Page 409 - Introduction to Microcontrollers Architecture, Programming, and Interfacing of The Motorola 68HC12

386 Appendix 1 Number Representations and Binary Arithmetic

PROBLEMS

1. Find the hexadecimal equivalents of the 8-bit two's-complement representations of
-44 and-121.

2. Explain why the 16's-complement technique works when used for calculations such as
Problem 1.

3. Suppose that you were going to add a 16-bit two's-complement representation with an
8-bit one. How would you change the 8-bit representation so that the 16-bit result would
be correct? This process is sometimes called sign extension.

4. Give a simple condition for signed overflow when two's-complement representations
are subtracted,

5. One textbook reason for preferring the two's-complement representation of integers
over the signed-magnitude representation is that the logic design of a device that adds and
subtracts numbers is simpler. For example, suppose that M and N have m-bit two's-
complement representations x and y. To subtract M from N, one can take the two's-
complement of x and then add it to y, presumably simpler from the logic design
viewpoint than dealing with signed-magnitudes. Does this always work? Try it with m =
8 N equal to -1, and M equal to -128. What is the condition for overflow? Does this
work when N and M are interpreted as unsigned numbers? Interpret.

6. Suppose that we add two m-bit representations x and y, where x is the unsigned
representation of M and y is the two's-complement representation of N. Will the answer,
truncated to m bits, be correct in any sense? Explain.