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# CO-Grand Test -Q10

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+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?

reshown Jun 18, 2020

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 (226,240 points)