4 ms·

From an information theory perspective, the most efficient base for computations would be e (2.718). And trinary is closer than binary.

mcdonje 5mo ago
True. Radix economy.
NooneAtAll3 5mo ago
I still don't get what is being optimized for to get e as the answer
- amavect 5mo ago
  Optimizing for "radix economy", an argument that attempts to balance the digit cost against the choice of base. When the cost per digit equals the base, e turns out optimal. But when the cost per digit equals the digit's information content (bit, trit, etc; 1 trit = log(3,2) bits), all bases turn out about equal. https://en.wikipedia.org/wiki/Optimal_radix_choice https://en.wikipedia.org/wiki/Optimal_radix_choice
amavect 5mo ago
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/1wfdtsuaav https://www.desmos.com/calculator/1wfdtsuaav I 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/Optimal_radix_choice https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Why_assume_that_the_cost_of_each_digit_is_proportional_to_b https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Why_...? https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Binary_has_the_lowest_radix_economy_for_humans https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Bina... https://youtu.be/rDDaEVcwIJM?t=701 https://youtu.be/rDDaEVcwIJM?t=701 timestamp 11:41

mcdonje 5mo ago

True. Radix economy.

NooneAtAll3 5mo ago

I still don't get what is being optimized for to get e as the answer

amavect 5mo ago
Optimizing for "radix economy", an argument that attempts to balance the digit cost against the choice of base. When the cost per digit equals the base, e turns out optimal. But when the cost per digit equals the digit's information content (bit, trit, etc; 1 trit = log(3,2) bits), all bases turn out about equal. https://en.wikipedia.org/wiki/Optimal_radix_choice https://en.wikipedia.org/wiki/Optimal_radix_choice

amavect 5mo ago

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/1wfdtsuaav https://www.desmos.com/calculator/1wfdtsuaav I 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/Optimal_radix_choice https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Why_assume_that_the_cost_of_each_digit_is_proportional_to_b https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Why_...? https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Binary_has_the_lowest_radix_economy_for_humans https://en.wikipedia.org/wiki/Talk:Optimal_radix_choice#Bina... https://youtu.be/rDDaEVcwIJM?t=701 https://youtu.be/rDDaEVcwIJM?t=701 timestamp 11:41