{
some of the square root energy developed recently in a way toward
making a formula for calculating square roots by hand.
the present draft is getting very close to usability.
some do not want to share it yet, but it seems it's time to share it a
little. (it's not time to share it, this could be shared in much
higher quality or more useful fun :/)
also, this was likely discovered thousands of years ago and is found
somewhere in textbooks under some name, but i haven't found it myself,
rather it was invented here.To Calculate Decimal Square Root from Square Let's call D the square. Describe D in terms of two summed parts of itself A and C such that D = (A + C)^2 . We'll have A represent the digits we have calculated already, and C represent the digits we have not yet calculated. To determine the first A to start with, look at the leftmost digits of D so as to find the largest smaller perfect square you already know. For example, if you know 1-digit perfect squares, and D is 532, then you would start with an A of 20, because 400 is a perfect square. If you know 2-digit perfect squares, and D is 5329, you could instead start with an A of 70, because 4900 is a perfect square. Following digits can be calculated by rote arithmetic via D = (A + C)^2 = A^2 + 2AC + C^2. For each digit or group of digits, first calculate A^2 and 2AC. Then, we will calculate (D - A^2)/(2A) to determine the value of C + C^2/(2A). Note that the smaller the magnitude of C, the less significant the value of C^2/(2A), so this value may be useable for quadratically more digits at a time as the procedure progresses. The calculation of (D-A^2)/(2A) can be significantly optimized to be performed by hand for repeated calculations that share the same leftmost digits. 1. square A To square a number when existing digits to the left have already been squared, separate the past digits and new digits into separate numbers, say A = A_old + A_new . Then, A^2 = A_old^2 + A_new * (2 * A_old + A_new) . This is much, much faster to calculate than resquaring the entire number. Notably, most of the left digits of A_new are zero, and most of the left digits 2 * A_old have already been calculated. A_new may have only 1 digit that is nonzero. 2. subtract A^2 from D As this proceeds with decimal, more and more of the leftmost digits of this subtraction become .9999 and can be simplistically reused. The subtraction tends to add many more .9999's from the last, and this becomes quick to recognise as e.g. 7 + 2 + carried-1 over and over again. There are only a few digits that are actually new. 3. Divide D-A^2 by 2A This is an approximation of C that can only be performed as accurately as the magnitude of the C^2/2A term that adjusts it. If only one digit is desired, there is no need for long division. One can eyeball the 1-3 largest magnitude digits to identify the largest magnitude digit of C in the result. The procedure then proceeds. Current neuron and behavioral research regards correctly calculating the potential C^2/2A accuracy representation to figure out how many digits can be extracted from a single number at once. We have tested so far mostly with the square root of 2, and correctly identified the square root as far as 1.41421 both digit-by-digit and by calculating accuracy in advance and then calculating many digits at once using partial long division. The work is challenged by usually happening when not having a calculator available due to being a coping strategy in non-device mental or physical consciousness situations, as well as by Karl not accurately remembering square roots to check arithmetic against or incorrectly remembering them as subsequences within pi, maybe similar to general mistake density requiring much work to discern what is real. Paths may change now this is posted. Fun to succeed at square root calculation! This was something karl was deriving approaches to some years ago to cope and try to protect logic skills, the derivation of which has somehow become heavily inhibited, and this approach is now novel to him and looks like it could be made to be quite successful :) there is imagination around implementing it in code
