Computer organization, design, and architecture by Sajjan G Shiva

By Sajjan G Shiva

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Hence, care must be taken to see that shifting does not result in either overflow or inaccuracy. Sign-Magnitude Shifting. When sign-magnitude numbers are shifted, the sign bit is not included in shift operations. 38. 2s Complement Shifting. When 2s complement numbers are shifted right, the sign bit value is copied into the vacant (MSB of the magnitude) bit position on the left, and a 0 is inserted in the vacant LSB position during the left shift. 39 illustrates this. 39 1 001010000 1100101000 Right shift (copy sign bit) 1110010100 Right shift (copy sign bit) 1100101000 Left shift (insert 0) 1001010000 Left shift (insert 0) ß 2007 by Taylor & Francis Group, LLC.

37 illustrates 1s complement addition. It is similar to that in 2s complement representation, except that the carry generated from the sign bit is added to the LSB of the result to complete the addition. 3 Shifting Revisited As we have seen earlier, shifting a binary number left 1 bit is equivalent to multiplying it by 2. Shifting a binary number right 1 bit is equivalent to dividing it by 2. 38 illustrates the effect of shifting an unsigned number. 1000 Inserted Shifting left again: ß 2007 by Taylor & Francis Group, LLC.

1 Binary System In this system, the radix is 2 and the two allowed digits are 0 and 1. BInary digiT is abbreviated as BIT. A typical binary number is shown in the positional notation in the following example. ß 2007 by Taylor & Francis Group, LLC. 2 N = (11010 24 23 22 21 20 16 8 4 2 1 Weights double for each move to left from the binary point . 1101)2 . 2−12−22−32−4 . 1 1 1 1 2 4 8 16 Weights Weights in decimal Weights are halved for each move to right from the binary point In polynomial form, this number is N ¼ 1 Â 2 4 þ 1 Â 23 þ 0 Â 22 þ 1 Â 21 þ 0 Â 20 þ 1 Â 2À 1 þ 1 Â 2À 2 þ 0 Â 2À 3 þ 1 Â 2À 4 1 1 1 ¼ 16 þ 8 þ 0 þ 2 þ 0 þ þ þ 0 þ (decimal) 2 4 16 1 1 1 ¼ 26 þ þ þ (decimal)  2  4 16 13 ¼ 26 16 10 As we can see from the polynomial expansion and summation shown here, the positions containing a 0 do not contribute to the sum.

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