Complement

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1's and 2's complements

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  • Given any positive integer n and any nonnegative integer a, if we divide a by n, we get an integer quotient q and an integer remainder r. In modular arithmetic we are only interested in the remainder (or residue) after division by some modulus, and results with the same remainder are regarded as equivalent. Two integers a and b are said to be congruent modulo n, if (a mod n ) = ( b mod n ) .
  • Complement

    1. 1. One’s and Two’s Complement Numbers and Arithmetic Operations 1 Digital electronics 13IDP14
    2. 2. 2 OutlineOutline  Number of binary digits (bits) used in computers for arithmetic operations  Positive and negative number representation using bits  1’s complement and 2’s complement  Arithmetic operations in 2’s complement  Carry and overflow concepts  See module 2a notes for more examples
    3. 3. Bits Used in Computers for Numbers and Implications  4, 8, 16, 32, 64  Using 4 bits we can only represent 24 = 16 numbers -8, -7, -6, -5, …-1, 0, 1, … 7  Using 8 bits we can only represent 28 = 256 numbers -128, -127, …-1, 0, 1, … 127  Since we can represent only a fixed number of positive and negative numbers, we may get wrong results when we compute arithmetic operations  We must determine when the results are wrong
    4. 4. One’s and Two’s Complement 0111 0110 0101 0100 0011 0010 0001 0000 1111 1110 1101 1100 1011 1010 1001 1000 +7 +6 +5 +4 +3 +2 +1 +0 -0 -1 -2 -3 -4 -5 -6 -7 Cryptography -Part -I 4 0111 0110 0101 0100 0011 0010 0001 0000 1111 1110 1101 1100 1011 1010 1001 1000 +7 +6 +5 +4 +3 +2 +1 +0 -1 -2 -3 -4 -5 -6 -7 -8 One’s Complement Two’s Complement
    5. 5. Two’s Complement Number Representation and Value (1)  MSB represents negative number with a value.  Example: 8-bit number say  1101 0011  Value in decimal  = -1*27 + 1*26 + 0*25 + 1*24 + 0*23 +0*22 + 1*21 + 1  =-128 +64+0+16+0+0+2+1  =-128+83  =-45 Cryptography -Part -I 5
    6. 6. Two’s Complement Number Representation and Value (2)  Example: 1111 1111  Decimal Value  = -1*27 + 1*26 + 1*25 + 1*24 + 1*23 +1*22 + 1*21 + 1  = -128+64+32+16+8+4+2+1 =-128+127 = -1  Example: 1000 0000 Value = -128  Example: 0111 1111 Value = 127 Cryptography -Part -I 6
    7. 7. Finding a Negative of a number  Assume 1’s complement  Number 8-bit: 0000 1111; value =15  Given 15 determine -15  Change all 0 to 1 and 1 to zero  Result: 1111 0000 = -127+64+32+16 =-127+112 = -15  Assume 2’s complement  Number 8-bit: 0000 1111; value =15  Given 15 determine -15  Change all 0 to 1 and 1 to zero then add 1  Result: 1111 0001 = -128+64+32+16+1 =-128+113 = -15 7
    8. 8. Arithmetic Operations in Two’s Complement (1)  Example:Find 7 – 5 assuming 4-bit numbers 7 = 0111 ; -5 =1011 in 2’s complement 0111 1011 ----- 10010 Carry in MSB 1, Carry out in MSB 1 Result ok value 2
    9. 9. Arithmetic Operations in Two’s Complement (2)  Example:Find 7 + 5 assuming 4-bit numbers 7 = 0111 ; 5 =0101 in 2’s complement 0111 0101 ----- 1100 Carry in MSB 1, Carry out in MSB 0 Result wrong; value -4. We can not represent real value 12. This is overflow. 9
    10. 10. Arithmetic Operations in Two’s Complement (2)  Example:Find 7 + 5 assuming 4-bit numbers 7 = 0111 ; 5 =0101 in 2’s complement 0111 0101 ----- 1100 Carry in MSB 1, Carry out in MSB 0 Result wrong; value -4. We can not represent real value 12. This is overflow. 9

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