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.

12 minutes ago, studiot said:

 

Indeed so, finally a response to what I said in my first reply.

 

You have put in a lot of good effort in creating this paper and I have tried to engage in constructive discussion but I find it very disappointing that you seem like a marathon runner who, have covered the first 26 miles has decided to sit on a milestone and admire the scenery rather than complete the course in not supporting your work.
So I will leave you and your thread there.

 

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. 

19 hours ago, studiot said:

 

Look carefully at the number of views of this thread, listed in the forum list.

378     

78 of these since this time yesterday.

 

Why do you think other people have stopped responding to you, even though they have been onsite and looked?

I think it is because you don't respond to other people's concerns.
You either dismiss them out of hand or just ignore them.

 

How many schools, colleges universities, industries   can or need to calculate to 15 digits?

How many manufacturers of calculators make 15 digit ones?
How many town centre shops have one in stock?

So yes, repeatedly stating you have a method that can calculate square roots to some arbitrarily large precision doesn't prove it.
Nor does a large number of examples.
And I have seen no mathematics to prove this.
There is a large body of mathematics concerned with precision and accuracy in calculation.
It should be possible to show mathematically these confidence levels so that any potential user of your system can demonstrate that his root extraction is sound and reliable for his purposes. This is especially important for the few that need to calculate to such levels.

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.

14 hours ago, studiot said:

 

You are carefully avoiding the question

How can any user trust your method or ascertain if the result answer is correct?

 

You method has clearly failed on my calculator compared to the proper method or using a different formula, both of which give the same ansswer, correct to the display capacity of the calculator.
 

Thus the error cannot be inherent in the calculator cpaacity, but must be inherent in your method.

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.

2 hours ago, studiot said:

I think don't you mean 'even decimals', but even zeros?
Both 123456 and 123456.0005 have an even number of digits.

The best I can do on my calculator is

tan (arcsin(123456/123456.005)) /10 = 351.3620004 which only gives me 5 digit accuracy compared with  351.3630601

 

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.

5 minutes ago, studiot said:

 

I am sorry you choose not to address my points, particularly about answering the question of demonstrating you have the right answer whn you have put numbers into your formula.

Checking is a vital part of any real world process.

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.

19 hours ago, studiot said:

Thank you for your answer.

Why do you think using radians would make any difference?
It doesn't on mine and is slightly slower.

Both using DMS or rads is much slower than using the log / antilog formula, which is effectively instantaneous.

However

I cannot enter 123456.0000005 into my caclulator.
Not many  calculators have that many digits.

Mine displays 10.
I 'm not sure if it calculates to further guard digits, but I think so.

If I use 123456/123456.0005 then I get 11110.42436, not the right answer.

I am not trying to dismiss your formula, or discredit it. Who knows, there may one day be a use for it.  Knowledge should not be disgarded because it is currently unwanted.

I am taking you up on your request to examine its use in real situations and have offered you the results of my simple trials.

 

So if I was sitting on top of some mountain in Arabia (as I have been) and wanted a square root (which I have) but have not lugged a heavy computer up there (which I didn't) my thoughts are.

1) Modern (small) calculators have a square root button.

2) This is faster than trigonometric solutions.

3) If you, like me, did not have a calculator with even a square root button, let alone trig and log functions, you would have to rely on human ingenuity (as I did).

4) My companion took a whole afternoon to extract a single root, using a version of Newton's method on his calculator.

5) My digit bracketing method takes a few minutes per root extraction and automatically confirms the answer at the end of the extraction. Checking is important in real life.

 

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.

5 hours ago, Enthalpy said:

I feel the comments are hard here.

It's not every day that we read a new method to compute square roots. Maybe the proposal is not faster than the algorithms running presently in computer libraries, but it may have other benefits. At least, it opens news windows.

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,

10 hours ago, John Cuthber said:

Well done.

You have rediscovered Pythagoras' theorem.

 

Of course my formula PATTERN is deduced using Pythagorean theorem

17 hours ago, studiot said:

Using John Cuthber's example

sqrt(123456) = 351.3630601  directly on my 30 year old TI calculator.

If I choose another formula

sqrt(123456) = antilog(0.5*log(123456)) = 351.3630601

There is no perceptible difference in speed.

What keystrokes would you recommend to achieve this in your system?

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,

23 hours ago, John Cuthber said:

It would be much quicker to use a Taylor series to calculate the square root.

So, what's the point of your formula?

I cannot imagine any circumstance where replacing "sqrt(2)" by " tan (asin( 2 / (2 + 0.5. 10^-36 )) / 10^18"

 or whatever would be an advantage (unless you were trying to hide the fact that you meant root 2.)

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.

On ٩‏/١٢‏/٢٠١٨ at 9:00 AM, Sensei said:

What advantages.. ? Trigonometric functions are extremely slow to calculate (by CPU/FPU)...

sqrt(x) is just generalization of power of x^^0.5

cube root(x) is just generalization of power of x^^0.3333333(3)

ps. Make function that is calculating something faster than currently existing methods, and you will have attention from scientific community, and more importantly from computer designer community (faster operation is always welcome by programmers and users of applications). You need to benchmark currently existing methods and you own method to verify your method is faster than native, and alternative implementations.

 

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

13 hours ago, John Cuthber said:

OK, without using a calculator (or computer), please calculate the square root of 123456 to 6 digits for me.

You will need to work out the tangents + arcsins etc by hand.

Let me know how you plan to do that. 

Do you see why I don't think your method is useful?

This is possible using Taylor series but  it will take a long time, again this is a formula not a method.

10 hours ago, studiot said:

 

Perhaps there is a language difficulty?

I didn't say you are imposing on anybody.

And I didn't say your proposition is wrong.

 

I did say your proposition is interesting and about an important subject.

 

I did identify two separate aspects of it.

Firstly the actual calculation of a square root, correct to a specified number of digits.

Secondly the use of that formula in place the square root in some larger formula.

Finally I noted some of the things others have been looking at.
Are you not interested in seeing what others have done as well as telling them what you have done?

 

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.

3 hours ago, studiot said:

 

Thank you for the link, John.  +1

 

 

This is not an answer to my question about surds.

How about a simple one eg

23–√−−−√

 

Why should anyone read it, whilst you choose not to read the forum rules?

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

1 hour ago, studiot said:

I would like to make it clear that the subject itself ((square ) roots) has been of great interest and importance for thousands of years and remains so today.

So I wish to clearly separate discussion about the subject from discussion about some associated claims you have made which are arguable.

Yes the babylonians are the first recorded people to have considered this

 

 

Fast forward to ultra modern times to look at it from a totally different point of view.

 

 

You seem to be implying that you are offering not a calculation of the numerical value of a square root, but an exact replacement for the conventional square root when it appears in a formula, and further claim it is somehow simpler than just leaving it as a square root.

In what way is this better?

We often leave irrational numbers that are infinite decimals if expressed in decimal form, as just that. For instance    π .

Similarly we leave    3–√ just as it is to avoid an infinite decimal and make it exact.

 

What could be simpler?

 

You have also indicated that you think series methods of obtaining trigonometric values are the only ones available.

There are others.

In fact there are proceedures for constructing exact tables starting with trig functions and angles that have exact values and using trig transformation formulae for sum and difference and multiple angles.

You would find these in Hobson for instance.

Finally, if you are allowed one press of a button to obtain these trig values, why is that preferable to one press of a button to obtain the square root directly from the same calculator?

After all, that calculator can never be more accurate than its digits and algorithms.
The more algorithms you employ the greater the compounding or build up of error.

 

 

 

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

3 hours ago, John Cuthber said:

Which words didn't you understand?

Most of the words are common enough

This might trouble some people
https://www.mathopenref.com/radical-sign.html

To write off the method as "jargon" seems a bit silly. My dad learned it as a schoolkid- so  did the rest of his classmates.

It has the enormous advantage over your method that you only need a pen and paper.
Your method requires an infinite set of trigonometric tables and is thus not any practical use.

 

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 

9 hours ago, studiot said:

You are new here and you obviously haven't read the rules like so many other don't.

So I will help you by posting the abstract of your thesis and commenting upon it.

But I will ask the moderators to explain the rules.

 

1)

Not all methods of root extraction require an initial 'guess'.

There is a perfectly satisfactory rote method a bit like long division, though admittedly it is multi step as is long division.
You would probably need to look up some algebra texts from 1890 to 1920 to find it though.

2)

Your methods claim a single step, but appear to require an inexhaustible supply of more advanced functions such as trigonometric ones to achieve this.
How many steps does the computer or calculator execute to obtain these, and what would do without them?

What about the accuracy obtainable for these?
For instance the tangent changes very rapidly indeed near its singularities.

Suppose you wanted the square root of the difference of two similar numbers with one known to a large number of decimal digits, so the difference looses many decimals?

How would your formulae be successful then?

3)

Suppose the real number you wish to extract the root from is a surd.

How do you handle those?

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.

You can still use the sophisticated program and methods to calculate the square root since the main purpose of the formula is not that

 

1)  I said "they requires an initial guess or starting value  and / OR many calculations steps or iterations"  "OR"   means the guess is not always needed since I have developed my own method  of square root calculation METHOD WITHOUT guess entitled

"square root and cubic root of any positive real number calculation methods without an initial guess or approximate value" 

READ IT at the same link

2) THIS IS NOT a METHOD it is a formula based on a pattern discovery with one step mean that you have just to replace the number to calculate the square root without iteration or multiple steps

The trigonometric function are certainly complicated to calculate using infinite series NOT STEPS, there is no known simpler method for the trigonometric function.

But you know mathematics is not calculation only, there is an idea behind the formulas which is to illustrate the incontestable relation between the square root and the trigonometric functions according to a specific pattern discovery

 

that is why I put the condition 

m   is the number of digits of x (which include decimals)

n  ≥  m  to ensure the accuracy

and that is why I introduced The PRECISION FACTOR 10^n to emphasis the pattern and to ensure that you deal always with integers. but you can use any number with any number of decimals 

No one or any computers in this planet can and will never calculate the infinite number of decimals and no one can use an infinite number

this is another idea behind my formulas

"the limited use of correct decimals is a calculation method adopted in all scientific and engineering fields for calculations"

You can use the number as a function as it is illustrated in square root replacement

READ IT at the same link

3) That is why I said "the trigonometric function should be calculated with high precision with  higher correct decimals calculation precision" using a high precision program and "the numerical accuracy of the functions "asin" near −π/2 and π/2 and "acos" near 0 and π is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of decimals)"

READ IT at the same link

I suggest that you work on the advantages of the formulas and explore their applications since it prove the relation between the square root and the trigonometric function according to a specific pattern and law, which do not depend on me or any one, this is HOW IT WORKS. the formulas of nature are very powerful you can not change it or complain about it. but you can critic and discard any calculation METHOD including mine.

 

 

 

8 hours ago, John Cuthber said:

The method is sort of interesting, but useless.

Not quite that far. My dad knows how to do it and he was born in the 30s
It seems the manual method is still available.
https://xlinux.nist.gov/dads/HTML/squareRoot.html

JARGON

10 hours ago, Sensei said:

Why do you think that doing tangent and arc sinus, will be faster than other, already present, methods... ?

 

The methods to calculate square root, you can find on this website:

https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

 

Your equations are missing parenthesis.. everywhere..

 

After entering your above equation to calculator, or wolfram alpha, we see result:

https://www.wolframalpha.com/input/?i=tan(asin(+2+%2F+(2+%2B+0.5+*+10^-36+))+%2F+10^18)

1.570796326794896618524215... × 10^-18

 

After adding missing parenthesis:

https://www.wolframalpha.com/input/?i=tan(asin(+2+%2F+(2+%2B+0.5+*+10^-36+)))+%2F+10^18

1.41421

 

 

 

8 hours ago, mathematic said:

The linking is too convoluted.  Can you supply an example directly?

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