Motorola M68000 User's Guide download pdf (Page 25)

Humber 5ystems 11

Representing Negative Values

So far in our discussion of number representations, we have only

been dealing with positive numbers. A method of representing negative

numbers in the computer must be introduced. You have already learned

that numbers are represented internally by binary digits. We must devise

a way of including the conventional minus sign used to indicate a

negative number, with the number itself. But how does a minus sign

translate into binary? Since numbers are either positive or negative, we can

indicate this fact by using a single binary digit. A negative number can be

indicated by using an “extra” bit rather than a minus sign. This may not work

as well on paper, but it is essential for a computer’s internal representations.

Numeric quantitites are normally restricted to fixed sizes—a single

byte, a word, or some multiple number of words or bytes. It is not

practical to append an extra “sign” bit to a fixed unit of storage such

as a byte: the central processing unit (CPU) normally is restricted to

manipulating integral numbers of bytes, and this extra bit would force

the use of an extra complete byte. The solution is to sacrifice one of the

bits of our number for use as the sign bit. The size of the largest number

we can represent is reduced, but we can now represent the same number

of positive numbers as negative numbers.

By convention, if a number is negative we indicate this fact by in

cluding a sign bit equal to one. The sign bit is normally the leftmost, or

high-order, bit of the number. The simplest technique would be merely

to indicate the magnitude of the number in the remaining bits, setting the

sign bit to either one or zero to indicate a negative or positive number.

This representation, called sign magnitude, has been used on older com

puters. It has a number of disadvantages, the most prominent to a pro

grammer being the fact that both a positive and negative zero exist—both

100000002 and OOOOOOOO2 are zero values for a single byte number. With

out going into additional detail, suffice it to say that a better method is

needed.

Virtually all modern computers, including microprocessors, use a rep

resentation called two’s complement. The sign bit is still used to indicate

whether a number is positive or negative, but the remaining bits do not

directly indicate the magnitude of the number if it is a negative number.

To represent a negative number in two’s complement, we first form the

one’s complement of the number in its binary form. The one’s comple

ment is merely the number with all the one bits converted to zeros, and

all the zero bits converted to ones. The one’s complement of 011000112

is 100111002. So far this is quite simple. We are almost finished. To get

the two’s complement we add one to the one’s complement. We perform

this addition just as we have done in the previous examples. To complete

the conversion of our example, we get:

1 2 ... 20 21 22 23 24 25 26 27 28 29 30 ... 255 256

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Motorola M68000 User's Guide Page 25

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