Si un número es irracional en base 10, es irracional en otras bases?

Question

Si un número es irracional en base 10, es necesariamente irracional en todas las otras bases? O es posible que un número irracional en sólo un par de bases?

Answer 1

Usted no entiende lo "irracional". Usted probablemente ha sido informado de que un número irracional es aquel cuya expansión decimal no se repita. Aunque este es el caso, es una propiedad secundarios. Un número irracional es aquel que no puede ser escrita en la forma $$a\over b$$ where $un$ and $b$ are integers; a rational number is what that can be written in that form. This definition has nothing to do with the base in which the numbers are written, and this is why your question, as you phrased it, does not really make sense. It's like asking if a verse of a song will still rhyme even if it is printed in colored ink.

Now it is the case that a number has a repeating base-10 representation if, and only if, it is a rational number, that is if it can be written as a fraction $\frac ab$. An irrational number always has a non-repeating base-10 representation.

And it is also the case that a number has a repeating base-$n$ expansion, for any base $n$, if, and only if, it is a rational number; an irrational number has a non-repeating representation in every base. This is probably the question you meant to ask.

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Suppose a number $x$ has a base-$n$ expansion that begins with some sequence of digits $a_1a_2a_3\ldots a_i = $, and then follows with $b_1b_2b_3\ldots b_j = b$ repeated forever. Then it turns out that $x$ is a rational number, and we can even find a fraction for it; the fraction is $$\frac{a}{n^i} + \frac1{n^i}\frac{b}{n^j-1}.$$

For example suppose we are working in base 8, and we want to find a fraction for the number 0.13456456456… where the digits are understood base 8. Then $i=2$, and $a_1a_2 =$ 13; and $j=3$, and $b_1b_2b_3 =$ 456. Then we can calculate that $$\begin{align}x & = \frac{13_{8}}{8^2} + \frac1{8^2}\frac{456_{8}}{8^3-1} \\ & = \frac{11}{64} + \frac1{64}\frac{302}{511} \\ &=\frac{5621}{32704} + \frac{302}{32704} \\ & = \frac{5923}{32704}\end{align}$$

and since this is a quotient of two integers, it is rational, because that is what a rational number is.

Its base-8 expansion is of course 0.13456456456456…, because that was how we constructed it, but it also repeats when written in any other base; for example in base 10 it is written $$0.181109\ 344422700587084148727984\ 34442270058708414872798\ \ldots.$$

Similarly, the base-10 decimal 0.13456456456… is equal to the rational number $$\frac{13}{10^2} + \frac1{10^2}\frac{456}{10^3-1} = \frac{13443}{99900} = \frac{4481}{33300}.$$