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ANphysics
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Dirichlet's Approximation Theorem not working for n=8 and α= pi?
I am reading a number theory textbook that states Dirichlet's Approximation Theorem as follows:
If α is a real number and n is a positive integer, then there exists integers a and b
with 1≤ a ≤ n such that |aα-b|< 1/n .
There is a proof of this theorem given in the text as well.
My question is if this theorem is suppose to be true for all real numbers α and positive integers n, then how come I cannot find integers a and b satisfying this inequality for α= pi and n = 8?
In this case since a is restricted to 1≤ a ≤ 8 it is easy to look for b by trying all 8 values for a:
1* pi = 3.14159...
2* pi = 6.28319...
3* pi = 9.42478...
4* pi = 12.5664...
5* pi = 15.708...
6* pi = 18.8496...
7* pi = 21.9911...
8* pi = 25.1327...
since "b" must also be an integer it is clear that the possibilities for b are 3,6,9,12,15,18,21,25.
However, none of these satisfy the inequality.
Since 1/8 = .125 the decimal values of the numbers I listed above for all the possible values of a should be less than .125 in one of the cases...but this does not happen. How can this be? I figure I must be overlooking something or computing something incorrectly because the theorem clearly states that such a, b can be found for all positive n and real α.
I am reading a number theory textbook that states Dirichlet's Approximation Theorem as follows:
If α is a real number and n is a positive integer, then there exists integers a and b
with 1≤ a ≤ n such that |aα-b|< 1/n .
There is a proof of this theorem given in the text as well.
My question is if this theorem is suppose to be true for all real numbers α and positive integers n, then how come I cannot find integers a and b satisfying this inequality for α= pi and n = 8?
In this case since a is restricted to 1≤ a ≤ 8 it is easy to look for b by trying all 8 values for a:
1* pi = 3.14159...
2* pi = 6.28319...
3* pi = 9.42478...
4* pi = 12.5664...
5* pi = 15.708...
6* pi = 18.8496...
7* pi = 21.9911...
8* pi = 25.1327...
since "b" must also be an integer it is clear that the possibilities for b are 3,6,9,12,15,18,21,25.
However, none of these satisfy the inequality.
Since 1/8 = .125 the decimal values of the numbers I listed above for all the possible values of a should be less than .125 in one of the cases...but this does not happen. How can this be? I figure I must be overlooking something or computing something incorrectly because the theorem clearly states that such a, b can be found for all positive n and real α.