Page 37 - Computer Science Class 11 With Functions
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Digital computers are based on the binary number system — a system that uses only two digits, 1 and 0, expressed
by the voltage levels high and low, respectively. As the binary system uses only two digits, it is also called a base-2
system. Some examples of binary numbers are (1101) , (101.011) , (1101010) , and (10001.001) . The binary number
2.
2
2
2
system is also a positional number system. So, the value of a digit in the binary number system depends on its position,
expressed as a power of 2 because the binary system is a base-2 system. Table 2.3 shows the decimal and its binary
equivalent. Recall that in the case of the decimal system, the integer and the fraction parts are separated by a decimal
point. Similarly, in the case of the binary system, the integer and the fraction parts are separated by a binary point. As
discussed above, while representing the whole numbers in the binary system, the values of digits positions increase
in powers of the base (2) from right to left. Similarly, for the fractional part, the values of digits decrease in powers of
the base to the right of the binary point. Table 2.4 shows the digit's place value, face value, and value represented by
the binary number: (1101.101) .
2
Binary Number: (1101.101) 2
Table 2.4: Digit's place value, face value, value represented by (1352) .
10
1
1
1
-1
-2
2
1
0
-3
3
Digit's place value 2 (=8) 2 (=4) 2 (=2) 2 (=1) 2 (= ) 2 (= ) 2 (= )
2 4 8
Binary digit 1 1 0 1 . 1 0 1
Face value 1 1 0 1 . 1 0 1
Value represented 1 × 2 3 1 × 2 2 0 × 2 1 1 × 2 0 1 × 2 -1 0 × 2 -2 1 × 2 -3
by digit
Now we are ready to compute the value of (1101.101) in decimal number system as follows:
2
0
1
-2
-1
3
2
(1101.101) = 1 × 2 + 1 × 2 + 0 × 2 + 1 × 2 + 1 × 2 + 0 × 2 + 1 × 2 -3
2
= 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 + 1 × 0.5 + 0 × 0.25 + 1 × 0.125
= 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125
= 13.625
As the leftmost bit has the highest positional value, it is called the Most Significant Bit (MSB). Similarly, the rightmost
bit has the least positional value and is called the Least Significant Bit (LSB).
Write the 3-bit binary equivalent of the following decimal digits:
(i) 7
(ii) 2
(iii) 9
2.3 Octal Number System (Base-8)
The octal number system (also called base 8 number system), comprises eight distinct digits: 0, 1, 2, 3, 4, 5, 6
and 7. Examples of octal numbers are (112.74) , (513) , and (714.007) . The octal number system provides a
8 8 8
compact representation of binary numbers. A sequence of three bits can take a value from (000) to (111) . In the
2 2
octal system (000) and (111) are expressed as (0) and (7) , respectively. So, an octal digit can be used to represent
2 2 8 8
a sequence of three binary digits (or 3 bits). Table 2.5 below shows the decimal and binary equivalent of each
octal digit.
Number Systems and Encoding Schemes 35

