Dirichlet's Approximation Theorem not working for n=8 and α= pi?

In summary, the speaker is questioning why Dirichlet's Approximation Theorem is not working for α=pi and n=8, despite the theorem stating that there should exist integers a and b satisfying the inequality for all real numbers α and positive integers n. Upon closer examination, the speaker realizes that they overlooked the absolute value sign in the theorem, which changes the interpretation of the inequality.
  • #1
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.
 
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  • #2
Note it says |aα-b|< 1/n instead of (aα-b) < 1/n.
The difference is subtle but crucial. Now look again at 7*pi.
 
  • #3
I completely overlooked the absolute value sign. Thank you for pointing this out!
 

1. Why doesn't Dirichlet's Approximation Theorem work for n=8 and α= pi?

Dirichlet's Approximation Theorem states that for any irrational number α, there exist infinitely many rational numbers p/q such that |α - p/q| < 1/q^n. However, this theorem does not hold for all values of n and α. In the case of n=8 and α=pi, the theorem fails because pi is a transcendental number, meaning it cannot be expressed as a root of a polynomial equation with rational coefficients. Therefore, there are no rational numbers that can approximate pi with a precision of 1/q^8, making the theorem not applicable in this case.

2. Can Dirichlet's Approximation Theorem be modified to work for n=8 and α= pi?

No, the theorem cannot be modified to work for n=8 and α=pi. As mentioned, pi is a transcendental number and therefore cannot be approximated by rational numbers with a precision of 1/q^8. Modifying the theorem would require changing its fundamental assumptions, which would ultimately result in a different theorem.

3. Are there any other values of n and α for which Dirichlet's Approximation Theorem does not work?

Yes, there are other values of n and α for which the theorem does not hold. In general, the theorem fails when α is a transcendental number or when n is too small to allow for accurate approximations. For example, with n=2 and α=√2, the theorem does not work because √2 is an irrational number and cannot be approximated by rational numbers with a precision of 1/q^2.

4. Is there a similar theorem that applies to transcendental numbers?

Yes, there is a similar theorem called the Liouville's Approximation Theorem. This theorem states that for any transcendental number α, there exist infinitely many rational numbers p/q such that |α - p/q| < 1/q^n, for any positive integer n. This theorem is more general than Dirichlet's Approximation Theorem, as it includes transcendental numbers as well.

5. Can you provide an example of when Dirichlet's Approximation Theorem does work?

Yes, an example of when the theorem works is when n=1 and α=√2. In this case, √2 is an irrational number, but it can be approximated by rational numbers with a precision of 1/q. For example, 1.41 is a rational number that approximates √2 with a precision of 1/100. Therefore, Dirichlet's Approximation Theorem holds for this case.

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