As an aside: a nice example of a number system with an irrational base appears in:
A number system with an irrational base by George M. Bergman. Mathematics Magazine 31 (1957/58), pp. 98-110. JSTOR link
which describes how to do arithmetic base $\phi$ (the golden ratio). But one should not confuse the numbers with their representations, as Ross notes above.
I'd like to point out that the above paper is the peer-reviewed paper with youngest author that I am aware of: it was submitted when Bergman was 12, and the accompanying note by his mother is printed in page 91 of the journal:
Dear Mr. James:
"The paper presented is the work of my twelve year old boy who took more than a year to gather courage to submit it for editorial scrutiny.
You may be interested to know that when my son first received a subscription to Mathematics Magazine about two and a half years ago, he was aghast to note that he couldn't understand a single thing in it. With each successive issue, however, his understanding unfolded (he is a self-taught mathematician) until now he awaits each issue with eagerness, and recently was able to submit his solution to one of the Proposals published in your last issue. He considers his subscription to Mathematics Magazine one of the finest presents he ever received!"
To quote Tom Lehrer: "It's people like this who make you realize how little you've accomplished..."