The n?bit two’s complement integer N is written an?1, an?2, . . . a1, a0. Prove that (in two's complement notation) the representation of a signed binary number in n + 1 bits may be derived from its representation in n bits by repeating the leftmost bit. For example, if n = ?12 = 10100 in five bits, n = ?12 = 110100 in six bits.
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In n bits the positive number N is represented by an?1, an?2, . . . a 1, a0. We can extend this to n+1 bits by appending a 0 to the left without changing its value; that is 0, an?1, an?2, . . . a 1, a0.
Now consider the value ?N in n bits. This is represented as 2n ? N. If we extend this to n + 1 bits, it becomes 2n+1 ? N or 2n + 2n ? N. This is, of course, the original negative representation with a leading 1 to the left. Consequently, a positive number is extended by appending a 0, and a negative number by adding a 1; that is, by extending the sign bit.
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