The Problem:
There are multiple "rooty" equations that can be simplified to a whole number, for example:
$$\sqrt{19 + 6\sqrt{2}} - \sqrt{18} = 1$$ Because: $$\sqrt{19 + 6\sqrt{2}} - \sqrt{18} = \sqrt{18+\sqrt{72} + 1} - \sqrt{18} = \\ = \sqrt{18+2\sqrt{18}+1} - \sqrt{18} = \sqrt{(\sqrt{18}+1)^2}-\sqrt{18} = \\ = \sqrt{18}+1-\sqrt{18} = 1$$
However, in the title, we have the expression of: $$\sqrt{93+63\sqrt{85}} - \sqrt{143} \approx 14$$ And using a scientific calculator you indeed get $14$ as the answer. But using a high precision online calculator you get the true answer of: $$\sqrt{93+63\sqrt{85}} - \sqrt{143} = 14.00000000005032...$$
The Question:
Is there a general way to prove that a "rooty" expression (like the one in the title) $\notin \Bbb{Z}$? Even if the expression is really close to a whole number, and even high precision calculators can't give you the correct answer?
Is there a general procedure or algorithm which tells you for sure that the number is or isn't $\in \Bbb{Z}$?
from Hot Weekly Questions - Mathematics Stack Exchange
Post a Comment