| I have an axe to grind. Radix economy makes a shallow argument when calculating the wrong per-digit information cost. I need some functions to show what I mean. Calculate logarithms, calculate the number of digits, and convert a base-n unit of information into base-2 units of information. Finally, calculate the information cost: the number of digits, times the information needed per digit. import ln, floor
define log := (num,base) -> ln(num) / ln(base)
define digits := (num,base) -> floor(log(num,base) + 1)
define tobits := (base) -> log(base,2)
define infocost := (num,base) -> digits(num,base) * tobits(base)
define infocost_wikipedia := (num,base) -> digits(num,base) * base
define infocost_tbwtc := (num, base) -> (digits(num,base) - 1) * tobits(base) + tobits(base- 1)
https://www.desmos.com/calculator/1wfdtsuaavI define a logarithmic per-digit information cost, following information theory. For example, 1 trit = log(3,2) bits. This results in no advantage for any base (in which case, choose base 2). Wikipedia uses a linear per-digit information cost equal to the base. This holds when communicating options takes linear time (Wikipedia's example of a phone menu). This results in advantage for base e (in which case, choose base 3). The video "The Best Way To Count" uses the logarithmic digit cost, and also notes that the leading digit carries less information (it excludes 0, like a IEEE floating point mantissa). This results in advantage for base 2. Therefore, know the context to apply the right cost analysis! https://en.wikipedia.org/wiki/Optimal_radix_choice https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Why_...? https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Bina... https://youtu.be/rDDaEVcwIJM?t=701 timestamp 11:41 |