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What is an "informational density" and why numeral system with the base of e (2,71828...) has the maximum informational density? How do you calculate "informational density" of a given numeral system?

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Let you want to represent integer values from 0 to 999. You need 3 digits, each has 10 states, 30 states overall. When using binary digits, you need 10 digits with 2 states, 20 states overall, So binary representation is more "dense". In general case, p*ln(N)/ln(p) states is required. This formula has minimum at p=e. However, this is pure theoretical speculation. Numeral system with the base 3 is slightly more dense than that of 2, and 3-based computers were really built, but appeared more complex than binary computers.

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  • $\begingroup$ Thank you Alexey! Actually, I´ve read it on the Wiki en.wikipedia.org/wiki/Radix_economy, topic called Radix Economy. What I wanted to figure out is how they construct numeral system for non-integer base, say e. And was disappointed with the answer :) $\endgroup$ – Roman Nov 4 '13 at 12:44

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