Is there a theorem on rational numbers in irrational bases? For example $100$ in base $\sqrt{2}$ is $2$, so it's an integer in both bases, but are there shared integers and rationals in base $\pi$? Or even is there a polynomial evaluated at $\pi$ that comes out as a rational number?
Asked
Active
Viewed 48 times
1
-
Constant polynomial? – Alex Vong Jun 07 '17 at 17:59
-
1What are the allowed "digits" in base $;\sqrt2;$ , for example?? – DonAntonio Jun 07 '17 at 18:00
-
1What does it mean to write a number in base $\pi$? There aren't any integers $a_i$ such that $\sum a_i\pi^i$ is an integer (other than trivial cases). – lulu Jun 07 '17 at 18:06
-
@DonAntonio Integers, so trivially 1 is 1, and then 10 is $\sqrt{2}$ and 100 is $\sqrt{2}^2$, and so on – Hovestar Jun 07 '17 at 18:06
-
@Hovestar So in base $;\pi;$ we can use the digits $;0,1,2,3;$ , or what – DonAntonio Jun 07 '17 at 18:06
-
@lulu perfect, what set of numbers is that that have that property? – Hovestar Jun 07 '17 at 18:06
-
@DonAntonio Arbitrary positive reals may be used as bases; phinary is the most common, but any number works. The digits are nonnegative integers $<$ the base, and the $i$th digit is interpreted as base$^i$, as usual. (See e.g. this paper.) – Noah Schweber Jun 07 '17 at 18:07
-
1The transcendental numbers. I mean, that's the definition. – lulu Jun 07 '17 at 18:07
-
@DonAntonio Yeah since 4 would be expressed as $10.something$ – Hovestar Jun 07 '17 at 18:07
-
@lulu, awesome can you make that an answer? – Hovestar Jun 07 '17 at 18:08
-
@NoahSchweber Well, then you (and the OP) seem to mean non-integer bases...Never heard of it, and I just can't understand what possible uses it can have. – DonAntonio Jun 07 '17 at 18:09
-
2Well, I'm not sure what it's an answer to. You can answer your own question if you like. The wiki article on Transcendental Numbers may be of use. – lulu Jun 07 '17 at 18:09
-
1@DonAntonio Irrational bases are a specific class of non-integer bases, so there's no inaccuracy here - the OP (it sounds like) is really interested in specifically irrational bases. "I just can't understand what possible uses it can have." Fair, but that doesn't mean it's unreasonable to ask questions about it. – Noah Schweber Jun 07 '17 at 18:11
-
1@NoahSchweber Who said it was unreasonable? It is just weird... for me . – DonAntonio Jun 07 '17 at 18:13