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Chapter 4: Problem 18

Write each of the following integers in two's complement representation. Here the results are eight-bit patterns. a) 15 b) \(-15\) c) 100 d) \(-65\) e) 127 f) \(-128\)

Short Answer

Expert verified
a) 00001111, b) 11110001, c) 01100100, d) 10111111, e) 01111111, f) 10000000

Step by step solution

01

Convert to Binary

The first step is converting each integer to binary form. To do this, each integer needs to be divided by 2 and the remainder to be recorded. The process continues with the quotient of each division again divided by 2, until the quotient is zero. The binary representation of the number is then obtained by reading the sequence of remainders backwards.
02

Two's Complement for Positive Numbers

For positive numbers, the two's complement is simply the binary representation. However, it must be an eight-bit pattern. If the binary representation of a positive number has fewer than eight digits, it needs to be left padded with zeros.
03

Two's Complement for Negative Numbers

For negative numbers, the two's complement process is different. The binary representation of the negative number is obtained as in Step 1, but then the 'One's Complement' (flipping all the bits) needs to be obtained, after which '1' is added to the result. This gives the two's complement of the negative number.
04

Answers

a) For 15, binary representation is 00001111, so the two's complement is 00001111. b) For -15, binary representation is 00001111, one's complement is 11110000, so the two鈥檚 complement is 11110001. c) For 100, binary representation is 01100100, so the two's complement is 01100100. d) For -65, binary representation is 01000001, one's complement is 10111110, so the two鈥檚 complement is 10111111. e) For 127, binary representation is 01111111, so the two's complement is 01111111. f) For -128, binary representation is 10000000, so the two's complement of negative 128 is also 10000000 (as it is a boundary case with a leading 1 already present).

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