(no subject)
Here is a problem I thought up on my own while trying to build a conceptual basis for my lesson plans on irrational numbers for my 9th graders:
Given two incommeasurable line segments. Describe a method to find the irrational ratio between these segments.
Now most of the irrational numbers I've encountered have been various kinds of roots. Then there is pi and e ...
So if I had two line segments and the ratio of their lengths is 2pisqrt(111) is there any rigorous way for me to find this out? Also is it possible to express "most" of the irrational numbers in terms of other irrationals? Or is it more likely that two random incommeasurable line segments have no "neat" ratio between them that we could write with known irrational numbers?
link
Given two incommeasurable line segments. Describe a method to find the irrational ratio between these segments.
Now most of the irrational numbers I've encountered have been various kinds of roots. Then there is pi and e ...
So if I had two line segments and the ratio of their lengths is 2pisqrt(111) is there any rigorous way for me to find this out? Also is it possible to express "most" of the irrational numbers in terms of other irrationals? Or is it more likely that two random incommeasurable line segments have no "neat" ratio between them that we could write with known irrational numbers?
link