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Number Theory/irrational numbers


Hello Sir,

I am knew to this site and you can help me out of the following two questions.

1) Prove that the number 0.1010010001100001  is irrational

2)Prove that for every positive rational number r satisfying the
 condition r^2 < 2 one can always find a larger rational number
 r + h (h > 0) for which (r +h) (r + h) < 2.

High regards and thanks in advance.
Vinod verma

Hello Vinod

(1) Your number is a decimal which is neither terminating nor recurring.  Therefore it is irrational.  What is more difficult to prove is that is non-algebraic.  That is,it is not the root of a polynomial equation with integer coefficients.  Liouville proved that, and you can find proofs on the internet, but they are too complex for me to type here.

(2) The infinite continued fraction for root 2 is [1,2,2,2,2...]
Its first convergents are 1/1, 1 + 1/2 = 3/2, 1+1/(2+1/2)=7/5, 1+1/(2+1/(2+1/2))=17/12.
ie. 1/1, 3/2, 7/5, 17/12.  The numerator and denominator of each fraction except the first two can be obtained by a simple rule. Double the previous and add the penultimate.  For example 17=2*7+3 and 12=2*5+2.  All the convergents in odd positions are less than root2, the others greater.  As the convergents get closer and closer to root2, you can find one better than r.

I also tried a do-it yourself method.  2-r^2 is a small number, so x = (2-r^2)^2 is very small.
Then r+x is a better approximation.  I've tested it and it certainly works, but I've not been able to prove it yet.



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