# Find a certain base for converting a rational to a irrational number

It is very clear that we could not propose any problem in Mathematics because that maybe does not make sence. However this is my question:

Can we find any base, in which, a rational number is written as a irrational number?

Thanks and sorry if it is not proper on the site.

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A number being rational if it is the ratio of two integers, and irrational otherwise. This is entirely independent of the base you write it in. – Rahul May 26 '12 at 11:45

I will suppose that with "rational with respect to a base" you mean that the "decimal" expansion in this base is finite or periodic and with "irrational" you mean the complement.

You can simply choose a transcendental number as base:

http://en.wikipedia.org/wiki/Non-integer_representation

Any periodic or finite representation will lead to a fraction involving $e$, but they are all irrational (in the ordinary sense), so a rational number (in the ordinary sense) between 0 and 1 cannot represented that way.

Note that irrational bases are somewhat tricky, the wikipedia link has some information on this.

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