Computer arthtmetic,,,
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The document discusses binary number systems and binary arithmetic. It begins by explaining that binary uses only two digits, 0 and 1, with a base or radix of 2. Each digit has a value depending on its position in the number. The document then provides reasons why binary is used in computers, such as being digital and using simple on/off circuits. It proceeds to explain the rules and processes for binary addition, subtraction, and provides examples of carrying and borrowing in binary arithmetic.
![Positive / Negative Combination
9 00001001
Positive / Negative
Positive Answer
+ (-5) + 11111011
4 1]00000100
1-Positive / 1-Negative
8th Bit = 0 : Answer is Positive
Take 2’s Complement Disregard 9th Bit
Of Negative Number (-5)
00000101
2’s
11111010 Complement
Process
+1
11111011 19](https://image.slidesharecdn.com/computerarthtmetic-121205134338-phpapp02/85/Computer-arthtmetic-19-320.jpg)
![Negative / Negative Combination
2’s Complement
(-9) 11110111 Numbers, See
Conversion Process
Negative / Negative
Negative Answer
+ (-5) + 11111011 In Previous Slides
- 14 1]11110010
2-Negative
Take 2’s Complement Of 8th Bit = 1 : Answer is Negative
Disregard 9th Bit
Both Negative Numbers Take 2’s Complement to Check Answer
11110010
2’s
Complement 00001101
Process +1
00001110 21](https://image.slidesharecdn.com/computerarthtmetic-121205134338-phpapp02/85/Computer-arthtmetic-21-320.jpg)
Computer arthtmetic,,,
- 2. Number System Used by Used in System Base Symbols humans? computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- 16 0, 1, … 9, No No decimal 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 Rules Rules: 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 = 15 Add 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 8415 So when the sum of a column is equal to or greater than the base, we subtract the base from the sum, record the difference, and carry one to the next column to the left.
- 10. Binary Addition Example 1 Col 1) Add 1 + 0 = 1 Write 1 Example 1: Add binary 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 Explanation What 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 Verification You can always check your Verification answer by converting the 1101112 5510 figures to decimal, doing the +0111002 + 2810 addition, and comparing the 8310 answers. 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 2 Example 2: Verification Add 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 00001110 Both Positive Numbers Use Straight Binary Addition 18
- 19. Positive / Negative Combination 9 00001001 Positive / Negative Positive Answer + (-5) + 11111011 4 1]00000100 1-Positive / 1-Negative 8th Bit = 0 : Answer is Positive Take 2’s Complement Disregard 9th Bit Of Negative Number (-5) 00000101 2’s 11111010 Complement Process +1 11111011 19
- 20. Negative / Positive Combination (-9) 11110111 Positive / Negative Negative Answer + 5 + 00000101 - 4 11111100 1-Positive / 1-Negative 8th Bit = 1 : Answer is Negative Take 2’s Complement Take 2’s Complement to Check Answer Of 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 Process Negative / Negative Negative Answer + (-5) + 11111011 In Previous Slides - 14 1]11110010 2-Negative Take 2’s Complement Of 8th Bit = 1 : Answer is Negative Disregard 9th Bit Both 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: 11101100 1) Start at the LSB and write down all zeros moving to 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 Subtraction By 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 = 1 Example 1: Subtract Col 2) Subtract 1 – 0 = 1 binary 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 5110 Subtract binary 11100 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 Verification Example 2: Subtract 1010012 4110 binary 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