mones

I have deduced a simpler formula with new aspect for the square root using only one function, and according to the same pattern, the square root is also proportional to the angle: √x = 1/(10n acos( x / (x + 0.5. 10-2n ))) You can calculate the function acos using this series expansion: with X=( x / (x + 0.5. 10-2n ) Using one function should be faster than the previous formula, Just calculate it correctly and precisely to get the correct √x result with 2n correct decimals. You can see my updates at : https://www.researchgate.net/profile/Mones_Jaafar I hope this will be useful.

I am engaged to support my work, I experienced the same problem when I was working on my formula, and I wanted to dismiss all the work, the low accuracy program is lost of time for my formula for n>3 accuracy it may return a mess of numbers, until I discovered that the problem is in the trigonometric function accuracy, I clearly and widely explained that in my paper ( see Paragraph 4. Discussion), this is not because of my work or formula. May be the development of more efficient method to calculate the trigonometric functions will resolve this problem, I am convinced that my formula has many aspects that will not be developed in just few days, it is a wonderful formula anyway, that still fascinating me.

You definitely must calculate the formula using high precision program such as (https://www.wolframalpha.com/) to ensure the accuracy of the trigonometric function, because the numerical accuracy of the function "asin" near −π/2 and π/2 is ill-conditioned, or you can not get accurate result using commercial program or calculator with relatively low accuracy.

My formula is an INITIALLY CONTROLLED DECIMALS precision output formula, THIS IS HOW IT WORKS, the problem is not in the calculator capacity. It is a very trusted calculation because you know in advance the correct decimals that you are calculating. example for the calculated 351.3630601 my formula control ONLY the correct DECIMALS (3630601) precision output, but it return ALWAYS a correct number digits (351) if you need 7 correct decimals and you choose n=1 you get 2n=2 correct decimals (36) you obviously never get 7 correct decimals, which is not wrong, it is a voluntary limited precision , the "error" is in your choice of n. if you can not enter the choice n=4 to get 2n=8 correct decimals in your calculator, it means my formula is not adequate for the calculator so use another method. You must choose FIRST just the correct DECIMALS (not all number digits) precision output which is equal to 2n by simply choosing n according to your need Most of the scientific calculation users want to use 15 correct decimals as a maximum, so you can choose n=8 to get 2n=16 correct decimals that will satisfy any need for less than 16 decimals. you can increase the precision of correct decimals as you want by simply increasing n.

Correct, I mean only even zeros"0" decimals, and add number 5 decimal at the end which is conform with 0.510^(-2n) this is true because your calculator has limited digits, the accuracy of the formula is 2n decimals which mean in your example 5 digits correct with n=1 and 2 correct decimals and.

I welcome your comments and attempt to discover more aspects, I never choose to ignore any question about my formula whatever is, ask me a specific question about my formula to answer, not general issues, I am not proficient in all science field, in spite of that I will try my best and will give an answer if I can may be it is useful.

the radians choice is imposed by the calculator you can choose degrees or grads it is the same, Since it is 0.510^(-2n) you have to enter an even decimals number of "0"(2,4,6,8,10...) it is better to just add "0" decimals than to calculate 10^(-2n) and consuming time, it is the same result. in your example you entered 3 "0" which is incorrect. I do not have facilities to benchmark a compared solution about square root calculation with my formula so I can not give a scientific reply, but you can calculate my formula with hand and trigonometric function by hands also using series expansion the speed of calculation depend also on the quality of program and techniques and how it is performed My formula is the result of hard work and high quality research with deep analysis resolving a very tough problem since the Babylonian, the speed of calculation is just an aspect of the formula that I wish to be the fastest formula, but I am not a computer scientist to work on this issue. This is what I am trying to say; the formula may have many aspects and applications since the square root is used every where in science for many purposes,

Of course my formula PATTERN is deduced using Pythagorean theorem If you mean just calculating using my formula on computer scientific calculator choose Radians calculate tan(antisin (123456/123456.0000005))/1000=351.3630601 when I tested your formula with log on my old computer it takes a tremendous time that I have to stop the operation without result. I am not proficient in computer science or all the science field, that is why I asked scientist to find out if there is possible benefits, but if I can give my opinion I will not hesitate My formula is like a tool you may not use it and you may need it,

It mean that the square root can be expressed as a Tangent of a specific angle and represented in a right triangle according to a specific pattern not arbitrarily, you can power the equation to enlarge the pattern to all number. Try this calculation method for X= x/(x+0.510-2n) You do not have to power X from the beginning each time use cumulative

This is possible using Taylor series but it will take a long time, again this is a formula not a method. I want every scientist to explore the advantages of my formula that is why I joined the forum, and this is exactly what I am looking for to identify all it's aspects and applications, and keep informing me on any new aspects or applications, it is very interesting to see what they will conclude because the square root has extremely vast application every where in science, I want to emphasis that it is a formula using trigonometric function may be complicated but have an incredible flexibility, and this will not happen in few days you have all the time.

I did not read the forum rules it was not a choice I was in hurry, but I respect the rules and I expected only scientific opinion since this is a respectful science forum I asked you to read so you will find answer to your question is that a mistake? As for surds As I told you use the square root function replacement described in paragraph 3.4. So the general formula for surds is tan (asin((tan (asin(( x / (x + 0.5. 10-2n ))))) / 10n ) / ((tan (asin( x / (x + 0.5. 10-2n ))) / 10n) + 0.5. 10-2n ))) / 10n for 2sqrt(sqrt(3)) choose for example n=18 the correct number of decimals is 2n=36 and replace 2tan (asin((tan (asin(( 3 / (3 + 0.5 10^(-36) )))) / 10^18 ) / ((tan (asin( 3 / (3 + 0.5 10^(-36) ))) / 10^18) + 0.5 10^(-36) ))) / 10^18 Test it on https://www.wolframalpha.com/input/?i=2tan+(asin((tan+(asin((+3+%2F+(3+%2B+0.5+10^(-36)+))))+%2F+10^18+)+%2F+((tan+(asin(+3+%2F+(3+%2B+0.5+10^(-36)+)))+%2F+10^18)++%2B+0.5+10^(-36)+)))+%2F+10^18 and you get the results with 36 correct decimals: 2.632148025904984921638437803593998110075341841792392364 Compared to the correct answer 2.632148025904984921638437803593998110320137180411644353 You can increase the correct decimals by just increasing n value I am not imposing any thing to scientific community, I said this is what I found and this is the proof if you find a proof that is wrong simply discard it I was talking about trigonometric calculation, table are not a method of calculation it is a pre-calculated values stored in book or computer and I do not have any problem to use it As I described for surds the formulas have the power for manipulation it can stand up alone without support, I say may be this is will be useful for science if not discard it If your methods are better then simply discard my formula and do not use it, I did not understand the relation between method and formula which are completely different concept, any method can not stand up alone including mine they need always support but the formula have the power of nature it can stand up alone

JARGON This is not a calculation method only it is a fundamental mathematical law and formulas that illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery and exact formulas, which is the sole pattern known for the square root according to my knowledge since the Babylonian. you may not consider it for square root calculation but it is not possible to ignore a scientific discovery and a fundamental mathematical law which is proved to perform unlimited mathematical operations. Your suggest method references is INCLUDED in my references READ IT I have restored the parenthesis notation according to norm that do not have any influence or change on the formulas according to the scientific context.

I have enabled the public profile at https://www.researchgate.net/profile/Mones_Jaafar If the link do not work properly let me know No jargon please just your scientific opinion. I have made updates and clarifications and restored the parenthesis notation according to norm that do not have any influence or change on the formulas according to the scientific context.

I have developed formulas to calculate the square root of real numbers discovered from a pattern you can read details at : https://www.researchgate.net/project/Square-root-of-positive-real-number-sequence-and-calculation-methods Looking for your scientific opinion