0

According to Cantor–Dedekind axiom, corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number. I learnt in high school, the method for finding the "exact" position of square-roots of rational numbers in a real number line. It basically uses this result: $$(\sqrt{x})^2 + (\frac{x-1}{2})^2 = (\frac{x+1}{2})^2$$ Is there any generalized way to find "exact" position of other irrational numbers like

  • $\text{n}^{th}$ roots of rational numbers
  • $\pi$ ?

This seems to be possible according to Cantor–Dedekind axiom but I cannot algebraically deduce it.

user961447
  • 319
  • 5
    What do you mean by exact position ? How do you find for example the exact position of $\sqrt 2$? Do you mean like this ? https://en.wikipedia.org/wiki/Constructible_number – InfiniteLooper Feb 02 '22 at 12:58
  • 1
    You will need to expand on what you think "exact position" refers to here, because as it stands, the question is meaningless. – Angel Feb 02 '22 at 13:06
  • The existence of the point on the real line does not imply that it can (at least in principle) be exactly constructed , if it is that what you mean. In practice, the accuracy is of course limited anyway. – Peter Feb 02 '22 at 13:08

0 Answers0