Arithmetic Computation using 2's Complement Notation

Arithmetic Computation using 2's Complement Notation

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  Binary Arithmetic Using SignedComplement Notation Akshay Kumar 1
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  Reference: Block 1Unit 2 Section 2.6.1 of MCS-012 2
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  Data Representation for Computation Themost common representation is 2’s Complement Notation. It is discussed Next:  Positive numbers are represented as the case with signed number, but negative numbers are represented in 2’s complement form  This is an efficient method for simple binary addition and subtraction. 3
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  2’s Complement Notation Positive integers have it’s sign bit as 0  Negative integers are represented as a 2’s Complement.  What is a Complement?  English Meaning: Balance to make a group complete. 4
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  Example – 1’sand 2’s Complement for 8 bit numbers (Please note first digit is sign bit) Decimal Number Equivalent Binary Place Value Sign Bit (0/1) 26 = 64 25 = 32 24 = 16 23 = 8 22= 4 21 = 2 20 = 1 +37 Value 0 0 1 0 0 1 0 1 -37 1’s 1 1 0 1 1 0 1 0 -37 2’s 1 1 0 1 1 0 1 1 •First find the magnitude of the Number in Binary as Positive Number (+37) •Complement each bit (1 by 0) and (0 by 1) to make 1’s complement of the negative number (-37) •Add 1 to 1’s complement to get 2’s complement of (-37) 5
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  Example 2 Decimal Number Equivalent Binary Place Value SignBit (0/1) 26 = 64 25 = 32 24 = 16 23 = 8 22= 4 21 = 2 20 = 1 +100 Value 0 1 1 0 0 1 0 0 -100 1’s 1 0 0 1 1 0 1 1 Add 1 c=1 c=1 1 -100 2’s 1 0 0 1 1 1 0 0 •Magnitude of the Number in Binary (+100) •Complement each bit to get 1’s complement of (-100) •Add 1 to 1’s complement to get 2’s complement of -100). Please note the carry bit on addition shown in Orange. 6
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  Conversion of BinaryIntegers to 2’s Complement Notation  For positive integers – no change is needed  For negative integers  Method 1: Complement all the bits individually and then add 1 to resultant, for instance complement of 65 will be: +68 in binary (using 8 bits) 0100 0100 - 68 will be obtained as 1011 1011 + 1 = 1011 1100  Method 2: Moving from least significant bit, leave all bits till the first 1 as it is, then complement all the remaining bits 7
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  Addition using 2’sComplement Notation  Four Cases:  Addition of two positive integers: +68 0 100 0100 +38 0 010 0110 ------------------------ +106 0 110 1010 ------------------------ 8
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  Addition using 2’sComplement Notation Addition of one positive and one negative integer (the positive integer is greater): +68 0 100 0100 -38 1 101 1010 ------------------------ +30 1 0 001 1110 ------------------------ Carry in to sign bit = Carry out of Sign bit => NO OVERFLOW - ignore the carry out of sign bit 9
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  Addition using 2’sComplement Notation Addition of one positive and one negative integer (the positive integer is smaller): -68 1 011 1100 +38 0 010 0110 ------------------------ -30 1 110 0010 ------------------------ +30 0 001 1110 10
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  Addition using 2’sComplement Notation Addition of two negative integers: -68 1 011 1100 -38 1 101 1010 ------------------------ -106 1 1 001 0110 ------------------------ +106 0 110 1010 11
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  Addition using 2’sComplement Notation Overflow: +68 0 100 0100 -68 1 011 1100 +60 0 011 1100 -60 1 100 0100 ------------------------ ------------------------ +128 1 000 0000 -128 1 1 000 0000 ------------------------ ------------------------ OVERFLOW NO OVERFLOW +127 0 111 1111 -127 1 000 0001 -128 1 000 0000 12
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  Check Your Progress Usingan 8 bit representation perform the following additions:  Add +92 with -85  Add -75 and -53  Add -92 and -39  Add +34 and -65  Add 75+53 You must indicate overflow, if any. 13
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  Queries  For queriesplease send mail to specified email id in the Programme Guide 14