272 DIGITAL BUILDING BLOCKS AND COMPUTER SYSTEMS
101 + 6 × 100 + 4 × 10 −^1 + 7 × 10 −^2. The radix or base is 10, whereas the most significant digit
or bit (MSB) is 2, the least significant digit or bit (LSB) is 7; the number of integer bits (digits)
is 4, and the number of fractional bits (digits) is 2.
Thebinary number systemhas a base of 2 with two distinct digits (bits), 1 and 0. A binary
number is expressed as a string of 0s and 1s, and a binary point if a fraction exists. To convert
the binary to the decimal system, the binary number is expressed in the polynomial form and the
resulting polynomial is evaluated by using the decimal-system addition. For example,
( 101101. 101 )^2 = 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 + 1 × 2 −^1
+ 0 × 2 −^2 + 1 × 2 −^3
= 32 + 0 + 8 + 4 + 0 + 1 + 0. 5 + 0 + 0. 125 =( 45. 625 ) 10
For digital processing it is also often necessary to convert a decimal number into its equivalent
binary number. This is accomplished by using the following steps.- Repeatedlydividetheinteger partof the decimal number by 2. Use the remainder after
each division to form the equivalent binary number. Continue this process until a zero
quotient is obtained. With the first remainder being the least significant bit, form the binary
number by using the remainder after each division. - Repeatedlymultiplythedecimal fractionby 2. If 0 or 1 appears to the left of the decimal
point of the product as a result of this multiplication, then adda0or1tothebinary function.
Continue this process until the fractional part of the product is zero or the desired number
of binary bits is reached.
For example, the decimal number (75) 10 is converted into its binary equivalent:
Quotient Remainder
75 ÷2 = 37 1 LSB
37 ÷2 = 18 1
18 ÷2= 9 0
9 ÷2= 4 1
4 ÷2= 2 0
2 ÷2= 1 0
1 ÷2 = 0 1 MSB
Stop
Thus, the binary number for (75) 10 is given by:
( 1001011 ) 2
↑↑
MSB LSB
Let us now convert the fractional decimal number (0.4375) 10 into its binary equivalent:
↓MSB of binary fraction
0.4375×2 = 0.8750
0.8750×2 = 1.7500
0.7500×2 = 1.5000
0.5000×2 = 1.0000
Stop
↑LSB of binary fraction
Thus, (0.4375) 10 =(.0111) 2.
For representing binary data, octal and hexadecimal numbers are used. Theoctal number
systemis a base-8 system and therefore has 8 distinct digits{0, 1, 2, 3, 4, 5, 6, 7}. It is expressed