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Question

Find three different irrational numbers between the rational numbers 59 and 57.
  1. 0.6020020002...
  2. 0.6010010001...
  3. 0.6020020002
  4. 0.6030030003...

A
0.6010010001...
B
0.6020020002
C
0.6020020002...
D
0.6030030003...
Solution
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59=0.555....=0.¯5 and 57=0.7142857142857.....=0.¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯714285
Here, the two decimal expansions are non-terminating recurring.
Hence, 59 and 57 are two rational numbers.

We know, between any two rational numbers, there are infinitely many irrational numbers.
An irrational number has non-terminating non-recurring decimal expansions.
Here, 0.6010010001..., 0.5020020002... and 0.62030030003... are non-terminating non-recurring,
while 0.6020020002 is terminating.
Hence, the required three irrational numbers areHere, 0.6010010001..., 0.5020020002... and 0.62030030003....
Thus, options A, B and C are the correct answers.

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Find three different irrational numbers between the rational numbers 59 and 57.
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