Mathematics

Problem

Consider the fraction

\dfrac{65}{84}

and the two real numbers

0.\overline{16}

and

\dfrac{\pi}{2}

. a. (i) Without actual division, determine whether the decimal expansion of

\dfrac{65}{84}

terminates or is non-terminating recurring. Show your reasoning clearly. b. (ii) Using long division, write the decimal expansion of

\dfrac{65}{84}

correct to six decimal places. c. (iii) For each of the three numbers

\dfrac{65}{84},\;0.\overline{16},\;\dfrac{\pi}{2}

, state whether it is rational or irrational and then name the smallest set out of

\mathbb N,\;\mathbb W,\;\mathbb Z,\;\mathbb Q,\;\text{irrational numbers},\;\mathbb R

that contains it, giving justification in each case.