of Microprocessors

648 MUSICAL ApPLICATIONS OF MICROPROCESSORS
rule in which raw numbers are fed in and raw answers come out. It is the
operator's responsibility to place the decimal point. Binary integers have the
binary point to the right of the magnitude bits. Binary fractions have the
binary point to the left of the magnitude bits. For unsigned numbers, this
means to the left of all bits, while signed numbers have the point between
the sign bit and the rest of the bits. Mixed numbers can have the binary point
anywhere in between.
A good way to think of mixed binary numbers is to consider their
resolution, which is the weight of the least significant bit, and their range,
which is the largest number that can be represented. We will also be referring
to their integer part, which is the string of bits to the left of the point, and
their fractional part, which is the string to the right. Thus, a signed number
with the binary points between bits 3 and 4 as illustrated in Fig. 18-2 has a
resolution of 1/16 or 0.0625 and a range of 127/16 or 7.9375 (which can be
called 8 for convenience). The integer part is bits 4-6 and the fractional part
is bits 0-3. Integers, of course, have a resolution of unity and a range
dependent on word size, while fractions have a range just short of unity and a
resolution determined by word size.
The last property of binary numbers is word size or simply the number
of bits allocated to the representation of the number. Arithmetic word sizes
are usually chosen to be integer multiples of the computer's word size, or in
the case of a sophisticated machine, a multiple of the smallest directly
addressable dara element, usually an 8-bit byte. When the word size of a
number is greater than the word size of the machine, the number is said to be
a double-precision or multiple-precision number. Thus, with the 6502 example
machine, 16-bit numbers are double-precision quantities, 24 birs is tripleprecision,
etc.
Before the advent of microprocessors, visualization of multiple precision
numbers on paper and in memory was a simple, unambiguous task. A
double-precision number, for example, would be visualized as consisting of a
high order, more significant parr and a low order, less significant parr. When
written on paper, it is natural and even necessary to write the high order parr
to the left of the low order part. If one listed a portion of memory containing
the number, it would be reasonable to expect it to look the same as it did on
the notepad. However, many of the most popular microprocessors (6502
included) handle double-precision unsigned numbers (namely memory addresses)
low byte first! Thus, the address bytes of instructions and indirect
address pointers all appear on a memory dump backward. Furrhermore, since
address arithmetic is done one byte at a time, the programmer must be aware
of this to get these operations done right.
The question, then, is: Should all numbers be stored and manipulated
backward or just addresses) The author has tried it both ways and found
neither to be completely satisfactory on the 6502. However, in signalprocessing
work, it is recommended that all numbers except addresses be