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(Incorrect magnitude 11110) (Erroneous sign bit 0) (Discard carry)

This result shows a positive sum of two negative numbers—clearly incorrect. We can extend the statement we made earlier about permissible magnitudes of sums to include negative as well as positive numbers.

N O T E

A sum of signed binary numbers must be within the range of 2n sum 2n 1 for numbers having n magnitude bits. Otherwise, there will be an overflow into the sign bit.

For an 8-bit signed number in 2’s complement form, the permissible range of sums is 10000000 sum 01111111. In decimal, this range is 128 sum 127.

N O T E

A sum of two positive numbers is always positive. A sum of two negative numbers is always negative. Any 2’s complement addition or subtraction operation that appears to contradict these rules has produced an overflow into the sign bit.

c. 9610 2210 d. 9610 4210

SOLUTION A sign bit overflow is generated if the sum of two positive numbers appears to produce a negative result or the sum of two negative numbers appears to produce a positive result. In other words, overflow occurs if the operand sign bits are both 1 and the sum sign bit is 0 or vice versa. We know this will happen if an 8-bit sum is outside the range ( 128 sum 127).

N O T E

The carry bit generated in 1’s and 2’s complement operations is not the same as an overflow bit. (See Example 6.12, parts c and d.) An overflow is a change in the sign bit, which leads us to believe that the number is opposite in sign from its true value. A carry is the result of an operation carrying beyond the physical limits of an n-bit number. It is similar to the idea of an odometer rolling over from 999999.9 to

1 000000.0. There are not enough places to hold the new number, so it goes back to the beginning and starts over.

6.4 Hexadecimal Arithmetic

(This section may be omitted without loss of continuity.)

The main reason to be familiar with addition and subtraction in the hexadecimal system is that it is useful for calculations related to microcomputer and memory systems.

SECTION 6.4 REVIEW PROBLEM

6.7Perform the following hexadecimal calculations:

a.A25F 74A2

b.7380 5FFF

6.5Numeric and Alphanumeric Codes

BCD Codes

K E Y T E R M

Binary-coded decimal (BCD). A code that represents each digit of a decimal

number by a binary value.

BCD stands for binary-coded decimal. As the name implies, BCD is a system of writing decimal numbers with binary digits. There is more than one way to do this, as BCD is a code, not a positional number system. That is, the various positions of the bits do not necessarily represent increasing powers of a specified number base.

Two commonly used BCD codes are 8421 code, where the bits for each decimal digit are weighted, and Excess-3 code, where each decimal digit is represented by a binary number that is 3 larger than the true binary value of the digit.

Table 6.2 Decimal Digits and

SOLUTION The binary value of 498710 can be calculated by repeated division by 2:

498710 1 0011 0111 10112