New Fastest Way to Find Perfect Square Roots By Hand.

[olvin dsouza]

Introducing The Most Fastest Way to Find Perfect Square Roots.

Creating Successive Series Beforehand. 
0 -- 1  has zero difference.
1 --- 3    has only as difference of one.
3 ---- 6     has a difference of two.
6 ---- 10     has a difference of three.
10 ---- 15    has a difference of four.
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.
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This series is a form of infinite successive series where integers at the right hand side of the below series, that is 1, 3, 6, 10 15…. (highlighted below in yellow color) is the result of addition of positive integers in following sequence such as  2, 3, 4, 5, 6, 7, 8….(highlighted below in green color) 

0 -- 1  
         + 2
1 --- 3 
         + 3   
3 ---- 6  
           + 4  
6 ---- 10    
             + 5 
10 ---- 15    

This above successive series is essential on finding key integer c, those key integer c is then used for further calculation of finding square roots. 
While finding square root solutions, the use of this series is to simply avoid calculation timing by creating it beforehand.
Therefore, while solving the problems of finding any square, non square or any higher roots we require this series beforehand.
eg., when we are about to find the square root of 40000. 
40000/ 72 = 555
So, we need check where the 555 appears at the successive series Therefore, we need a ready successive series.
So, as shown above, one can continue go on creating same series by adding 15 + 6 = 21, 21 + 7 = 28 ….. keep record of it and use it whenever required to calculate key integer ‘c' of any other problems of finding square roots. 

 

Example Solution - ( this method works for all square and non square roots. For finding higher roots check the PDF paper attached).
To find perfect square root of √3249

1) Divide 3249 by 72 to get m.

3249 / 72 = 45.125 …..
Ignore the decimals, we get  45 as answer.

2) Using Successive Series For Finding the Key Integer c .
 0 -- 1  has zero difference. ( ignore this step).
1st Step    1 --- 3    as difference of one.
2nd Step    3 ---- 6     has a difference of two.
3rd Step      6 ---- 10     has a difference of three.
4th Step   10 ---- 15    has a difference of four.
5th Step   15 ---- 21   has a difference of five.
6th Step    21 ---- 28    has a difference of six.
7th Step    28 ---- 36    has a difference of seven.
8th Step    36 ---- 45     has a difference of eight.
9th Step  45  ----  55     has a difference of nine.

To Check, at Which Step 45 Appears.
Now, as per the above successive series,45 appears at ninth step i.e. at 45 to 55 and the series has a difference of nine  therfore, we can take c = 9 
(Note - though 45 appears at both 8th and 9th steps but we must always consider only the second step. So in this case we will consider 9th step).

3) Multiplying c by 6.
From above, we get c = 9
Therefore, 9 × 6 = 54
Now c = 54
Checking whether c is final answer by dividing 3249 by 54 . We found it is not divisible.
Therefore, we will proceed to below 4th step of checking rules.

4) Checking the Rules of Finding ‘c’ .
√X = 3249 is the integer divisible by 3. Therefore it follows the fourth rule of finding c.
Fourth Rule - If the √ X is any odd integer and is divisible by 3, then the final answer c will be adding c by 3 i.e. ‘c + 3'.
54 + 3 = 57
Since 3249 is divisible by 57
Therefore, √3249 = 57
 Note - Rules are available at the paper, check below attachment. Paper also contains explanation on finding higher roots, non square roots.
 

squareroot (1).pdf

Edited November 1, 2022 by olvin dsouza