If a number is irrational in base 10, is it irrational in other bases?

Question

If a number is irrational in base 10, is it necessarily irrational in all other bases? Or is it possible for a number to be irrational in only a few bases?

Answer 1

You don't understand what "irrational" means. You have probably been told that an irrational number is one whose decimal expansion does not repeat. Although this is the case, it is a secondary property. An irrational number is one that cannot be written in the form $$a\over b$$ where $a$ 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 $a_1a_2a_3\ldots a_i = a$, 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 $$\frac{13}{10^2} + \frac1{10^2}\frac{456}{10^3-1} = \frac{13443}{99900} = \frac{4481}{33300}.$$