MY SQUAREROOT DISCOVERY by sirjon PART 5

This is the final video to complete the tricks on how to get the square roots of numbers in a quick and easy way. Learn and have fun.

WANT TO LEARN SOMETHING NEW?

Kids, try asking these questions to your parents;

1) Are they familiar with the word “square root”?

2) Have they ever tried taking the square root of an eight-digit number such as 38,775,529 without the aid of a calculator?

3) Are they familiar with the “long hand division method”?

If they answer “yes” to all my questions, maybe some how, they will agree with me, along with the other parents and teachers that such a kind of mathematics, is a “real burden” for grade school kids like you.

But I devised a much easier method that I called EASY SQUARE ROOTING designed specially, for kids like you.

WELCOME TO THE WORLD OF "ESR"


AUTHOR'S NOTE:

For parents and Math teachers who have  at least the basic idea of what a square or a square root of numbers are, it is advisable to read first,  "Easy Way of Number Squaring For Grade School Kids"

Take note: 

ESR is the revised 'edition' of my earlier blogs. I did a lot of modifications to make it much simple and easy to read - 

Easy Square Root Method for Grade School kids  

Square Edging: A New Format of Square Root Method 

Square Root: A Digit Per Digit Approach

 

I) TRICKY ESR FOR THE SQUARE ROOT OF 13

Q: What is the square root of 13?

..__
√13    = ?  

Okay, let's apply ESR in taking the square root of 13... 

.._3._                                We choose 3 because 13 is in-between 16 (the square of 4)
√13.00'00'00'00                  and 9 (square of 3) 
________________

4.0²  = 16.00                        Do the Square Averaging Method
9/8<<
           14.---
7/6<<
3.5² =  12.25  .
           28.---   / 2
           14.---
_________________

3.7²   =  9.49
3x14  =  4 2    .
            (Blank) OVER


3.6²        =     9.36
3x12       =     3 6    .
                 < 12.96 >
____________________

.._3. 6                      (Write down 6 after 3...)        
√13.00'00'00'00        

___________________

3.6²           =     9.36
3x12          =     3 6    .
                   < 12.96 >
                          36    .
3.65²        =    13.32       <<< Middle Square
4/3<<
                       13.---        <<< Lower  Quarter Square
2/1<<  
3.6²          =    12.96    .  
                       26.---  ./ 2
                       13.---
_________________

3.62²   =  12.96'04
36x4  =         14'4    .
                    (Blank)  OVER


3.61²   =  12.96'01               <<< Later, you'll need to compute this...
36x2  =          7'2  .
                   (Blank)  OVER

3.60²  = < 12.96'00 > 

 Take Note:  
Sampling the digits 2 and 1 give us a sum values  which are 'greater than' 13. In this case, we must  resort into choosing  the digit "0" as the next digit after 3.6. don't forget to paste '00' after 12.96 
_________________

.._3. 6   0                 .    (Write down 0 after 3.6 ...)        
√13.00'00'00'00  

_________________

3.60²      = < 12.96'00 > 
                          3 60   .
3.605²   =      12.99'60
7/6 <<<
                     13.---
9/8 <<<
3.61²      =     13.0321  . <<< Solve the sampling for 3.61  (See Above, Left Side))
                     26.---   / 2
                     13.---
_________________

3.607²   =  12.96'00'49
" x 14   =          5 04 0   .
                      (Blank)    OVER


3.606²   =  12.96'00'36           <<< Later, you'll also need to compute this...
" x 12  =           4'32'0  .
                      (Blank)    OVER


 3.605²   =    < 12.99'60'25 > 

Take Note:
Again, the two digits 7 and 6 failed to fit as the next candidate, so we must choose the digit '5" as the required digit. Simply copy the value for the middle square  "3.605²  = 12.99'60" but DON'T FORGET TO PASTE '25', so you'll not go wrong in computing for the next digit we're looking for. 

____________________

.._3. 6   0   5           (Write down 5 after 3.60 ...)        
√13.00'00'00'00  
___________________

3.605²     =   12.99'60'25
                            36'05  .
3.6055²    =   12.99'96'30
7/6 <<<
                     13.---
9/8 <<<
3.601      =     13.00'32'36  .
                      26.---     / 2
                      13.---
____________________

3.6057²    =  12.99'60'25'49
" x 14      =           50'47'0    .
                            (Blank)   OVER


3.6056²    =  12.99'60'25'36
" x 12     =            43'26'0    .
                            (Blank)   OVER

3.6055²   = < 12.99'96'30'25 >   <<< Don't forget to paste '25'
______________________

.._3. 6   0   5    5     (Finally, write down 5 after 3.605 ...)        
√13.00'00'00'00            

H) HELPFUL TIPS IN USING ESR

 
ESR EXPRESS

If you think that you already mastered this ESR Method, then it would be much easier to speed up your way of taking the square roots of numbers by doing some kind of modifications. This technique is much effective if the number problem involved is in small value (single or two-digit numbers such as √3, √7, √13, √37, √88,... etc), but as you become familiar with this ESR-Express, then you can probably apply it also for large valued numbers. 

Let's try again to take the square root of two...


...1......             .   (Put a decimal point after 1)
 √2.00'00'00'00        

_______________________________
1.5²     = 2.25
4/3 {           
...........   1.---
2/3 {
1.0²     = 1.00  
..........    3.25 / 2
..........    1.---   
________________________________
 
1) Find the middle square (2.25) and add it to the square of 1 (that is, 1.00)
2) Divide the sum by 2
 3) But this time, don't complete the digits involved in the quotient. The value, 1 followed by a point and ' ---' will be enough. The 'one point something' is good enough to give you an idea that the value you taken is less than 2.00
4) There's also no need to write down 1.25² ( quarter-square).. Simply write down 1.-- and on it's left side, the notation 4/3 above 1.--- and write down the notation 2/1 on the lower left side.
5) Encircle 4/3, which is the area you're are looking for. Do the sampling  
6) Once you located the 'required' digit ( at this time, the digit 4), write 4 next to 1 in your  'answer line'. Do this every time you came up with the required digit to complete your answer.



...1.  4                  .  <<< This is your 'Answer Line'
 √2.00'00'00'00       
 _______________________________________


Let's advance a little...
 ...
 ...
 ...

1.42²  = 1.96'04
14x4 =       5'6   .
(Do not write anything here)
____________________________________________

7) As much as possible, check if the sum will become 'more than' 2.00. Skip this stage by not writing any value for the supposed to be the 'sum'

Take Note:
In doing the sampling, pick first the higher digit (4 first before 3, or 2 first before 1) and if the supposed sum become greater than the number problem (2.00), then try the next lower digit number.  



  
ESR-EXPRESS FOR SQUARE ROOT OF 7

...                         . 
 √7.00'00'00'00 

Step 1:
The number problem given (that is "7"),  is in-between 9 (which is the square of 3) and 4 (which is the square of 2). We can't choose 9 because it is over or beyond 7. We then prefer 4. Therefore the first digit for our answer will be 2. (Reason: The square root of 4 is 2 or √4 = 2)

...2.                        .  (Don't forget the decimal point)
 √7.00'00'00'00 

Step 2:
Find the middle square value. Since it is between 2² and 3², the middle square will then be 2.5², which is equivalent to 6.25


LOWER QUARTER- SQUARE

If you will notice, in taking the square root of 2, the middle square is above or greater than the given number problem (2.00). That is the reason why we choose 1 (or 1.00) as the next lower value ( that supposed to be, or must always be 5 units  below the middle square). We call that quarter square below the middle square as the lower quarter-square. 

UPPER QUARTER-SQUARE

Now, in this this given problem (taking the square root of 7), the middle square value of 6.25 (which is the square of 1.5), is much lower than 7 (or 7.00). So we must consider the next higher value to that middle square ( that supposed to be, or must always be 5 units above the middle square). We will then call that quarter square above the middle square as the upper quarter-square. 


Step 3: 
Perform the Square Averaging Method for 3² and 2.5². 

3.0² = 9.00
2.5² = 6.25
........ 15.25 / 2
.......... 7.---

 INSERT the value ' 7.--- ' in-between 9.00 and 6.25

3.0² =  9.00
.......... 7.--    (Insert Here) 
2.5² =  6.25  
......... 15.25 / 2
........... 7.---

Step 4:
Notice that the quarter square value 7.--- ( seven point something), is considered above or greater than the given number problem (7.00). So, write down between that quarter square and the lower value the notation " 7/6 ".


..... 3.0² =  9.00 
9/8{
................ 7.--    (Insert Here)
7/6 {                     (Either 7 or 6 is the digit we're looking)  
.... 2.5² =   6.25   
............... 15.25 / 2 
...............   7.---

Step 5:
Perform 'SAMPLING'

2.7²    = 4.49
2x14 =   2 8   . 
..........  (Skip)

2.6²    = 4.36
2x12  =  2 4   . 
......... < 6.76 >


Step 6:
Write down 6 as the second digit in our answer

 ..2. 6                      .  (Don't forget the decimal point)
 √7.00'00'00'00 

Step 7: 
Find the next middle-square 

2.6²    = 4.36
2x12  =  2 4   . 
......... < 6.76 >
............    26  .
2.65²   = 7.02

Step 8:
The value 7.02 is greater than 7.00, so consider the lower value 2.6² = 6.76. Perform again the second 'Square Averaging Method' and so on and on...


TRY TO DO IT YOURSELF and check if you come up with the same pattern values below :

..2.  6  4   5   7     .  (Don't forget the decimal point)
 √7.00'00'00'00
_______________________
3.0² =  9.00
.......... 7.--    (Insert Here) 
7/6 }
2.5² =  6.25  
......... 15.25 / 2
........... 7.---
________________________  
 2.7²    = 4.49
2x14 =    2 8   . 
..........  (Skip)

2.6²    = 4.36
2x12 =   2 4   . 
......... < 6.76 >
............    26  .
2.65² =   7.02
4/3 }
............. 6.---
2.60²  =  6.76  . 
........... 13.76 / 2
............. 6.---
________________________

2.64²    = 6.76'16               In the pink shaded area, you will notice that the middle square 
"x8       =   20'8               is below 7.00. we must choose the next higher value above it and that
.........   < 6.96'96 >          is the 2.65². In our previous computation (see blue shaded  
........    .       2 64   .          number above left), we just wrote down 7.02. But to avoid 
2.645² =  6.99'60   .       confusion, simply paste 25 after 7.02
7/6 }                         .            Also, as a guide, draw an arrow going up on the right side of    .............. 7. ---        .       of that pink shaded area to indicate that it is a 'reverse' representation
9/8 }                          .      where the lower 'digits notation' such the 7/6 notation is in the upper
2.65²    =  7.02'25    .    portion (see green shade)
              14. --- / 2   .      
............... 7. ---         .
_________________________

 2.647²   = 6.96'96'49       Suggestion: Leave the space for the sum, "blank" and on the  
"x14     =       3 69 6  .      right side of it, draw a half circle going clockwise with the arrow 
                (Don't Add)     pointing downward to indicate the possible value is 'over'
_________________________


 2.646²   = 6.96'96'36        
"x12     =       3 16 8  .       
                (Don't Add) (Same suggestion)
__________________________

2.645²   = < 6.99'60'25 >      (No need to do SSQ at this time, simply copy the value for   
...............           26'45  .       2.645²  above. Don't forget to paste 25) 
2.6455²  =    6.99'86'70      ( Take Note: The next higher value to 2.6455² is  
.................  6.---                    2.6460² or 2.646²  . It's time to go back and compute for the 
9/8 }                                        sum of 2.646² . That's the reason why I suggest to leave the 
2.646²    =    7.00'13'16         the space for the sum of  2.646²  'blank'.)                  
                   13. ---          / 2        Also by visual check, there's no need to complete to
...................  6.---                    add 6.99'86'70 to 7.001316. Just add the first two digits of
                                                the required values (6.--- + 7.--- = 13.---) and then divide by 2
__________________________

2.6459²  = 6.99'60'25'81                     Take Note: There are instances that the 'sampling'  
" x18      =       47'61'0   .                             procedure will be more than two attempts.
.................. (Leave It Blank) OVER              As a final step, we consider the digit 7 as the
                                                                        final digit to be included in our answer
 2.6458²  = 6.99'60'25'64 
" x 16    =         42'32'0    .
................. (Leave It Blank) OVER

2.6457²  = 6.99'60'25'49 
" x 14    =        37'03'0   . 
..............< 6.99'97'28'49 > 

                (END)  
_________________________
  

G) HOW ESR WORKS: PART II

Now that we already knew the next digit after 1 (in trying to get the square root of 2 ), the next thing to do is to determine the next digit after "1.4... "

Actually, the basic rules in ESR (Easy Square Rooting) are just limited into the following simple instructions:

1) Find a single digit which index square is nearest but less than the given number problem as our first digit.
2) Find the middle square value. Determine if the middle square is greater or lesser than the given number problem.
3) Find the quarter-square value (either located below or above the middle-square). 
4) Determine which next digit to consider by sampling the 'known' digits.


In Part I, we did the 'sampling' of digits and we choose 4 as the next digit after 1. The question then is, what to do next?


MIDDLE SQUARE: (Phase 2)

The next thing that we must do is to determine again the middle square between 1.4²  and 1.5².

The middle square between 1.4² and 1.5² is 1.45² 

1.5²    = 2.25
1.45²  = ?
1.4²    = 1.96


Digit Per Digit SSQ

We can determine the value for 1.45  by doing the Three-digit SSQ

1.45²    = 1.96'25
14x10 =      14'0   .
.............. 2.10'25
 

COPY AND PASTE

But there is this more practical way of determining the next middle-square value by a technique which I called "copy and paste" (Sounds familiar?). How it works?

 We already knew the next digit after 1 is the digit 4 (The partial answer as 1.4... ). The equivalent square value for 1.4²  is 1.96. 

Activity 1: Write down 1.4² on the left side of the equal sign  and its corresponding equivalent value on the right side of the equal sign.

1.4²  = 1.96

Activity 2: Copy the digits on the left side of the equal sign, except the square sign ( ² ) and the decimal point (if any). Put it under 1.96. 

1.4²  = 1.96
............. 14

Activity 3: Add 14 and 1.96

1.4²  = 1.96
.........     14 .
...........2.10

Activity 4: Paste 25 next to the sum that we got from adding 14 to 1.96 (Optional)

1.4²  = 1.96
..........    14 .
.........  2.10'25

If you will notice, the technique that we used in determining the 'first' middle-square is based on Two Digit SSQ Ending in Five. Now that we have  "1.4... " as our partial answer, determining the 'second' middle-square needs a different approach and this technique (Copy and Paste) is the most effective and practical way we can apply.


NEXT QUARTER-SQUARE

Again, after determining that the middle-square value  (2.10) is above 2.00 (the given number problem), we can now say that the next digit to consider (the third digit), is below the digit 5. Still, we need to know the 'Quarter-Square Value' to cut down our search from four digits into two candidate digits. 

Actiivity 1: Add the upper limit (2.10) to the lower limit (1.96)

1.45²  =  2.10
1.4²    =  1.96  .
............. 4.06

Activity 2: Divide the sum by 2.

1.45²  =  2.10
1.4²    =  1.96  .
............. 4.06  ./ 2
............  2.03

Take Note: At this point, there's no need to subtract six. Why? If you complete the value for 1.425²,  what you got is:

1.425²  = 1.96'06'25              (Applying Groupee SSQ)
14x50² =      7'00      .
.............  2.03'06'25

On the other hand, if we complete the representation for both 1.45² and 1.4², what we have is:

1.45²  = 2.10'25
1.40²  = 1.96'00  .
..........  4.06'25 / 2
..........  2.03'12 remainder 1

With the value 2.03'12, we can apply "the always minus six" notation

20312 - 6 = 20306  (Take note 20312 is the floating value for 2.0312) 
1.425²  = 2.03'06'25  (Paste the digits 2 and 5)

Applying the Square Averaging Method...

1.45²   = 2.10
4/3<< 
1.425² = 2.03    (Insert the quarter square value here)
 2/1<<              (either 2 or 1 is the next digit)
1.4²     = 1.96  .
............  4.06 ./ 2
              2.03 


SAMPLING (Part 2)

Again, we determined that the quarter-square value of 2.03 is above or higher (greater), than the given number problem (2.00). So, we must consider the digits below it and that are the digits 2 and 1. Now it's time to do the Sampling, the second time around...

1.42²  = 1.96'04
14x4 =       5'6   .
............ 2.01'64   (This value is over 2.00, not accepted)

1.41²   = 1.96'01
14x2    =     2'8   .
............<1.98'81 >  (This value is accepted)


Now, we determined the third digit as 1.... 

.. 1. 4  1                
 √2.00'00'00'00

 
FOURTH AND FIFTH DIGITS

Looking for the fourth and fifth digits, all you have to do is to repeat the pattern for getting the middle square (phase 2), that is the 'Copy and Paste' technique. To know the quarter values, do the 'Square Averaging' Method

1.41²   =  1.96'01
14x2    =      2'8   .
............ <1.98'81 >  (This value is accepted)
..........          1'41  . 
1.415²=    2.00'22      (Above 2.00)  
4/3 <<                      (either 3 or 4 is the next digit)
1.4125²  =  1.99'51 
2/1 << 
1.41²     =   1.98'81  . 
..............   3'99'03 / 2
..............   1.99'51



SAMPLING (Part 3)


1.414²  =  1.98'81'16
" x 8    =        112'8   . 
............< 1.99'93'96 > (This value is accepted)




We choose 4 as the fourth digit...

1.414²  =    1.98'81'16
" x 8    =          112'8   . 
............   < 1.99'93'96 >
..............           14'14   
1.4145²  =   2.00'08'10    (Above  2.00)
 4/3 <<
1.41425² =   2.00' 01'03
 2/1<<                               (either 1 or 2 is the next digit)
1.414²     =   1.99'93'96  
................    4.00'02'06 / 2
..................  2.00' 01'03      


SAMPLING (Last Part)

1.4142²  =  1.99'93'96'04
" x 4      =          5'65'6          
............. < 1.99'99'61'64 >  (This value is accepted)


The final digit will then be 2....

.. 1. 4  1   4   2    
 √2.00'00'00'00