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Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177

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Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177

Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177

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CO-Grand Test -Q10


Warning: count(): Parameter must be an array or an object that implements Countable in /home/customer/www/thegatebook.in/public_html/qa/qa-include/qa-theme-base.php on line 177
+1 vote

Consider the following decimal rational numbers : 

\frac{1}{10}, \frac{2}{10}, \frac{3}{10}, \frac{3}{17}, \frac{53}{64}, \frac{9}{12}, \frac{9}{10}

How many of the above decimal numbers have an exact representation in binary notation?

asked Jun 14, 2020 by gbeditor-2 (72,840 points)
reshown Jun 18, 2020 by gbeditor-2

1 Answer

0 votes
 
Best answer

Whether or not a rational number has a terminating expansion depends on the base.  For example, in base-10 the number 1/2 has a terminating expansion (0.5) while the number 1/3 does not (0.333...). 

In base-2 only rationals with denominators that are powers of 2 (such as 1/2 or 3/16) are terminating(provided that numerator and denominator have no common divisor other than 1). Any rational with a denominator that has a prime factor other than 2 will have an infinite binary expansion. 

So, 

\frac{1}{10} , \frac{2}{10}, \frac{3}{10}, \frac{3}{17}, \frac{9}{10}   can not be represnted finitely in binary.

\frac{53}{64}, \frac{9}{12}   can be represented finitely. 

http://web.cse.ohio-state.edu/~reeves.92/CSE2421au12/SlidesDay33.pdf

https://math.stackexchange.com/questions/413035/why-are-only-fractions-with-denominator-2-and-5-non-repeating

http://www.cas.mcmaster.ca/~qiao/courses/cs4xo3/presentations/SK2.pdf

answered Jun 20, 2020 by deepak-gatebook (226,240 points)
Answer:

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