C) BASIC THINGS ABOUT SQUARE ROOTS

REVIEWER

In the INTRODUCTION, under the articles entitled, "Easy Way of Number Squaring For Grade School Kids", I discussed about the squares and square roots of numbers. As a starting point, let me refresh you back about that discussion.


SQUARE OF A NUMBER

Getting the ‘square value’ of a number is like doing a special kind of multiplying a number, in which both the multiplicand and the multiplier are equally the same values. Sometimes, it is described as product of a number multiplied by itself.

Examples:

2 x 2 = 4
27 x 27 = 729
146 x 146 = 21,316

The value 4 is sometimes called the ‘square value’ of 2, or simply, “square of 2”. The same way, 729 is the “square of 27” and 21,316, the “square of 146”. Sometimes, instead of writing 2x2, 27x27 or 146x146, “a small number 2 in upper right side” of a given number is used as a symbol, telling you to multiply that number by itself. So, instead of 2x2, we write 2² = 4 and 27x27 as 27² = 729, while 146x146 as 146² = 21,316.

Maybe, you are wondering why it is called ‘square’. Probably, early mathematicians noticed that the measure of the area of a square is always equal to a certain ‘number multiplied by itself’, so they named it, that way.

SQUARE ROOT

On the other hand, getting the square root of a number, need a very different way, of dividing a number. Unlike in ordinary division at which you need to mention the value of the divisor, in getting the square root of a number, both the divisor and the quotient are unknown and the difficult thing is, both divisor and the quotient must be equally in the same values. 
Examples:  
36 ÷ 3 = 12               36 ÷ 4 = 9           36 ÷ 6 = 6

In the above examples, 36 can be divided by 3 or 4 but the quotient would not be equal or the same with the divisor. Dividing 36 by 6, we can get a quotient equal to 6, which is the same exact value as to the divisor. In this situation, we can say then, that, 6 is a square root of 36.

In doing this special kind of division, the symbol √ is used before a given number (example √144, read as, “the square root of one hundred forty-four’), to tell you to look for a divisor that will give a quotient, equal to that divisor. Dividing 144 by 12, we come up with a quotient equal to 12 (144 ÷ 12 = 12). Showing equal values for both divisor and quotient we can say then, that √144 = 12.

But there are occasions that the given numbers are in large values (example, √139,876). Getting the square root of such large valued numbers requires a very tedious and tricky method called ‘long hand division’. But as a practice, small valued numbers are introduced for grade school children, to make them easier to memorize.

 
TABLE OF SQUARE ROOTS

√1 = 1
√4 = 2   
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√81 = 9   
√100 = 10
 

PERFECT SQUARES

Not all numbers from 1 to 100 give a square root in whole exact values. Most of them are in decimal values. Below is a list of examples of numbers, having no whole exact square root values:

√ 2, √3, √10, √99 , √28 , √50

Counting from 1 to 100, there are only ten numbers having square roots in ‘exact whole values’ and they are called perfect squares (or simply call them “perkies”).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

(Using a calculator, find the square roots of each numbers from 1 to 100 and write down which numbers have an exact whole numbers)
 
SET OF “PERFECT SQUARES” = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}


HOW TO DIFFERENTIATE SQUARE FROM SQUARE ROOT

A square of a number can be considered a 'special type' of a product by multiplying two identical numbers, meaning - both the multiplicand and the multiplier must be the same number. On the other hand, the square root of a number can be considered a 'specific type' of a quotient, by which a number must be divided by a a 'required' divisor, so that the quotient will become exactly the same value as to the divisor. Squaring a number, in a way, is  direct and forward, much a simpler process of choosing a number and multiply that number by itself. But square rooting (I think there's still no such 'term' that you can find in any English Dictionary), is something that one cannot easily be done. It is not the same as to process of 'Division' by which a given number (Dividend) can be divided by any assigned or chosen 'divisor' such as shown below:

Examples: 

144 ÷ 8 = 18        144 ÷ 6 = 24       144 ÷ 16 = 9       144 ÷144 = 1       144 ÷3 = 48        etc...

In the above examples, we can divide 144 in so many ways by replacing the 'divisor' by any number (such as 1, 2, 3, 4, 6, 8,... etc) but the only difference is that, the divisor and the resulting quotient will not be 'identical' or with the same numbers.

In taking (or getting) the square root of a number, the ultimate rule is that, the divisor must be exactly 'identical' to the final quotient. In this case, the process of getting the square root of a number is not at all an easy task. (Example of equal values for the divisor and quotient shown below)

144 ÷ 12 = 12 

There are instances that we confuse things - such as the number 4. Don't be confused, if you will consider the number 4, as a square root or a square.  The number 4 can either be a square or a square root. It is a fact that 4 is indeed, the square of 2 and that the same time, the square root of 16. It is important that you must 'declare' a number as the square of such number or a square root of a certain number.

4 is the square of 2 since 2 x 2 = 4  (can be represented as 2² = 4)
4 is also the square root of 16 since 16 ÷  4 = 4  (can be represented as √16 = 4)