How many bits to a trit?

bits_in_a_trit.json

https://arbital.com/p/bits_in_a_trit

by Nate Soares May 28 2016 updated Jun 3 2016

[summary: $~$\log_2(3) \approx 1.585.$~$ This can be interpreted a few different ways:

  1. If you multiply the number of messages you might want to send by 3, then the cost of encoding the message will go up by 1.58 bits on average.
  2. If you pack $~$n$~$ of independent and equally likely 3-messages together into one giant $~$3^n$~$ message, then the cost (in bits) per individual 3-message drops as $~$n$~$ grows, ultimately converging to $~$\log_2(3)$~$ bits per 3-message as $~$n$~$ gets very large.
  3. The infinite expansion of $~$\log_2(3) = 1.58496250072\ldots$~$ tells us not just how many bits it takes to send one 3-message $~$(\approx \lceil 1.585 \rceil = 2)$~$ but also how long it takes to send any number of 3-messages put together. For example, it costs 2 bits to send one 3-message; 16 bits to send 10; 159 bits to send 1000; 1585 to send 10,000; 15850 to send 100,000; 158497 to send 1,000,000; and so on. ]

There are [3nd $~$\log_2(3) \approx 1.585$~$] bits to a Trit. This can be interpreted a few different ways:

  1. If you multiply the number of messages you might want to send by 3, then the cost of encoding the message will go up by 1.58 bits on average. See [+marginal_message_cost] for more on this interpretation.
  2. If you pack $~$n$~$ of independent and equally likely 3-messages together into one giant $~$3^n$~$ message, then the cost (in bits) per individual 3-message drops as $~$n$~$ grows, ultimately converging to $~$\log_2(3)$~$ bits per 3-message as $~$n$~$ gets very large. For more on this, see [+average_message_cost] and the GalCom example of encoding trits using bits.
  3. The infinite expansion of $~$\log_2(3) = 1.58496250072\ldots$~$ tells us not just how many bits it takes to send one 3-message $~$(\approx \lceil 1.585 \rceil = 2)$~$ but also how long it takes to send any number of 3-messages put together. For example, it costs 2 bits to send one 3-message; 16 bits to send 10; 159 bits to send 1000; 1585 to send 10,000; 15850 to send 100,000; 158497 to send 1,000,000; and so on. For more on this interpretation, see the [log_series_of_ceilings "series of ceilings"] interpretation of logarithms.