Difference between revisions of "Rational approximation"
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The main theme of this article is the question how well a given [[real number]] <math>x</math> can be approximated by [[rational number]]s. Of course, since the rationals are dense on the real line, we, surely, can make the difference between <math>x</math> and its rational approximation <math>\frac pq</math> as small as we wish. The problem is that, as we try to make <math>\frac pq</math> closer and closer to <math>x</math>, we may have to use larger and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how well can we approximate <math>x</math> by rationals with not too large denominators. | The main theme of this article is the question how well a given [[real number]] <math>x</math> can be approximated by [[rational number]]s. Of course, since the rationals are dense on the real line, we, surely, can make the difference between <math>x</math> and its rational approximation <math>\frac pq</math> as small as we wish. The problem is that, as we try to make <math>\frac pq</math> closer and closer to <math>x</math>, we may have to use larger and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how well can we approximate <math>x</math> by rationals with not too large denominators. | ||
==Trivial theorem== | ==Trivial theorem== | ||
− | Every real number <math>x</math> can be approximated by a rational number <math> | + | Every real number <math>x</math> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\frac 1{2q}</math>. |
==Proof== | ==Proof== | ||
Note that the closed interval <math>\left[qx-\frac12,qx+\frac12\right]</math> has length <math>1</math> and, therefore, contains at least one integer. Choosing <math>p</math> to be that integer, we immediately get the result. | Note that the closed interval <math>\left[qx-\frac12,qx+\frac12\right]</math> has length <math>1</math> and, therefore, contains at least one integer. Choosing <math>p</math> to be that integer, we immediately get the result. |
Revision as of 17:52, 23 June 2006
Contents
Introduction
The main theme of this article is the question how well a given real number can be approximated by rational numbers. Of course, since the rationals are dense on the real line, we, surely, can make the difference between
and its rational approximation
as small as we wish. The problem is that, as we try to make
closer and closer to
, we may have to use larger and larger
and
. So, the reasonable question to ask here is how well can we approximate
by rationals with not too large denominators.
Trivial theorem
Every real number can be approximated by a rational number
with a given denominator
with an error not exceeding
.
Proof
Note that the closed interval has length
and, therefore, contains at least one integer. Choosing
to be that integer, we immediately get the result.
So, the interesting question is whether we can get a smaller error of approximation than . Surprisingly enough, it is possible, if not for all
, then, at least for some of them.
Dirichlet's theorem
Let be any integer. Then there exists a rational number
such that
and
.
Proof of Dirichlet's theorem
Consider the fractional parts . They all belong to the half-open interval
. Represent the interval
as the union of
subintervals
.
to be continued
Liouville Approximation Theorem
We can generalize Dirichlet's theorem as follows: If is an algebraic number of degree
, then there are only finitely many rational numbers
satisfying the following inequality:
. This gives us the following corollary:
is a transcendental number, known as Liouville's constant.