I was wondering, if it were possible if you were given a continued fraction expansion for a number, to be able to determine if the number was rational or irrational? I know that if it is never repeating the number is irrational but not quadratic, but if it is periodic can you tell whether it’s rational?

  • $\begingroup$ If the continued fraction terminates at a point its rational ,otherwise it is irrational $\endgroup$ – The Integrator Mar 26 '18 at 12:37
  • $\begingroup$ A rational number has a finite continued fraction. An irrational does not. $\endgroup$ – B. Goddard Mar 26 '18 at 12:37

The continued fraction expansion of any rational number terminates. The continued fraction expansion of any irrational number has infinitely many terms. If the continued fraction is periodic, then it does not terminate, and it is irrational.

This is a basic result on continued fractions that you should be able to find in any elementary treatment of the topic. It's also fairly straightforward to show:

If the expansion terminates, then it is really just a fraction, and all we have to do is simplify it. Conversely, suppose the number $x$ is rational. Then, when we write $x=a_0+\frac{1}{x_1}$, we obtain a rational number $x_1$ with a smaller denominator than $x$. The denominator is a positive integer, and these can't keep getting smaller forever. Eventually, the denominator reaches $1$, and we're done.

If you desire, you could formalize the above argument using Strong Induction.

  • $\begingroup$ Very clear, thank you. $\endgroup$ – Tyler6 Mar 26 '18 at 13:32

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