What was the reason to extend the number system into rational numbers
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Not only rational but even irrational numbers can be number bases if one so wishes. Bergman published a paper called Number System with an Irrational Base in Mathematics Magazine back in 1957. It represents numbers as sums ∑∞i=−∞aiφi, where φ=1+5√2 is the golden ratio, and ai are 0 or 1 (Bergman denotes φ as τ and calls it the "Tau System"). This presentation is non-unique but Bergman derives nice transcription rules (related to the Zeckendorf decomposition of integers into Fibonacci numbers) which make conversions tractable, e.g. 2=10.01=1.11. He also developed paper and pencil algorithms for addition, multiplication and division. Even so, already expressing 1/2 is not so straightforward, and "with 1/10 I had to work it out 5 or 10 times before I got the correct answer, as there is much room for error".