Sunday, August 15, 2010

Square Roots of 3-Digit/4-Digit Numbers

What is Square Edging?


The S.E. Method, in its original form, was intended as an alternative method beside the traditional long hand division method (search in NIST Square Root

http://www.itl.nist.gov/div897/sqg/dads/HTML/squareRoot.html

).

The only limitation of Square Edging is that it is only applicable in taking the square roots of numbers called perfect squares. The square roots of perfect squares are always in a “whole numbers”, never as fractions or decimal numbers. In fact, from 1 up to 100, there are only 10 perfect squares. While from 101 up to 10,000, there are only 90 perfect squares. The rest (9,900 other numbers) are useless in dealing with this format of S.E. But TRUST ME, it will WORTH A LOT.


I divided the topics into three;


1) 3D/4D.SE

2) 5D/6D.SE

3) 7D/8D.SE



Four Digit Square Edging (3D/4D.SE)


WARNING:

To easily understand this method of Square Edging, I highly recommend that you first study the SSQ Method.


Let us start by taking the square root of a four-digit number.


Given Problem: What is the square root of 2,304?


√2,304 = ?


Step 1

Count the digits of the given number. Re-group them by twos, starting from the last digit.


√23’04


Step 2

Find an index square, equal to or nearest to but less than the first group of digits of the given number. Write down the equivalent square root, below this first group of digits.


√23’04

4


Step 3:

Find a pair of index squares ending with the same last digit, as to the last digit of the given number. Write down below the last group of digits, their corresponding square roots.


04 is the last group of 23’04. The last digit of 04 is 4. There are two index squares that end with 4, these are 04 and 64. Their equivalent square roots are, 2 and 8.


√23’04

4 2

_ 8


It would be helpful, if you memorized the pairs of index squares having the same last digits. I provided one for you.


Table of Complementary Index Squares


12 = 01

92 = 81

1 + 9 = 10


22 = 04

82 = 64

2 + 8 = 10


32 = 09

72 = 49

3 + 7 = 10


42 = 16

62 = 36

4 + 6 = 10


02 = 00

52 = 25

NO PAIRS


Step 4

Copy the first digit of the upper square root to complete the lower square root.


√23’04

4 2 ← first possible square root

4 8 ← second possible square root


Now, you determined the two possible square roots, only one of them is the ‘true’ square root of 2,304


Final Step

One way to find out which of the two is the true square root, apply the 2D.SSQ


... 422 = 16’04 ← PSL

+4x2x2 = 1'6 ← SP1 (provided that 6 of 16 aligned to 0 of PSL)

............. 17’64 ← T-Sum (provided that T-Sum aligned to PSL)


....482 = 16’64 ← PSL

+4x8x2 = 6'4 ← SP1 (provided that 4 of 64 aligned to the second 6 of PSL)

............ 23’04 ← T-Sum (provided that T-Sum aligned to PSL)


Comparing the results, the second equation matches the given number. We are now sure that 48 is the correct answer.


√2,304 = 48


Helpful Tips:


There is another way of knowing which of the two square roots, is the true square root.


Tip 1: Check the first digit of any of the two possible square roots. Multiply it to a number next to it, higher by 1. Consider the product as our “square root indicator”.


The first digits of of both possible square roots of 42 and 48 are the same and that is 4

The number next to 4 is 5. 4 x 5 = 20

Sq. Rt. Indicator = 20


Tip 2: Compare the ‘square root indicator’ to the first group of digits of the given number


First Condition:

If the first group of digits is less than the square root indicator, pick the square root with lower value as your final answer


Second Condition:

If the first group of digits is greater than the square root indicator, pick the square root with higher value as your final answer.


The first group of digits of the given number is 23. It is greater than 20. So, pick 48 as the final answer.


√2,304 = 48


Three-Digit Square Edging (3D/4D.SE)


Now, let’s try a three digit number


Given Problem: What is the square root of 729?


√729 = ?


Step 1:

Re-group by twos


Important:

Take note, that if we re-group 729, it will appear as 7’29. But in the general rules of SSQ - in the process of squaring a number, the count of digits of a number must be doubled. So, a two-digit number must become four-digit number. To obey this rule, we should write 729 as 07’29.


√07’29


Step 2


√07’29

2


Step 3


√07’29

2 3

_ 7


Step 4


√07’29

2 3

2 7


Final Step

Next to 2 is 3.

2 x 3 = 6

Sq,Rt, Indicator = 6


07 > 6 (or 7 > 6)

27 > 23


√729 = 27


Author’s Note:


The truth is, this 3D/4D.SE in not really new. There are similar ideas that were posted in google and you.tube (this is one good example by Z-Math

http://www.ehow.com/how_2322332_square-root-number-mentally.html

).
I wish to give the credit to a certain Prof. Barbosa, for the Helpful tips (watch:

http://www.youtube.com/watch?v=WNJ2dCavUrA&feature=related

). I will admit, I got that from him.


But still, most of these ideas presented were limited only in taking the square roots of perfect squares in three or four digits. I will extend this method. I will show you how to take the square roots of perfect squares, even up to eight digits.

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