A. INTRODUCTION

Kids,do you know that besides adding, subtracting, multiplying and dividing numbers, there are two other very interesting math operations that are also very useful in many activities? If you wish to know the area of a square or the measures of the sides of a right triangle, two special math operations - getting the square values or getting the square roots of numbers, are some of the more advanced math operations that you must learn.

They are advanced because unlike ordinary multiplication and division, their operations (methods of getting the answers), are indeed, much more tricky and difficult.

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, needs 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 “PERKS”).

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}
 
For fun way of naming things, let's call them Perkies, pertaining to "squares of whole numbers".


TABLE OF SQUARES

1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81 
10² = 100

G. Squaring Numbers Ending in Five

Squares of Two-Digit Numbers Ending In Five      


Look at the list of two-digit numbers below and find out a pattern that you might notice:    

05² =      25

15² =    225

25² =    625

35² = 1,225

45² = 2,025

55² = 3,025

65² = 4,225

75² = 5,625

85² = 7,225

95² = 9,025

1) All the given numbers end with 5 (the last digit is always 5)
2) Their corresponding square values are also, always end with '25' (the last two digits are always the digits 2 and 5)
3) With another pattern that might 'amuse' you. You want to find it out?  


Let us rearrange the appearance of the square values of the two-digit numbers ending in 5 according to the first golden rule of SSQ - "the count of digits doubles as you square a number".


INDEX SQUARES OF TWO-DIGIT NUMBERS ENDING IN FIVE 

05² = 00'25

15² = 02'25

25² = 06'25

35² = 12'25

45² = 20'25

55² = 30'25

65² = 42'25

75² = 56'25

85² = 72'25

95² = 90'25

Check This Out

1) The first two digits of the square of 05 are 0 0 and of course, its last two digits are 2 and 5. Take note that 0 in 05² when "multiplied to a number next to it, higher by 1", will still give a product equal to zero (0). (Take note: Any number multiplied by zero is always equal to zero)

0 x 1 = 0      (05² = 00'25)  (1 is next to 0, higher by 1)

2) The first two digits of the square of 15 are 0 2 and of course, its last two digits are 2 and 5. Take note that 1 in 15² when "multiplied to a number next to it, higher by 1", will give us a product equal to 2

1 x 2 = 2    (15² = 02'25)  (2 is next to 1, higher by 1)


3) The first two digits of the square of 25 are 0 6 and of course, its last two digits are 2 and 5. Take note that 2 in 25² when "multiplied to a number next to it, higher by 1", will give us a product equal to 6

2x 3 = 6     (25² = 06'25)  (3 is next to 2, higher by 1)


Repeating the same pattern for the other numbers, we can now easily get the square of any two-digit number ending in 5.

95²  = ?

Step 1: Simply write down the last two digits as 2 and 5

95²  = _ _'25

Step 2: Multiply the first digit (in this case, 9) by a number higher by 1 to it

Since 9 + 1 = 10

9 x 10 = 90

Step 3: The product (90) will be the first two digits that will complete the square value

95²  = 90'25


Exercise: (Do It Yourself)

1) Prove that 45  is indeed equal to 20'25 using the 'Digit Per Digit SSQ Method'

2) 257²  = ?

3) 45,65²  =  ?

4) 35, 257² = ?

(Hint:  For questions 2 - 4, Try to use Groupee Squaring)

F. Multiple-Digit Number Group Squaring

Now that you have the idea of squaring numbers 'digit per digit' using SSQ, it is also an advantage if you can do things mentally or in short cuts.

In INTRODUCTION, I required you to remember the "index squares" of the single-digit numbers from 0 to 9. But this time, it will also be an "added advantage" if you can also memorize the squares of other numbers, to at least up to 15.( To be frank, I myself can only memorize the squares of numbers up to 15)

10²  = 100       11²  = 121        12²   = 144        13²  = 169        14²  =  196        15²  = 225

 
In my third article (Three-Digit Number Squaring), I introduced to you to the word "groupee" as the fun way of calling a group of digits considered as one unit. The above numbers from 10 up to 15 can be considered as groupees, They consist of two digits that act as one number. We can no longer say 11 as "one-one" but rather, as, "eleven", neither 10 as "one-zero" but rather as "ten".

In SSQ, there are instances (if you know the technique), that we can divide a multiple-digit number into groups of digits (groupees) as a "short way" of squaring  numbers. To further understand what I'm talking about, it is important that I must first, show you an example


Question:

124²  = ?


There are two ways of getting the square value of 124


Digit Per Digit SSQ Approach

(Example 1)
 

124² = (Blank)
12..²  = 01'04..
1x4   =      4         . 
124² =   01'44'16
" x 8  =        9'6   .
.............01'53'76 

Groupee SSQ Approach

If you look closely at the given number 124, you may notice that it consist of a groupee (12) and a single digit 4. So, we can simplify our way of squaring that number by...

(Example 2)

 124²  =  01'44'16 
12x8    .        9'6   .          
 .............01'53'76                         


Groupee Index Squares    

10²  = 01'00       
11²  = 01'21        
12²  = 01'44        
13²  = 01'69        
14²  = 01'96        
15²  = 02'25


If you are gifted enough, that you can memorize the squares of numbers more than the listed numbers above, then it will be really an easy thing for you to do "groupee squaring"  

(Example 3)

412²  = ?

If you will notice, the given number 412, is again, a combination of a single digit followed by a groupee...

 412²  = 16'01'44
 4x24 = .    96     . 
.............16'97'44

Again, if you look at them closely, 124 and 412 are somehow, related, in a sense that we use the same index squares (16 and 01'44), except their positions are interchanged


Adjust SP Two Digits to The Left


412²  = 16'01'44 ← PSL
 4x24 =.    96^^ ← SP ( you might notice that 6 of 96 is aligned to second 1 of 16'01'44) 
.............16'97'44


It is very important that you 'be more careful' in aligning the sub-product  to the PSL when doing 'groupee squaring'. Take note that in Example 2, the last digit is a single digit 4, so, we simply adjust the SP one digit to the left. While in Example 3, the last digit is not exactly the single digit 2 but the groupee '12', which happens to be a two-digit groupee. In that case, adjust the last digit of the SP, two digits to the left.


(Example 3)

1,413²  = ?


1413²   = 01'96'01'69
14x26  =        3'64     .
...............01'99'65'69




 

E. Squaring Multiple-Digit Numbers

Four-Digit Number Squaring

Question: What is the square of 2,384?

2,384² = ?

Following the rules we just learned...

2,384²   = (leave this space blank) 
23..²     = 04’09.. ← PSL 1 
+2x3x2=   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)  
238..²  = 05’29'64   ← PSL 2   
23x16  = .   36'8  .   ← SP 2  (8 of 36'8 is aligned to 6 of 05'29'64)
2384²=   05'66'44'16   ← PSL 3 (new PSL) 
" x 8 =.        1'90'4   .  ← SP3 (new SP) (4 of 1'90'4 aligned to 1 of 05'66'44'16)
..............05'68'34'56   ← T-Sum

Write in the 'blank space' above 23..² , the actual number value of  2,384²  

2,384²   = 5,683,456   

You might noticed that the same procedures are applied in squaring a multiple-digit number such as in the four-digit number 2384. The only difference is that as the digits increases, so do the way we do the SSQ method.

As a summary:

1) Do the two-digit squaring for first two digits 2 and 3.
2) The partial sum become part of the new PSL (PSL2) and we insert the next digit 8 along with its equivalent index square 64.
3) Add the SP2 to  PSL2 to get the new partial sum.
4) The partial sum become part of another new PSL (PSL3) and we, again insert the next digit 4  along with its equivalent index square 16.
5) Add the new Sub-product (SP3) to the PSL3 to get the Total Sum (T-Sum)

The " Notation

You might also noticed that instead of writing, 238 x 8 at the left side of the SP3 line (look the yellow shade below), we simply indicated the " notation. We do that so that we can save time and effort in repeating to write numbers.Just makes it sure that you properly multiply the "all to the left" of the last digit, to the double value of the last digit itself ( remember Dou-LAL?) 

23..²     = 04’09.. ← PSL 1 
+2x3x2=   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)  
238..²  = 05’29'64   ← PSL 2   
23x16  = .   36'8  .   ← SP 2  (8 of 36'8 is aligned to 6 of 05'29'64)
2384²=   05'66'44'16   ← PSL 3 (new PSL) 
" x 8 =.        1'90'4   .  ← SP3 (new SP) (4 of 1'90'4 aligned to 1 of 05'66'44'16)
..............05'68'34'56   ← T-Sum

Underlined Last Digit

You might also noticed that the 'involved' last digit is always 'underlined'. As a beginner, you are required to do this but as you become familiar with the method of Systematic Squaring (SSQ), there's no need to do this. Just make sure that you're doing the right thing.  


23..²     = 04’09.. ← PSL 1 
+2x3x2=   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)  
238..²  = 05’29'64   ← PSL 2   
23x16  = .   36'8  .   ← SP 2  (8 of 36'8 is aligned to 6 of 05'29'64)
2384²=   05'66'44'16   ← PSL 3 (new PSL) 
" x 8 =.        1'90'4   .  ← SP3 (new SP) (4 of 1'90'4 aligned to 1 of 05'66'44'16)
..............05'68'34'56   ← T-Sum




Exercise: (Do It Yourself)

Now that you have the clear idea of how SSQ works, try to square the following given numbers:

1) 45,896

2) 398

3) 978,675

C. Squaring Three-Digit Numbers

Squaring a three-digit number using SSQ is as simple as squaring a "two-digit" number. It is my advise that you must first read my first two articles (entitled, "Introduction" and "Squaring Two Digit Numbers") to fully appreciate how SSQ works for three-digit numbers.

Question: What is the square of 238?

238² = ?


General Rule:

In  squaring number using SSQ,  no matter how many digits it has (or involved) - you must always 'remember' that it is much easier to deal it by 'doing things' two separate digits at a time.


238 is a three-digit number. By general rule, we must deal (or consider) 238 as three separate digits as 2 - 3 - 8.

Now, focus your attention on the first two separate digits 2 and 3...

23..² = (leave this space blank)
23..²      = 04’09.. ← PSL  
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
................05’29..   ←Partial Sum (remember, this is not yet the final answer)

You might notice that the above computation is the same as to the example given on Squaring Two-Digit Numbers.


Groupee

What we actually done is the same process or procedure as we learned in squaring two-digit number. Now, in this case, there is a remaining digit 8, in the given value 238 that we must deal with.

At the moment, don't bother yourself about that remaining digit. Let's first  consider the square of 23.

23..² = 05'29.. 

In reality, 05'29 is the 'actual square' of 23. But in SSQ, as fun way of memorizing words, let's call 05'29 as the square of the 'groupee' 23.

Groupee, is a group of digits considered as one unit. 2 and 3 as separate digits in the first procedure that we had done (Two-Digit SSQ)., will now be a groupee known as 23.

Its corresponding 'square' will then become part of the new PSL (Partial Square Line) along with a new 'index  square' of the remaining digit 8.

238² = 05'29'64

Now, as a general rule of SSQ, we must get the sub-product (new SP), using the DouLAL Multiplication Pattern

Activity 1: "DouL" means Double the Last digit...

8 x 2 = 16  

Activity 2: "AL" means Multiply to All in Its Left

You may notice that the 'all to the left' of 8 is 23 in the given value 238, so...

23 x 16 = 368

The last procedure will then be, add the new sub-product (SP) to the new Partial Square Line (PSL) to get the T-Sum.

238²   = 05'29'64   ← new PSL 
23x16 =     36'8  .   ← new SP1 
............05'66'44    ← T-Sum


Practical Technique for Three-Digit SSQ

Step 1: Consider squaring the first two digits using SSQ method..


238² = (leave this space blank)
23..²      = 04’09.. ← PSL  
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
................05’29..   ←Partial Sum 

Step 2:  Create a new PSL

Activity 1:  To conserve time and effort  (as well as your ball pen's ink), write down at the left side of the partial sum, 23 along with the third digit 8 and underline 8. Don't forget the square sign ( _ ² ).

Activity 2: At the right of the partial sum, write down the index square for 8. 


238² = (leave this space blank)
23..²      = 04’09.. ← PSL 1
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
238²     =  05’29'64   ← PSL 2  (the new PSL)

Step 3: Find the new Sub-product. Add it to the new PSL to get the T-Sum.

238² = (leave this space blank)
23..²      = 04’09.. ← PSL 1
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
238²   =    05’29'64   ← PSL 2  (the new PSL)
23x16 =    .   36'8  .   ← SP 2  (the new SP)
.............   05'66'44    ← T-Sum

Step 4: As the final step, write down the actual value of T-Sum.

 Remember that we left blank the space after we wrote down 238²   
 Now, we can use this space to write down the 'actual value' of 05'66'44

238²     = 56, 644  ←  "Final Answer"  
23..²      = 04’09.. ← PSL 1
+2x3x2 =    1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
238²   =   05’29'64   ← PSL 2  (the new PSL)
23x16 =   .   36'8  .   ← SP 2  (the new SP)
.............  05'66'44    ← T-Sum


"Double Dots" Sign

Maybe, you noticed that there are those 'double dots' after 23 in 23..² and also after 04'09 in  04’09.. . We simply use this sign to indicate that there is a following digit or digits after 23. 
As part of the rule, make it as a practice to include the 'double dots' in performing SSQ. 


 Now, it's time to level-up. You are now ready to do the squaring of numbers in more than three digits. Next topic, the multiple-digit squaring, enjoy it.

                                                                                                      

B. Squaring Two-Digit Numbers

Common Way of Multiplying Numbers

Squaring a number is the same as multiplying two numbers having identical values.

Example:

Square the number 743 = 743x 743.

1) Multiply 743 by 3. Put the carries above 743 and the partial product = 2229

2) Then multiply 743 by 4. The partial product = 2972. Put the last digit 2 on the tens decimal place.

3) Lastly, multiply 743 by 7. the partial product = 5201. Put the last digit 1 on the hundreds decimal place.

4) Add the partial products and what we get is = 552049 or 552,049

That is how we commonly get the square of a number.

Systematic Squaring Method (SSQ)

This time, I’ll teach a new way of getting the squares of numbers in an easier and orderly manner. But first, you must also know some new things.

Digit Number

A digit (what I’m talking about here is the numeric digit), is either any of the following;

0, 1, 2, 3, 4, 5, 6, 7, 8 or 9

A number such as 743 has three digits, 7, 4 and 3. Sometimes it is called a three-digit number. All you have to do is to count the digits. Counting the digits of 4,569,742, we can then, name that number, as a seven-digit number. In SSQ, the “count of digits of a number is important”. Later, you will realize the reason why it is important. But for now, giving you the idea of what a digit of a number is all about, would be enough.

Meaning of SSQ

SSQ stands for Systematic Squaring. It is based on a popular algebraic equation, (X + Y)². It is much different from the common method of multiplying two identical numbers.

SSQ has only three main parts, namely:

1) PSL (Partial Squares Line)

2) Sub-product

3) Total Sum (TSum)

Index Squares

Always remember that there are only ten basic digits (numeric digits) and these are;

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

An index square is a product of a ‘basic digit’ multiplied by itself:

0x0 = 0 / 1x1 = 1 / 2x2 = 4 / 3x3 = 9 / 4x4 = 16

5x5 = 25 / 6x6 = 36 / 7x7 = 49 / 8x8 = 64 / 9x9 = 81

It is safe to call 0, 1, 4, 9, 16, 25, 36 , 49, 64 and 81 as index squares but in SSQ, an index square must be expressed as “two-digit square”. So the proper way of writing them are as follows:

Table of Index Squares

0² = 00

1² = 01

2² = 04

3² = 09

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

Two-Digit SSQ

Let's start by squaring a two-digit number, using the SSQ method

Question: What is the square of 23?

23² = ?

Step 1: Create a PSL

Partial Squares Line (PSL)

The PSL is simply, the “two-digit squares” representation of each, individual digits of a certain number. In 23², the two-digit squares representation of the digits, 2 and 3 are 04 and 09, respectively. So we simply write it this way:

23² = 04’09 ← PSL

But don’t forget to also include this sign - ’ (a special character called single close quote). It will easily give us a clue of how many index squares are there in a PSL.

Step 2: Solve the sub-product

Sub-Product  (SP)

Don’t think that the value we’d taken from the PSL is already the correct answer. The value 04’09 is still incomplete. We must add a sub-product to come up with the ‘true’ square value of 23. But to get the sub-product of 23, we must multiply the digits 2 and 3 in a special kind of pattern.

General Rules in Dealing with the Sub-product

Rule 1: Look for the last digit of the given number

Rule 2: Double its value, meaning, multiply it by 2

Rule 3: Then multiply that value to the remaining digits on its left.

 

DOU-LAL Multiplication Pattern

Dou-LAL stands for "Double the Last Digit and All to It's Left". It is simply an easy to memorize acronym which is kind of multiplication pattern that is, effective in getting the sub-product. How it works?

HOW DOU-LAL WORKS?

In the given number 23, you may notice that 3 is the last digit, and on the left of 3 is 2.

Activity 1: Simply double the value of the last digit  (which in this case, is 3)

 3 x 2 = 6

Activity 2: Now, the "all to the left" of the last digit is in this case, is a single digit which is 2.

The next thing to do is to multiply 2 by 6, which happens to be the 'double value of the last digit' 3 of the given value 23

2 x 6 = 12

Officially, 12 will be the value of our Sub-product


Step 3: Find the 'total sum' by simply adding the Sub-product to the PSL with some little adjustment needed to perform.

 Adjust SP One Digit to The Left

In strict ruling in Math, the sub-product of 23 should be:

20 x ( 2 x 3)  = 120

But it is a 'short cut' way of writing number (for the value of SP in a two-digit squaring), if we try to omit the zero at the end of the SP

Instead of 120, we should simply write 12,  by which the last digit of SP, must be aligned to the next digit to the left of the last digit of the PSL.

 23²       = 04’09 ← PSL
 +2x3x2 =   1'2   ← SP1  (2 of 12 is aligned to second 0 of 04'09)

................05’29 ← T-Sum

Total Sum

In the above example, the value 05'29  will be our T-Sum, which is simply the result of adding the SP to the PSL. T-Sum reflects the true value of 23²  or the square value of 23.

But in 'reality', we can't simply say that the square of 23 is 05'29. Instead, we must omit the zero (0) before the 5 and erase the ' (  single close quote). 

So, the final answer will be:

23² = 529