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Is there a difference between using nats and bits to express entropy?

It seems to me like questions involving decimal vs binary representations of some number are not particularly interesting: for instance $\pi$ or $\sqrt{2}$ are conjectured to be normal in every base, ...
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Entropy, magnitude, diversity of finite metric spaces in number theory

I was reading the article by Tom Leinster, (Maximizing diversity in biology and beyond, arXiv link), and find it very interesting. Since I was searching for entropies of finite metric spaces I found ...
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