Square Roots of Big Numbers

Now, it’s time to deal with perfect square numbers in millions. It involved 7-digit or 8-digit numbers.

Given problem:

√ 22,877,089

Take note that this time, the number is in million (read as – twenty two million, eight hundred seventy seven thousand, eighty nine)

The instructions are the same except that there is an added letter – the letter M.

As an initial instruction, I request you to separate your paper into two columns – the left column and the right column

Step 1

Determine the two IPS (initial possible square roots). Write them on the left column

√ 22’87’70’89

4 M N 3

4 M N 7

Step 2

Create a Parameter Checker (P-Chk). Write them on the right column next to step 1

Helpful Tips:

1) Write first, Let P = 22’87..

2) Follow it up by writing the middle numbers (45..² = 20’25..). Leave two spaces above and below.

3) Check the middle square value (20’25..)

First Condition: If P is less than the middle square value, write down below it the lower numbers ( 40..² = 16’00..). Enclose them and write 4↓ on the left side. Write the letter P on the right side.

Second Condition: If P is greater than the middle square value, write down above it the higher numbers ( 50..² = 25’00..). Enclose them and write 4↓ on the left side. Write the letter P on the right side.

Third Condition: If you choose 4 , simply write down M = 4, 3, 2, 1, 0

Fourth Condition: If you choose 4 , simply write down M = 4, 3, 2, 1, 0

Optional: You could ignore writing the whole data (see example in 5D/6D.SE). Leave a blank or write it as “ . . . = . . . ”

P Chk

...... Let P = 22’87…

..... / 50..² = 25’00.. \

..5↑\ 45..² = 20’25.. / P

.......... . . . = . . .

M = 5, 6, 7, 8, 9

Step 3:

Find out the missing digits

Left Column, Second Row

4MN3² = ..09

. Nx3x2 = ..8 .

………… ..89

Nx6 = ..8 …. N = 3, 8

4M33

4M83

Right Column, Second Row

4MN7² = ..49

..Nx7x2 = ..4 .

………… ..89

Nx4 = ..4 …. N = 1, 6

4M17

4M67

Left Column, 3^rd Row

4M33² = ..09’09

... 3x3x2 = ..1’8 .

..……..…..10’89

..Mx3x2 = ..0

................. ..0’89

Mx6 = ..0 …. M = 0, 5

4533

Right column, 3^rd row

4M17² = ..01’49

... 1x7x2 = ..1’4 .

............... ..02’89

..Mx7x2 = ..8

................. ..0’89

Mx4= ..8 …. M = 2, 7

4717

Left Column, 4^th row

4M83² = ..64’09

... 8x3x2 = ..4’8 .

............... ..68’89

..Mx3x2 = ..2

................. ..0’89

Mx6 = ..2 …. M = 2, 7

4783

Right Column, 4^th row

4M67² = ..36’49

... 6x7x2 = ..8’4 .

............... ..44’89

..Mx7x2 = ..6

................. ..0’89

Mx4= ..6 …. M = 4, 9

4967

Left Column, 5^th row

Sq. Rt. Loc 1

↓4533 4717 … 4783 4967↑

....../ H = 25’00.. \

.. ↑ \M = 22’56.. / P

….... L = 20’25..

Right Column, 5^th row

Sq. Rt. Loc 2

↓4783 …. 4967↑

....... H = 25’00..

. ↓ / M = 23’78..\ P

.......\ L = 22’56.. /

Left Column, last row

√ 22,877,089 = 4,783

Question: How I got the Middle Numbers of the Sq. Rt. Loc 1?

… H = 25’00..

… M = ?

…. L = 20’25..

……… 45’25 / 2

……… 22’62 – 6 = 22’56

1) Add H and L

2) Divide by 2 (I use this “___/2” to represent division by 2)

3) Subtract the quotient by 6. Why? Doing it makes the middle square value much nearer to its ‘true’ square value.

Important

You do this only on the first square root locator.

On Sq, Rt. Loc. 2, do the same procedures except the last one.

Note from the Author:

The use of up and down arrows and the letters H, M and L will simplify the equations. Also, omitting the "nn" and "mm" notations save your time and effort in writing unnecessary things. But for beginners, I strictly request to follow the procedures on the topic – “Five/Six-Digit Equare Edging”