Page 36 - Computer Science Class 11 Without Functions
P. 36

We have seen that while representing the whole numbers in the positional system, the values of digits increase from
        right to left in the powers of the base. Similarly, while representing the fractional part, the values of digits decrease in
        powers of the base to the right of the radix point. Note that the rightmost digit remains the least significant digit. In the
        decimal system, the radix point is called the decimal point. Thus, in the decimal system, the values of digits increase in
        powers of ten on the left and decrease in powers of ten on the right of the decimal point. For example, consider the
        number 56.91. having 56 as the integer part and 0.91 as the fraction part. The value represented by different digits in
        the number 56.91 is described below:
                                       1
        ●  Digit 5 represents 5 tens (5 × 10  = 5 × 10)
                                        0
        ●  Digit 6 represents 6 units (6 × 10  = 6 × 1)
        ●  Digit 9 represents 9 tenths  9 × 10  -1    = 9 ×   1
                                                 10
                                             -2
        ●  Digit 1 represents 1 hundredth  1 × 10  = 1 ×   1
                                                    100
        The value of the number (56.91)  can be calculated as follows:
                                     10
               (56.91) 10
                           0
                   1
            = 5 × 10  + 6 × 10 + 9 × 10 + 1 × 10 -2
                                   -1

            = 5 × 10 + 6 × 1 + 9 ×   1   + 1 ×   1
                               10      100
        Table 2.2 shows the digit's place value, face value, and value represented by the decimal number (56.91) .
                                                                                                     10
                               Table 2.2: Digit's place value, face value, value represented by (56.91) .
                                                                                        10
         Digit's place value    ten (10)         unit (1)        radix point      tenth   1     hundredth   1
                                                                                        10                 100
         Digit                     5                6                ,                9                1
         Face value                5                6                .                9                1

         Value represented      5 × 10 1         6 × 10 0                          9 × 10 –1        1 × 10 –2
         by digit




                 A number is the sum of products of its digits' place values and corresponding face values.




        2.2 Binary Number System (Base-2)

                                Table 2.3: Binary Equivalent value of Digits of Decimal number system
                      Decimal Number                          Binary Equivalent

                             0          (0)  = 0
                                           2
                             1          (1)  = 1
                                           2
                                                   1
                                                          0
                             2          (10)  = 1 × 2 + 0 × 2   = 2 + 0 = 2
                                            2
                                                   1
                                                          0
                             3          (11)  = 1 × 2 + 1 × 2  = 2 + 1 = 3
                                            2
                                                    2
                                                           1
                                                                  0
                             4          (100)  = 1 × 2  + 0 × 2 + 0 × 2  = 4 + 0 + 0 = 4
                                             2
                                                    2
                                                                  0
                                                           1
                             5          (101)  = 1 × 2  + 0 × 2 + 1 × 2  = 4 + 0 + 1 = 5
                                             2
                                                                  0
                                                           1
                                                    2
                             6          (110)  = 1 × 2  + 1 × 2 + 0 × 2  = 4 + 2 + 0 = 6
                                             2
                                                                  0
                                                           1
                             7          (111)  = 1 × 2  + 1 × 2 + 1 × 2  = 4 + 2 + 1  = 7
                                                    2
                                             2
                                                                           0
                                                                    1
                                                            2
                                                     3
                             8          (1000)  = 1 × 2  + 0 × 2  + 0 × 2 + 0 × 2  = 8 + 0 + 0 + 0 = 8
                                              2
                                                                           0
                                                                    1
                                                            2
                                                     3
                             9          (1001)  = 1 × 2  + 0 × 2  + 0 × 2 + 1 × 2  = 8 + 0 + 0 + 1 = 9
                                              2
          34   Touchpad Computer Science-XI
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