Project Euler/198: Difference between revisions
From charlesreid1
(Created page with "==Problem Statement== A best approximation to a real number x for the denominator bound d is a rational number <math>\frac r s</math> (in reduced form) with <math>s \le d</ma...") |
No edit summary |
||
| Line 7: | Line 7: | ||
We shall call a real number x ''ambiguous'' if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational. | We shall call a real number x ''ambiguous'' if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational. | ||
How many ambiguous numbers <math>x=\frac p q, 0 | How many ambiguous numbers <math>x=\frac p q, 0 < x < \frac 1 {100}</math>, are there whose denominator q does not exceed <math>10^8</math>?</p> | ||
Revision as of 00:36, 16 April 2025
Problem Statement
A best approximation to a real number x for the denominator bound d is a rational number $ \frac r s $ (in reduced form) with $ s \le d $, so that any rational number $ \frac p q $ which is closer to x than $ \frac r s $ has $ q > d $
Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. $ \frac 9 {40} $ has the two best approximations $ \frac 1 4 $ and $ \frac 1 5 $ for the denominator bound 6. We shall call a real number x ambiguous if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational. How many ambiguous numbers $ x=\frac p q, 0 < x < \frac 1 {100} $, are there whose denominator q does not exceed $ 10^8 $?
Flags
