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Signed Numbers

Signed Numbers. Up till now we've been concentrating on unsigned numbers. In real life we have to represent signed numbers ( like: -12, -45, 78). The difference between signed and unsigned numbers is the sign. A scheme is needed to represent the sign as part of the binary representation.

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Signed Numbers

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  1. Signed Numbers • Up till now we've been concentrating on unsigned numbers. In real life we have to represent signed numbers ( like: -12, -45, 78). • The difference between signed and unsigned numbers is the sign. A scheme is needed to represent the sign as part of the binary representation. • There are a number of schemes for representing signed numbers in binary format. • sign-magnitude representation • the twos-complement representation.

  2. Sign-Magnitude Representation In this representation, the leftmost bit of a binary code represents the sign of the value: • 0 for positive, • 1 for negative; the remaining bits represent the numeric value.

  3. Sign-Magnitude Representation To Compute negative values using Sign/Magnitude (signmag) representation. Begin with the binary representation of the positive value, then flip the leftmost zero bit.  

  4. Sign-Magnitude Representation Ex 1. Find the signmag representation of -610 Step1: find binary representation using 8 bits 610 = 000001102 Step2: if the number is a negative number flip left most bit 10000110 So: -610 = 100001102 (in 8-bit sign/magnitude form)

  5. Sign-Magnitude Representation Ex 2. Find the signmag representation of -3610 Step 1: find binary representation using 8 bits 3610 = 001001002 Step 2: if the number is a negative number flip left most bit 10100100 So: -3610 = 101001002 (in 8-bit sign/magnitude form)

  6. Sign-Magnitude Representation Ex 3. Find the signmag representation of 7010 Step 1: find binary representation using 8 bits 7010 = 010001102 Step 2: if the number is a negative number flip left most bit 01000110 (no flipping, since it is +ve) So: 7010 = 010001102 (in 8-bit sign/magnitude form)

  7. Sign-Magnitude Representation What is this signmag number? 100000002 The machine will think of it as - 0 , which is a non valid value.

  8. Two’s Complement Representation • Another scheme to represent negative numbers • The leftmost bit serves as a sign bit: • 0 for positive numbers, • 1 for negative numbers.  

  9. Two’s Complement Representation To Compute negative values using two’s Complement representation, begin with the binary representation of the positive value, complement (flip each bit if it is 0 make it 1 and visa versa) the entire positive number, and then add one.

  10. Two’s Complement Representation Ex. Find the two’s complement representation of –610 Step1: find binary representation in 8 bits 610 = 000001102

  11. Two’s Complement Representation Step 2: Complement the entire positive number, and then add one 00000110 (complemented) -> 11111001 (add one) -> + 1 11111010 So: -610 = 111110102 (in 2's complement form, using any of above methods)

  12. Two’s Complement Representation Alternative method for step 2  Scan binary representation from right too left, find first one bit, from low-order (right) end, and complement the remaining pattern to the left. 00000110 (left complemented) --> 11111010

  13. Two’s Complement Representation Ex 2: Find the Two’s Complement of -7610 Step 1: Find the 8 bit binary representation of the positive value. 7610 = 010011002

  14. Two’s Complement Representation Step 2: Find first one bit, from low-order (right) end, and complement the pattern to the left. 01001100 (left complemented) -> 10110100 So: -7610 = 101101002 (in 2's complement form, using any of above methods)

  15. Two’s Complement Representation Ex 3: Find the Two’s Complement of 7210 Step 1: Find the 8 bit binary representation of the positive value. 7210 = 010010002

  16. Two’s Complement Representation Step 2: Since number is positive do nothing. So: 7210 = 010010002 (in 2's complement form, using any of above methods)

  17. Two’s Complement Representation The most important characteristic of the two’s-complement system is that the binary codes can be added and subtracted as if they were unsigned binary numbers, without regard to the signs of the numbers they actually represent.

  18. Two’s Complement Representation For example, to add +4 and -3, we simply add the corresponding binary codes, 0100 and 1101: 0100 (+4) +1101(-3) 0001 (+1) A carry from the leftmost column has been ignored. The result, 0001, is the code for +1, the sum of +4 and -3.

  19. Twos Complement Representation Likewise, to subtract +7 from +3, we add the code for -7, 1001, to that of +3, 0011: 0011 (+3) +1001(-7) 1100 (-4) The result, 1100, is the code for -4, the result of subtracting +7 from +3. 

  20. Two’s Complement Representation Benefits of Twos Complements: • addition and subtraction simplified in the two’s-complement system, • In 8 bits, -0 has been eliminated, replaced by -128, for which there is no corresponding positive number.

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