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  • 1. ComputerArithmetic
  • 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. Binary?– Uses only two digits, 0 and 1– It is base or radix of 2 In State 0 In state 1
  • 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. 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. Why Binary?• Recall: we can use numbers to represent marital status information: • 0 = single • 1 = married • 2 = divorced • 3 = widowed
  • 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. 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. 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. 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. 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. 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. 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. Binary subtraction By compliment method
  • 15. 1’S Complement 01010011 Invert All Bits 10101100 15
  • 16. 2’S Complement 01010011 Invert All Bits 10101100 +1 Add One 10101101 16
  • 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. Positive / Positive Combination 9 00001001 Positive / Positive Positive Answer + 5 + 00000101 14 00001110Both Positive NumbersUse Straight Binary Addition 18
  • 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. 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. 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. 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. Binary SubtractionBy borrow method
  • 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. 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. 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. 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. Binary Multiplication