Bound for Difference of Irrational Number with Convergent
Jump to navigation
Jump to search
The validity of the material on this page is questionable.
this needs Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
 It has been suggested that this article or section be renamed.
|
Theorem
Let $x$ be an irrational number.
Let $\sequence {C_n}$ be the sequence of convergents of the continued fraction expansion of $x$.
Then $\forall n \ge 1$:
- $C_n < x < C_{n + 1}$ or $C_{n + 1} < x < C_n$
- $\size {x - C_n} < \dfrac 1 {q_n q_{n + 1} }$
Proof
this needs Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
To discuss this proof in more detail, feel free to use the talk page.
If you are able to fix this page, then when you have done so you can remove this instance of {{Questionable}} from the code.
If you would welcome a second opinion as to whether your correction is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Immediate.
Note that:
- $\size {x - C_n} < \size {C_{n + 1} - C_n} = \dfrac 1 {q_n q_{n + 1} }$
$\blacksquare$
