Are you tired of using the traditional method of getting the square root of a number that involved six-digits or eight-digits numbers? Try this new exciting and challenging method of getting the square root of a special group of numbers in an easier way. Welcome to square edging!
MSM -3 is the simplest and direct method of extracting the square root of any number, big or small, perfect or non-perfect.
The problem of long hand division (algorithm) is the ‘uncertainty’ of knowing the next digit. When I was in high school, all we had to do is to try a digit from 0 to 9, guessing by mere ‘chance’, and if you’re fortunate enough, make things a little bit easier.
The easiest thing in doing square rooting (is there such a word?), using the traditional method is the “first step”- looking for a square value, equal or nearest to the first group of digits of the given problem.
The next step seems to be, the ‘real’ problem.
In MSM-3, we can avoid that ‘trial and error’ method by knowing the middle square value of the “already known digit”
Example:
√20
1) Like in any rule, look for a square, equal or nearest the given number. Write down its square root value
....4.
√20.00
Our first digit is 4..
2) Now, our problem is knowing, the next digit
There are nine possible digits to choose: (1, 2, 3, 4, 5, 6, 7, 8 or 9) and only one is the correct digit.
If you will notice, 20.25 is greater than 20. It follows then, that 20 is within the area of below 4.52, (that is, within the range of 4.42 and 4.12).
3) To avoid squaring individually, the numbers from 4.1 up to 4.4, we use the digit locator.
Add the middle square value to the lower square value:
4.52 = 20.25
4.02 = 16.00
…….. 36.25
Divide the sum by 2
36.25 ÷ 2 = 18.12
To make it accurate and nearer to the square value of 4.25, always subtract 6 (ignore the decimal point)
18'12 – 6 = 18'06
4.52 = 20.25
4.42 = ?
4.32 = ?
4.252 = 18.06
4.22 = ?
4.12 = ?
4.02 = 16.00
You will notice that 20 is in-between 20.25 and 18.06, so we miniminized our choice of digits from 9 into only two – 4 and 3.
4) To know which of the two remaining digits is the proper digit, do the Digit Sampling
4.42 = 16.16
4x8 = ...3 2 .
.......... 19.36≤ 20
\4.32 = 16.09
4x6 = ...2 4 .
.......... 18.49≤ 20
Both 19.36 and 18.49 are near to 20, but the nearest is 19.36, so we choose 4 as our next digit.
4.42 = 19.36
Again, to know the next digit after 4.4, repeat the process by knowing the middle square value (4.452)
4.4_2 = ?
IMPOTANT TIP
“Knowing the Square of Numbers Ending in Five”
4.452 = ?
1) Put 25 next to the square value of already known digits
19.36’25
2) Simply copy the already known digits (44) and add to that ‘modified’ square value but make sure that the last digit of the known digits (44), is align to the third digit, from the last digit of the modified square value. (the modified square value is 19.36'25)
4.52 = 19.36’25
+ ‘’ = ..... 44 ....
.......... 19.80’25
The same rules are applied for 4.4752 and 4.47252
4.4752 = 19.98’09’25
‘’x10 = ........ 4’47 .... (Ignore the decimal point)
.............. 20.02’56’25↓ (the down arrow shows that P: 20 is in lower area)
4.47252 = 19.99’87’84’25
.......................... 44’72 ...
................ 20.00’32’56’25 ↓
I believe that one’s you master this technique, extracting the square roots of numbers will be much easier to explain to any one, including grade school children.
At the beginning, I thought only of creating or designing a method that would help grade school children, in their studies involving square roots of large numbers. The fact that I was been challenged by these following statements:
“For some reason, extracting the square roots of large numbers was more difficult than expected. For forty minutes or so, I labored to explain and demonstrate the proper method in square root extraction. The process can be very tedious. Attention to detail is necessary. Patience is definitely helpful. Yesterday was undoubtedly one of the harder days for this class”… http://www.mygba.org/extracting-square-roots/
“…many school books seem to think that since calculators can find square roots, that kids don't need to learn how to find square roots using any pencil-and-paper method.”
Somehow, I agree with them. Why do you think a child should take (or extract), the square root of a large number, in millions, only to find out that there is “no end” in their attempt to find out the square root of such number?
Square Edging, in a sense, is more ‘logical’ and ‘practical’. Children are more interested in knowing “results” which are more reasonable than ‘trivial’ answers (such as, knowing the square root of 6,4967,028). So, MSM-1 and MSM-2 are practically designed, obviously, for “perfect square” numbers in large values only, or else, use a calculator.
MSM-3, has its own “special” place.
Lets take this example:
Given:
√20
Using the traditional method, we come up with the following procedures;
Repeating the same process over and over again makes the number of digits involved to further increase (this time, from two digits, 44, into three, 447), while guessing the next possible digit seems to become very difficult to attain
. (447x20x ?) + ?2 ≤ 19100
This is the “main problem” of the traditional method of extracting the square roots of numbers ;
1) The repeated “trial and error” method to come up with the right digit.
2) As you try to add another digit, the longer the multiplication and subtraction of numbers involved
APPLYING THE UNIVERSAL SQUARE EDGING METHOD
Given:
√20
Step 1:
Like in any rule, look for a square, equal or nearest the given number. Write down its square root value.
....4.
√20.00
Step 2:
Create a “P-Chk”
P: 20
4.5 : 20.25 ↓
4↓
The down arrow, indicates that the next digit is below 5 (or 4↓)
Step 3:
Create A Digit Locator. (Note: it is similar to the Square Root Locator, except this time we’re simply looking for the next digit after 4) Please review :
R.A.R. stands for, Reference and All the digits to the Right but this time, you will learn another technique:
L.A.L. stands for Last digit and All the digits to the Left. I will not explain here how I got it (except if requested). It is still based on the same equation (A+B)2 = A2 + 2AB + B2.
We know that 4.42 = 19.36
If we put a digit next to 4.4, example, a digit “2”
4.422 = ?
Instruction 1:
Write down the square of the original value (4.42) = 19.36 then write next to it the square of 2
4.422 = 19.36’04
Instruction 2:
Following the LAL multiplication technique, multiply the last digit (2) to all to the left digits ( that is, 44, ignore the decimal point).
44 x 2 = 88
Instruction 3:
Don’t forget the DTP, "double the product”
88 x 2 = 176
As a practice, always double the value of the last digit before multiplying to all digits to the left.
44x4 = 176
Instruction 4:
To make it clear, the square of 44.2 must end in 4. So, the last digit of 176 must be aligned to the next digit to the left of the last digit of19.36’04
.... 4.422 = 19.36’04
(+) 44x4 = .... 17’6 .
.... 4.422 = 19.53’64 (which is the true square root of 4.42, check in your calculator)
Now, let’s go back to our main discussion.
We already know that the second digit of our square root is 4, so our “partial answer”will be equal to “4.4”. The next question is…what will be the next digit?
4.4 ?
Step 6:
Create a 2nd P-Chk for the next digit after 4.4
In data above, the square value for 4.4 is
4.42 = 19.36
2nd P-Chk
P: 20.00 00
4.452 = 19.36’25
‘’x10 = .... 44 .... (Ignore the decimal point)
............ 19.80’25 ↑ (the up arrow shows that P: 20 is in upper area)
5↑
The 5↑ indicates that the next digit is above 5 (There are four possible digits, 6, 7, 8 or 9 and only one is acceptable).
Step 7:
To easily determine which group the next digit belongs (9, 8 group or 7, 6 group), use a 2nd Digit Locator
2nd Digit Locator
(4.502) = 20.25’00 (already given, it is the same as to 4.52 = 20.25)
(M.V.) = 20.02’62
(4.452) = 19.80’25
..............40.05’25 / 2
.............. 20.02’62
The M.V. (Middle Value) shows that it is greater than 20.
Therefore, the possible digit belongs to “7, 6”group.
Let us simplify the equation of the 2nd Locator
2nd Digit Locator
9, 8 / 20.25
.......\ 20.02 \ ↓
........19.80 / 7, 6 (20 is between 20.2 and 19.80)
........ 40.05’/ 2
........ 20.02
The 2nd digit locator gives us a clue that the next digit could either be 7 or 6
Step 8:
Again, use 7 or 6 as sample, (do theDigit Sampling)
.... 4.472 = 19.36’49
(+)’’x14 = .... 61’6 .
.... 4.472 = 19.98’09
“There is no need to solve for 6, because its answer would be much lesser than 4.47”
Making Sure of Your Choice of Digit:
To avoid the possibility that the 7 is not yet the correct digit, choose 8 as a ‘sample’
.... 4.482 = 19.36’64
(+)’’x16 = .... 70’4 .
.... 4.482 = ... w/out completing the addition, you will notice that if we add 3 to 7, the carry 1 makes the 19 to become 20, giving us a clue that the 4.482 > 20.
This result give as an assurance that 7 is the correct, next digit after 4.4
Therefore, we accept this equation;
4.472 = 19.98’09
Step 9:
For the next digits, simply repeat these three basic rules of square edging
1) Create a parameter checker (P-Chk)
2) Determine the correct digit by using the Digit Locator
3) To make sure which digit is the appropriate digit, do the Digit Sampling
3rd P Chk
P: 20.00 00
4.4752 = 19.98’09’25
‘’x10 = ........ 4’47 .... (Ignore the decimal point)
.............. 20.02’56’25↓ (the down arrow shows that P: 20 is in lower area)
4↓
3rd D Loc
20.02
20.>.\ ↓ (The symbol “20.>” means, 20 point “something”)
19.98 / 2, 1 (the digit locator is giving us a hint that the next digit could either be 2 or 1)
40.00 / 2
20.00
Digit Sampling
4.4722 = 19.98’09’04
....’’x4 = ...... 1’78’8 .
4.4722 = 19.99’87’84 (acceptable)
To make sure, check digit “3”
4.4732 = 19.98’09’09
....’’x6 = ...... 2’68’2 .
4.4732 = 20...0.......... > 20 (over)
We are now sure that the next digit is 2, therefore;
4.4722 = 19.99’87’84
P-Chk
4.47252 = 19.99’87’84’25
.......................... 44’72 ...
.................20.00’32’56’25 ↓
4↓
D-Loc
20.00’33 (round-off value)
20.00’10 \ ↓
19.99’88 / 2, 1
40.00’21 / 2
20.00’10
D. Sampling
4.47222 = 19.99’87’84’04
‘’x 4..... = ..........17’88’8 .
................ 20. ... 04’ ...... > 20 (over)
4.47212 = 19.99’87’84’01
‘’x 2..... = ........... 8’94’4 .
................ 19.99’96’78’41 (acceptable)
Therefore,
√20 = 4.4721 (approximately)
If you want, you can continue the process by repeatedly doing the parameter checking, digit locating and digit sampling.
You will notice that the middle missing digit (N). could either be 1, 3, 6 or 8(The given problem belongs to Group 1 as indicated by the third to the last digit of √40’32’25).
Take note too, that the P-Chk also give us a hint that the ‘true square root’ is below 650 (Due to the notation 4↓),
Step 3
Underline the 1 and 3, indicating that 6 and 8 are eliminated:
P: 40’32
65: 42’25↓
4↓N : {0, 1, 2, 3, 4}
Step 4
Create A Square Root Locator
↓ 615 ..... 635 ↑
42’25 \
39’06 / ↑
36 .....
78’25 / 2
39’12
Therefore:
√ 403,225 = 635
MSM-2 seems to be simple when dealing with “squares of 5”in five or six digits. All you have to do is to determine which group the missing “middle-digit” belongs and slim down the possibility by using the P-Chk.
“SQUARES OF FIVE” IN EIGHT DIGITS
Case No. 1
Given Problem:
√ 21,949,225
Step 1:
Use the SE telegram procedure as initial instructions:
√ 21’94’92’25
.... 4 ........... 5
N : 1, 3, 6, 8
Step 2:
Create A P-Chk
P : 21’94
45: 20’25 ↑
5↑ M : 5, 6, 7, 8, 9
The N notation is telling us that ‘the second to the last digit’ of the possible square root (which is still unknown), could either ‘any’ of the indicated digits on its right side (Only one of them, 1, 3, 6 or 8 is the correct digit).
The M notation also tells us, that the second digit of the possible square roots could be either be 5, 6, 7, 8 or 9. So we have “4x5” combinations, starting with:
4515, 4535, 4565, 4585, 4615, 4635, 4665, 4685, 4715, 4735, 4765, 4785, 4815, 4835, 4865, 4885, 4915, 4935, 4965 and 4985 and only one of the 20 possible square root is the true square root
Step 3:
To slim down the possibilities, we must use the1st Square Root Locator .
25 .....
22’56 \ ↓
20’25 /
45’25 / 2
22’62
Take note that the ‘down arrow’ ↓ indicates that the possible square roots are within the M: “5, 6 and 7” range.
Modify Step 2:
1) As a rule, underline 5, 6, 7(or draw a box, enclosing 5, 6 and 7 of the P-Chk )
2) Insert the notation “/_750/” on the second line of P-Chk . The purpose of this is determine the limit of set of possible square roots
P : 21’94
45: 20’25 ↑ /4750/
5↑ M : 5, 6, 7, 8, 9
Step 4:
On the next column, write down the set of possible square roots
... ↓ ........ ↑
4615 ..... 4735
4585 ..... 4715
4565 ..... 4685
4535 ..... 4665
4515 ..... 4635
This is the “tricky part”:
1) Starting with 4515, write it down by starting from the bottom part, going upward up to 4615 (five elements at each set). Put a ‘down arrow’ symbol on top of the first set
2) Write down the next batch of possible square roots on the next column, starting from 4635 up to 4735, following the same procedures. This time, put an ‘up arrow’.
3) Adding the notation /4750/ on the second line of the P-Chk give us the idea that the ‘upper limit’ only include 4735. Therefore, 4765 and above, are eliminated.
4) Indicate a down arrow for the left group and an up arrow for the right group
Step 5:
Use the 2nd Square Root Locator, to determine which one of the two groups should be eliminated
22’56 \
21’40 / ↑
20’25
42’81 / 2
21’40
The ‘up arrow’ indicates which group should remain. Enclose inside a square the second group, indicating that it is where the true square root belongs. So the left group is eliminated.
4735
4715
4685
4665
4635
Important:
Enclose and put an ‘up arrow’ ↑ for {4685, 4715, and 4735} while a ‘down arrow’↓ for {4635, 4665 and 4685}
... ↑
4735
4715
4685
... ↓
4685
4665
4635
Step 5:
Continue the process of averaging the square values ( 3rd Square root locator)
22’56
21’98 \ ↓
21’40 /
43’96 / 2
21’98
The down arrow ↓ eliminates - 4685, 4715, and 4735. There are still 3 remaining square roots:
..↑ / 4685
.... \ 4665 \
...... 4635 / ↓
Step 6:
Continue the averaging and elimination process (4th square root locator)
21’98 \
21’69 / ↑
21’40
43’38 / 2
21’69
4635 is eliminated.
↓4665 ....... 4685↑
21’98 \
21’83 / ↑
21’69
43’67 / 2
21’83
The up arrow indicates that the true square root is 4685
Want to learn the easy and fastest way of taking the square roots of big numbers (even in millions but provided that they are in perfect squares and not ending in 25), without the painstaking trial and error method of the traditional long hand division?
