a The number 1213 0923076923076923 has no exact represen

(a) The number 12/13 = 0.923076923076923 . . . has no exact representation in any decimal floating point system ( = 10) with finite precision t. Is there a finite floating point system (i.e. some finite integer base and precision t) in which this number does have an exact representation? If yes then describe such a system. (b) Answer the same question for the irrational number e = exp(1).

Solution

a) The number 12/13 can be represented in the base-13 representation where

0,....,9 -> represents 0--> 9

A->10, B->11,C->12,D->13

The number can be represented as 0.D

According to the number system rule, it will be equal to D * (1/13) = 12/13

Hence we can represent the number 12/13 with complete precision in the base 13 representation

The value of e cannot be write in the form of p/q since it is just the summation of 1/n!, hence we can\'t write the number e in terms of some base with finite precision