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Mathematics

Find three different irrational numbers between the rational numbers 57\dfrac{5}{7} and 911\dfrac{9}{11} .

Number System

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Answer

57\dfrac{5}{7} = 0.714285‾0.\overline{714285}

7)0.7142857)50‾7)−49 ‾7)−108−−7‾8)−8301330−28‾13303332013303)−14‾13303333)601330333)−56‾13303333334013303333)−35‾13303333333513303333311111‾\begin{array}{l} \phantom{7)}{\enspace 0.714285} \ 7\overline{\smash{\big)}\enspace 50\qquad\quad} \ \phantom{7)}\underline{-49\space} \ \phantom{7)-}10 \ \phantom{8-}\underline{-7} \ \phantom{8)-8}30 \ \phantom{1330}\underline{-28\enspace} \ \phantom{1330333}20 \ \phantom{13303)}\underline{-14\enspace} \ \phantom{13303333)}60 \ \phantom{1330333)}\underline{-56\enspace} \ \phantom{1330333333}40 \ \phantom{13303333)}\underline{-35\enspace} \ \phantom{13303333333}5 \ \phantom{133033333}\overline{\phantom{11111}} \end{array}

911\dfrac{9}{11} = 0.81‾0.\overline{81}

11)0.818111)90‾11)−88 ‾11)−2011)1−11‾8)−819013301−88‾133033320133033−11‾13303333913303311111‾\begin{array}{l} \phantom{11)}{\enspace 0.8181} \ 11\overline{\smash{\big)}\enspace 90\qquad} \ \phantom{11)}\underline{-88\space} \ \phantom{11)-}20 \ \phantom{11)1}\underline{-11} \ \phantom{8)-81}90 \ \phantom{13301}\underline{-88\enspace} \ \phantom{1330333}20 \ \phantom{133033}\underline{-11\enspace} \ \phantom{13303333}9 \ \phantom{133033}\overline{\phantom{11111}} \end{array}

Irrational numbers between the rational numbers 57\dfrac{5}{7} and 911\dfrac{9}{11}

0.75075007500075000075. . .

0.767076700767000767. . .

0.808008000800008. . .

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