WHY MSM-3 SO EFFECTIVE?

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.

5.0² = 25.00

4.9² = ?

4.8² = ?

4.7² = ?

4.6² = ?

4.5² = ?

4.4² = ?

4.3² = ?

4.2² = ?

4.1² = ?

4.0² = 16.00

By using the technique we learned involving the “squares of five” (See: SQUARES ENDING IN 5 http://easysqrtsforkids.blogspot.com/2010/08/five-digitsix-digit-square-edging.html );

4.5² = 20.25

5.0² = 25.00

4.9² = ?

4.8² = ?

4.7² = ?

4.6² = ?

4.5² = 20.25

4.4² = ?

4.3² = ?

4.2² = ?

4.1² = ?

4.0² = 16.00

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.