How to find irrational num
ber between 5/7 and 7/9
Answers
Many answers in this thread rightly state that there are infinitely many irrational numbers between 57 and 79; I wish to provide an explanation of that fact.
It is a well known fact explained, in many countries, to schoolchildren of age about 12 or 13, that rational numbers are represented by decimal fractions which become periodic from some place on; for example,
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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Answer:
45/63 & 49/63
Step-by-step explanation:
take LCM of 5/7 & 7/9
LCM=63
=[5/7×9/9] , [7/9×7/7]
=45/63 , 49/63
45/63 & 49/63
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