find an irrational number between 5/7 and 7/9 how many more there may be
Answers
Answer:
5.3 5.2 5.1 and many more
Answer:
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 72100 is 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 57 or 79 as you wish: just start with something like
0.714285714286( slightly bigger than 57)
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 .