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Unformatted text preview: ½ Ü ½ Ü ¾ Û Û Û Ü¼ µ ¾Ì Û ´ ÜÛ ½ Ü ¾ Û Ü¼ µ where in the expression on the left-hand side, we have made additional copies of bit ÜÛ ½ . The proof follows by induction on . That is, if we can prove that sign-extending by one bit preserves the numeric value, then this property will hold when sign-extending by an arbitrary number of bits. Thus, the task reduces to proving that ¾Ì Û·½ ´ ÜÛ ½ Ü ½ Ü ¾ Û Û Ü¼ µ ¾Ì Û ´ ÜÛ ½ Ü ¾ Û Ü¼ µ Expanding the left-hand expression with Equation 2.2 gives ¾Ì Û·½ ´ ÜÛ ½ Ü ½ Ü ¾ Û Û Ü¼ µ Ü Ü Ü Ü Û Û ½¾ Û ½ ¼ · ܾ Û Û Û ½¾ Û · Ü ½¾ ½ · Û ¾ ¼ ܾ Û ½ ´¾ Û ¾ Û ½ µ · ¾ ¼ Û ¾ ¼ ܾ Û ½¾ Û ½ · Û Ü¾ ܼ µ ¾Ì Û ´ ÜÛ ½ Ü ¾ Û The key property we exploit is that ¾Û · ¾Û ½ ¾Û ½. Thus, the combined effect of adding a bit of weight ¾Û and of converting the bit having weight ¾Û ½ to be one with weight ¾Û ½ is to preserve the original numeric value. One point worth making is that the relat...
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