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            

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   

F) HOW ESR WORKS: PART I

If you will notice, 2.00'00'00'00 (or simply 2.00), is in-between the square of 1 (or 1² ) and the square of 2 (or 2² )

2²  = 4.00
1²  = 1.00

The middle half square of 1.0² (which is also equivalent to 1²)  and 2.0² (another way of representing 2²) is 1.5²

2.0²  = 4.00   Upper Limit 
1.5²  = 2.25   Middle Square 
1.0²  = 1.00   Lower Limit



MIDDLE SQUARE: Phase 1

In this new method of 'taking the square root of a number', we can 'use' the method of getting the squares of numbers ending in five (Review Here) as a 'tool' to make it easier for us to 'detect' where the next digit could be located. To be candid, I thought learning the technique on how to get the squares of two-digit numbers ending in five is just a 'fancy' way of showing off. I was wondering what is the significance or use of it. Now, I realized that it is indeed, very important.

Comparing the square of 1.5² (which is equivalent to 2.25) to 2.00 (given problem), we can say...

 2.25 > 2.00

The value 2.00 is above 1.00, which is the square of 1 but below 4.00 (the square of 2). Determining the middle square between 1.0 and 2.0 will give us instantly, an idea of where the value 2.00 could be located. Actually, there are nine digits to select (from 1, 2, 3... up to 9) and it is not practical that we randomly choose any of the these digits to suspect as the next digit to consider. Determining that the 'middle square' is above (higher or greater than 2.00), we eliminated four digits above 5 (these are the digits 6, 7, 8 and 9). We then limited our selection into only four digits (4, 3, 2 and 1) and only one of them is the required digit. (See the shaded area below). 

2.0² =  4.00   .
1.9² =  ?
1.8² =  ?
1.7² =  ?
1.6² =  ?
1.5² =  2.25   .      
1.4² =  ?                                       . 
1.3² = ?                                        .     (2.00 is possibly located in this shaded area)
1.2² =  ?                                       .  
1.1² =  ?                                       .
1.0² =  1.00   . 

WE still have four digits to select to know the next digit. But we can still trim it down into a more 
comfortable set of selection.


QUARTER SQUARE

If you will notice, 2.00 is in-between 2.25 ( the square of 1.5) and 1.00 (the square of 1). The 'middle' value between them is the 'square of 1.25'... 

1.5² =  2.25                                  .                             
1.4² =  ?                                       . 
1.3² = ?                                        .    
1.25² = ???                                      Middle portion between 1.5²  and 1.00²
1.2² =  ?                                       .  
1.1² =  ?                                       .
1.0² =  1.00                                  .   

There are three ways to get the square value of 1.25 ...

 1) Digit-per-Digit SSQ

1.25²    = 1.44'25
12x10 =      12'0  
.............. 1.56'25

2) Ending in Five SSQ

Next  to 12 is 13 (that is, 12 + 1 = 13)

1.25²  = (12x13)'25
.........  = 1.56'25       (Don't forget the proper place for the decimal point)
 

Square Averaging Method   

The third method is somehow, the more practical way we can use. Doing this, we will further limit the possible next digit to a much manageable shaded area.

Activity 1: Add the square of 1.5 and the square of 1 (or written as 1.0²) 

1.5²  =  2.25
1.0²  =  1.00  .
...........   3.25

Activity 2: Divide the sum by 2

1.5²  =  2.25
1.0²  =  1.00  .
...........   3.25/ 2
...........   1.62 remainder 1

Activity 3: Ignore the remainder and subtract 6 to the floating value of 1.62 (Note: When we say floating value, we must ignore temporarily the decimal point as part of the true value of 1.62)

162 - 6 = 156  or 1.56 (don't forget to insert back the decimal point at its proper place)

Activity 4: If you want, just paste 25 after 1.56 so that value we got is exactly equivalent to the square of 1.25

1.5²   =  2.25
1.25² =  1.56'25
1.0²   =  1.00  .
...........   3.25/ 2
...........   1.62 remainder 1  (Take note: Ignore the remainder then 162 - 6 = 156 => 1.56)


"The Always Minus Six" Notation  

In ESR (Easy Square Rooting), it is more convenient to use the third method (Square Averaging Method),  to get the 'quarter square value' of the numbers involved. But simply getting the average between two square values involved (this time, 1.00 and 2.25), will not reflect the actual true value of what we're trying to determine. So we must always follow the rule that, "we must always subtract six" to the qoutient we got from dividing the sum of the two numbers involved.   


REVIEW 

Let's review back the steps in doing ESR:   

Question:  What is square root of  2 ?

......
√2  = ?
 
 Step 1: Instead of  putting two zeroes after 2 (see this), let's put 8 zeroes grouped by in twos

 ..........               .
√2.00'00'00'00

Step 2 : Find a number that when "being multiplied by itself" will bring out a product  that is near but lesser to that asked number
  ..1......               .   (Put a decimal point after 1)
 √2.00'00'00'00      
   we choose 1 because 1 x 1 = 1 ( or 1² = 1)

Step 3: Determine the 'middle square value" between the upper limit (that is, 2² ) and the lower limit (that is, 1² )

2.0²  = 4.00   Upper Limit 
1.5²  = 2.25   Middle Square 
1.0²  = 1.00   Lower Limit  

Step 4: Check if the 'middle square' is higher or lower than to the  given 'number' problem.

2.25 > 2.00, so the next digit could be, either 1,2,3 or 4  

Step 5: Determine the 'quarter square value' by doing the 'square averaging method'...

1.5²   =  2.25
1.25² =  1.56 
1.0²   =  1.00  .
...........   3.25/ 2
...........   1.62      (Take note: 162 - 6 = 156 => 1.56)



SAMPLING    


If you will notice, there are four digits that is in-between 1.0² and 1.5² (see table below)

1.5²  = 2.25
.. 4  = ?
.. 3  = ?
.. 2  = ?
.. 1  = ?
1.0²  = 1.00

Now, determining the value of the quarter-square, our selection from four digits is now cut down into two digits only


1.5²  = 2.25
.. 4  = ?               (2.00 is in-between 2.25 and 1.56'25, therefore,
.. 3  = ?                    it is either the digit 3 or 4 is the possible next digit to consider)
2.25  = 1.56'25
.. 2  = ?
.. 1  = ?
1.0²  = 1.00

To become sure of which of the two (between the digits 3 and 4),  is the rightful or precise digit to consider, we need to 'sample' them one at a time using the Two-Digit SSQ

1.4²  = 1.16
1x8 =     8     
........< 1.96 >   (Note: Enclose your answer in < > to indicate that it is the nearest value)

1.3²  = 1.09
1x6 =     6   .
.......... 1.69     (cannot be considered nearest to 2.00)

In reality, there's no need to do the squaring for 1.3², for reason that 1.4² can be considered the 'nearest' and yet lesser value than 2.00. So we choose 4 as the next digit to consider.


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