 2Example 2: VerificationAdd 1111 to 111010. 1110102  5810 +0011112 + 1510 7310 1 1 1 1 1 64 32 16 8 4 2 1 1 1 1 0 1 0 1 0 0 1 0 0 1+ 0 0 1 1 1 1 = 64 + 8 +1 = 7310 1 0 0 1 0 0 1
  14. 14. Binary subtraction By compliment method
  15. 15. 1’S Complement 01010011 Invert All Bits 10101100 15
  16. 16. 2’S Complement 01010011 Invert All Bits 10101100 +1 Add One 10101101 16
  17. 17. Add/Sub : 4 Combinations 9 (-9)Positive / Positive Negative / Positive Positive Answer + 5 Negative Answer + 5 14 -4 9 (-9)Positive / Negative Negative / Negative Positive Answer + (-5) Negative Answer + (-5) 4 - 14 17
  18. 18. Positive / Positive Combination 9 00001001 Positive / Positive Positive Answer + 5 + 00000101 14 00001110Both Positive NumbersUse Straight Binary Addition 18
  19. 19. Positive / Negative Combination 9 00001001Positive / Negative Positive Answer + (-5) + 11111011 4 1]000001001-Positive / 1-Negative 8th Bit = 0 : Answer is PositiveTake 2’s Complement Disregard 9th BitOf Negative Number (-5) 00000101 2’s 11111010 Complement Process +1 11111011 19
  20. 20. Negative / Positive Combination (-9) 11110111Positive / Negative Negative Answer + 5 + 00000101 - 4 111111001-Positive / 1-Negative 8th Bit = 1 : Answer is NegativeTake 2’s Complement Take 2’s Complement to Check AnswerOf Negative Number (-9) 11111100 00001001 2’s 2’s Complement 00000011 11110110 Complement Process Process +1 +1 00000100 11110111 20
  21. 21. Negative / Negative Combination 2’s Complement (-9) 11110111 Numbers, See Conversion ProcessNegative / Negative Negative Answer + (-5) + 11111011 In Previous Slides - 14 1]111100102-NegativeTake 2’s Complement Of 8th Bit = 1 : Answer is Negative Disregard 9th BitBoth Negative Numbers Take 2’s Complement to Check Answer 11110010 2’s Complement 00001101 Process +1 00001110 21
  22. 22. 2’S Complement Quick Method Example: 111011001) Start at the LSB and write down all zeros movingto the left.2) Write down the first “1” you come to.3) Invert the rest of the bits moving to the left. 0 001 0 1 0 0 22
  23. 23. Binary SubtractionBy borrow method
  24. 24. Binary Subtraction Explanation In binary, the base unit is 2 So when you cannot subtract, you borrow from the column to the left.  The amount borrowed is 2.  The 2 is added to the original column value, so you will be able to subtract.
  25. 25. Binary Subtraction Example 1 Col 1) Subtract 1 – 0 = 1Example 1: Subtract Col 2) Subtract 1 – 0 = 1binary 11100 from 110011 Col 3) Try to subtract 0 – 1  can’t. Must borrow 2 from next column. But next column is 0, so must go to column after next to borrow. 2 1 Add the borrowed 2 to the 0 on the right. 0 0 2 2 Now you can borrow from this column (leaving 1 remaining). 1 1 0 0 1 1 Add the borrowed 2 to the original 0. Then subtract 2 – 1 = 1- 1 1 1 0 0 Col 4) Subtract 1 – 1 = 0 1 0 1 1 1 Col 5) Try to subtract 0 – 1  can’t. Must borrow from next column. Add the borrowed 2 to the remaining 0. Then subtract 2 – 1 = 1 Col 6) Remaining leading 0 can be ignored.
  26. 26. Binary Subtraction Verification Verification 1100112  5110Subtract binary11100 from 110011: - 111002 - 2810 2310 2 1 0 0 2 2 64 32 16 8 4 2 1 1 0 1 1 1 1 1 0 0 1 1 = 16 + 4 + 2 + 1- 1 1 1 0 0 = 2310 1 0 1 1 1
  27. 27. Binary Subtraction Example 2 VerificationExample 2: Subtract 1010012  4110binary 10100 from 101001 - 101002 - 2010 2110 64 32 16 8 4 2 1 0 2 0 2 1 0 1 0 1 1 0 1 0 0 1 = 16 + 4 + 1 = 2110- 1 0 1 0 0 1 0 1 0 1
  28. 28. Binary Multiplication

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