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| author | Adrien Hopkins <adrien.p.hopkins@gmail.com> | 2025-09-22 16:13:38 -0500 |
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| committer | Adrien Hopkins <adrien.p.hopkins@gmail.com> | 2025-09-22 16:13:38 -0500 |
| commit | d3fdb797db81ab81c4f91dd88d24723a88b39c2e (patch) | |
| tree | 905a34d4d36659c13b0dca550bdbc36a63bfe8c2 | |
| parent | ac766f4f6ab9db683e29b093bbc48cfe3b27b2b7 (diff) | |
Add regular complexity documentation
| -rw-r--r-- | README.org | 14 |
1 files changed, 14 insertions, 0 deletions
@@ -86,6 +86,20 @@ The MTC is an estimate of how difficult it is to learn a radix's multiplication The radix's base-2 logarithm indicates its information density - how much information it can fit into a digit. Radices with higher logarithms will be able to count higher with fewer digits, and approximations will be more accurate for the same number of decimal places. Specifically, one digit in radix n is equivalent to log_2(n) bits (base-2 digits). The reciprocoal of the logarithm is proportional to the number of digits in a number, so it is also shown in the extended view. It uses a scale where decimal is 1, so when you see the value of 3.3219 in binary, you know that numbers in binary require ~3.32 times as many digits as decimal. +** Prime Regular Complexity +The *regular complexity* of regular /n/ in radix /r/ RC(n, r) is defined as: +\[\textrm{RC}(n, r) = \lim_{e \rightarrow \infty} \sqrt[e]{\frac{\textrm{MPM}(r, n^e)}{n^e}}\] +where MPM(x, y) is the smallest power of x that is a multiple of y. + +Note that RC(n^e, r) = RC(n, r)^e. + +This number measures how easy it is to work with powers of /n/ in radix /r/ - the smaller, the better. Two specific applications of this are: +- In order to be able to test for divisibility by n in radix /r/, you need to memorize approximately the first RC(n, r) multiples of n^e. +- You can divide by n^e by multiplying by approximately RC(n, r) then shifting the decimal point left. + +(The exact value for both of these is \(\textrm{MPM}(r, n^e) / n^e\) - the RC simply condenses all the powers into a single number.) + +The Radix Info Script shows the complexity for all of the base's prime factors, not every number, in order to keep the output reasonably small. The regular complexity of a product of prime powers will be smaller than the largest of the multiplicands' complexities - prime powers are the worst case. ** Copyright & Contact Info radix_info: gives some information about number radices Copyright (C) 2023 Adrien Hopkins |
