Find an irrational number between 5/7 and 7/9 . How many more there may be? The answer is 5/3 root of 5 . Give me step by step. Please help me.
Answers
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ANSWER:
An irrational number between57and79 is (a) 0.75 (b) √6.
Step-by-step explanation:
57=0.714285714285;714285;714285⋯
where the periodic bits 714285 start immediately after the decimal point. similarly,
79=0.7777777777777777777777⋯
has period made of just one digit 7, and
72100=0.72=0.72000000000000⋯
has period 0 which keeps repeating starting from the 3rd digit after the decimal point. Please observe that
57<72100<79,
but 72100is rational; how to change into an irrational number? Very simple: keep digits 0.72 intact, but continue by writing someting very non-periodical, say
0.7201001000100001000001⋯
increasing the number of 0’s in every segment ending in 1. Or just toss, at every step, 9 coins and take the number of heads for your next digit. Or use a random numbers generator (I bet you can find a plenty of them on the Internet).
You can make your irrational numbers as close to 57or 79 as you wish: just start with something like
0.714285714286(slightly bigger than57)
or
0.777777777777776(slightly less than 79)
and then continue in a non-periodic way.
This is an illustration of the general fact which can be formulated, using a bit more advanced mathematics, in a rigorous form: rational numbers are rare, a random real number is irrational with probability 1.
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