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32 CHAPTER 2 / NUMBER SYSTEMS, BINARY ARITHMETIC, AND CODES

be readily understood by humans. A minimum number of identifiable characters (say 1 and
0, or true and false) is not practical or desirable for direct human use. If this is difficult
to understand, imagine trying to complete a tax form in binary or in any number system
other than decimal. On the other hand, use of a computer for this purpose would not only
be practical but, in many cases, highly desirable.

2.2 POSITIONAL AND POLYNOMIAL REPRESENTATIONS

The positional form of a number is a set of side-by-side (juxtaposed) digits given generally
in fixed-point form as

Radix
MSD Point LSD
I I I
N r = (a n-\-

Integer Fraction
where the radix (or base), r, is the total number of digits in the number system, and a is
a digit in the set defined for radix r. Here, the radix point separates n integer digits on the
left from m fraction digits on the right. Notice that a n-\ is the most significant (highest
order) digit called MSD, and that a_ OT is the least significant (lowest order) digit denoted
by LSD.
The value of the number in Eq. (2.1) is given in polynomial form by

n-\
tt l
2
N r=^ fl,-r' = (a n-ir ~ H ----- h a 2r + a,r' + aor° + a-\r~ l
i=—m
m
2
+ a- 2r~ + "'+a- mr- ) r, (2.2)
where a t is the digit in the z'th position with a weight r' .
Applications of Eqs. (2.1) and (2.2) follow directly. For the decimal system r = 10,
indicating that there are 10 distinguishable characters recognized as decimal numerals
0, 1, 2, . . . , r — \(= 9). Examples of the positional and polynomial representations for the
decimal system are

= 3017.528

and
n-l
#,<,= £) <// iff
(=-3
1
2
2
3
= 3 x 10 + 0 x 10 + 1 x 10' + 7 x 10° + 5 x 10' + 2 x 10" + 8 x 10
= 3000 + 10 + 7 + 0.5 + 0.02 + 0.008,