Page 94 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

Binary Arithmetic 75

'Unfortunately, all of the previous examples will return incorrect results if
a larger value is subtracted from a smaller value; that is, for these techniques
to work, the final result must be greater than or equal to zero. In the case of
unsigned binary numbers, the reason is clear because, by definition, an
unsigned binary number can only be used to represent a positive value, but
subtracting a larger value from a smaller value results in a negative value. It
would obviously be somewhat inconvenient if computers could only be used to
generate positive values, so we need some way to represent negative numbers.

Signed Binary Numbers
Signed binary numbers can be used to represent both positive and negative
values, and they do this in a rather cunning way. In standard decimal
arithmetic, negative numbers are typically represented in a form known
as ~ign-magnitude,~ which means prefixing values with plus or minus signs,
For example, values of plus and minus twenty-seven would be shown as +27
and -27, respectively. However, for reasons of efficiency, computers rarely
employ the sign-magnitude form, and instead use the signed binary format, in
which the most significant bit is also called the sign bit (Figure 8-8).
The least significant bits continue to represent the same positive quantities
as for unsigned binary numbers, but the sign bit is used to represent a negative
quantity. In the case of a signed 8-bit number, a 1 in the sign bit represents -27
= -128, and the rernaining bits are used to represent positive values in the range
O,, chrough +127,,. Thus, when the value represented by the sign bit is corn-
bined with the values represented by the remaining bits, an 8-bit signed binary
number can be used to represent values in the range -l28,, through +1271,.
To illustrate the differences between the sign-magnitude and signed binary
formats, first consider a positive sign-magnitude decimal number and its negative
equivalent: for example, +27 and -27. As we see, the digits are identical for
both cases and only the sign changes. Now consider the same values represented
as signed binary numbers (Figure 8-9).
In this case, the bit patterns of the two binary numbers are very different.
This is because the sign bit represents an actual quantity (-12810) rather than
a simple plus or minus; rhus, the signed equivalent of -2710 is formed by com-
bining -128,, with ~101,~. Now pay attention because this is the clever part:

3 Sometimes written as sign+rnugnitude.