Question

Find three different irrational numbers between the rational numbers $5 over 7$ and

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Solution

7 space right parenthesis space 5.000000 space left parenthesis space 0.714285......... space space space space bottom enclose space space 49 space space space space end enclose space space space space space space space space 10 space space space space space space bottom enclose space space space 7 space space space end enclose space space space space space space space space space 30 space space space space space space space space bottom enclose space space 28 space space end enclose space space space space space space space space space 20 space space space space space space space space bottom enclose space 14 space end enclose space space space space space space space space space space 60 space space space space space space space space bottom enclose space space 56 space space end enclose space space space space space space space space space space space 40 space space space space space space space space space space bottom enclose space space 35 space end enclose space space space space space space space space space space space space space 5 Thus comma space space 5 over 7 equals 0.714258.... equals space 0. top enclose 714285

11 space right parenthesis space 9.0000 space left parenthesis space 0.8181..... space space space space space bottom enclose space space 88 space space end enclose space space space space space space space space 20 space space space space space space space bottom enclose space 11 space space end enclose space space space space space space space space space 90 space space space space space space space space bottom enclose space space 88 space space end enclose space space space space space space space space space space 20 space space space space space space space space bottom enclose space space 11 space space end enclose space space space space space space space space space space space 9 Thus comma space space 9 over 11 equals 0.8181...... equals 0. top enclose 81 space space space space space space

Three different irrational numbers between the rational numbers $5 over 7$ and $9 over 11$ can be taken as

0.75 075007500075000075......
0.7670767000767.......
0.808008000800008.......

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