Page 62 - Engineering Digital Design

2.3 UNSIGNED BINARY NUMBER SYSTEM 33

where d/ is the decimal digit in the z'th position. Exclusive of possible leading and trailing
zeros, the MSD and LSD for this number are 3 and 8, respectively. This number could have
been written in a form such as N\Q = 03017.52800 without altering its value but implying
greater accuracy of the fraction portion.

2.3 UNSIGNED BINARY NUMBER SYSTEM

Applying Eqs. (2.1) and (2.2) to the binary system requires that r = 2, indicating that there
are two distinguishable characters, typically 0 and (r — 1) = 1, that are used. In positional
representation these characters (numbers) are called binary digits or bits. Examples of the
positional and polynomial notations for a binary number are

= (b n-i • • • b 3b 2b\b 0 • b-\b- 2b-3 • • • b- m) 2
= 1 0 1 1 0 1 . 1 0 1 2
t t
MSB LSB

and

5
2
4
3
= 1 x 2 + 0 x 2 + 1 x 2 + 1 x 2 + 0 x 2 1
2
+ 1 x 2° + 1 x 2~' + 0 x 2~ + 1 x 2~ 3
= 32 + 8 + 4+1+0.5 + 0.125
= 45.625,0,
where n = 6 and m = 3, and £, is the bit in the z'th position. Thus, the bit positions are
weighted,... 16, 8, 4, 2, 1, 1/2, 1/4, 1/8,.. .for any number consisting of integer and fraction
portions. Binary numbers, so represented, are sometimes referred to as natural binary. In
positional representation, the bit on the extreme left and extreme right are called the MSB
(most significant bit) and LSB (least significant bit), respectively. Notice that by obtaining
the value of a binary number, a conversion from binary to decimal has been performed. The
subject of radix (base) conversion will be dealt with more extensively in a later section.
For reference purposes, Table 2.1 provides the binary-to-decimal conversion for two-,
three-, four-, five-, and six-bit binary. The six-bit binary column is only halfway completed
for brevity.
In the natural binary system the number of bits in a unit of data is commonly assigned a
name. Examples are:

4 data-bit unit — nibble (or half byte)
8 data-bit unit — byte
16 data-bit unit — two bytes (or half word)
32 data-bit unit — word (or four bytes)
64 data-bit unit — double-word