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For example, number such as $\pi$ and $e$ cannot be represented as rational numbers in our number study and extend in decimal places to infinity.

QUESTION:

Is there a possibility that some other number system can represent fundamental constants present in nature correctly?

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Since $e=\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}$, you can represent it as $10.111111\dots$ using the factorial number system.

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