A New Method of Squaring 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 four main parts, namely:

1) PSL (Partial Squares Line)

2) Sub-products (SP1, SP2, SP3…)

3) Sub-totals (ST1, ST2, ST3…)

4) 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 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 Systematic Squaring (2D.SSQ)

Let 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 (SP1)

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 cross multiply the digits 2 and 3 in a certain kind of pattern.

General Rules in Dealing with Sub-products

Rule 1: Cross multiply the individual digits of a given number using the R.A.R. multiplication pattern

Rule 2: Don’t forget the DTP reminder, “Double The Product”

Rule 3: Follow the decimal place of the reference digit

R.A.R. Multiplication Pattern

R.A.R. stands for “Reference Digit and All the Digits to the Right”. It is a multiplication pattern that is effective in solving the sub-products. How it works?

First Pattern:

If we pick 3 from 23 as our reference digit, ‘the all to the right’ of 3 is nothing, null, empty or zero. Multiplying 3 by 0 is futile, so, we can skip this pattern.

Second Pattern (SP1):

If we pick 2 as our reference digit, ‘the all to the right’ of 2 is 3.

Rule 1: 2 x 3 = 6

Rule 2: DTP reminder 6 x 2 = 12

If you wish, you can skip rule 2 as long as you directly multiply 2 x 3 by 2

2x3x2 = 12 ← SP1

Rule 3: Our reference digit 2 is in the tens decimal place, therefore, the last digit of SP1 must be also, in the tens decimal place

2x3x2 = 12 ← SP1 (provided that 2 of 12 is aligned to the tens decimal place)

Step 3: Get the total sum, add the sub-product to the PSL

Total Sum

The total sum is the final answer. It reflects the ‘true’ square value of a given number. If you multiply 23 by 23, using the common method of multiplying numbers, you will discover that the total sum of SSQ method is exactly equivalent to the product of 23 x 23.

T-Sum = PSL + SP1

... 23² = 04’09 ← PSL
+2x3x2 = 1'2 ← SP1

..............05’29 ← T-Sum

Squares of Three-Digit Numbers

Multi-Digit SSQ

SSQ is applicable to any number. But SSQ becomes a little bit complicated when the digits of a number increases. The 2D.SSQ is the easiest but when the digits of numbers increases, such squaring 54,675², it involves many sub-products and many sub-totals and the process of multiplying and adding the set of numbers also become very complex. BUT DON’T WORRY, we will never to do that 5D.SSQ. But I will just give some new general rules of SSQ so that you will have an idea, how to tackle our next topics, the 3D.SSQ and 4D.SSQ.

General Rules of SSQ

General Rule No.1:

If you square a number, make sure to count the digits. The count of digits doubles as you square a number.

35² = 1,225

(Take note, a from two-digit number it becomes a four-digit number)

But in real life, this rule seems to be not true.

12² = 144

Well in SSQ, it always agrees because we write 12² as 01’44.

12² = 01'44

General Rule No. 2

The number of sub-products increases as the digit of numbers increases. But the number of sub-products depends on the count of digits. The sub-products are always less than one to the count of digits.

In 2D.SSQ, there is only one sub-product (SP1). But be prepared, in 3D.SSQ, you will not only be dealing with one sub-product but two sub-products (SP1 and SP2). While in 4D.SSQ, you need three sub-products (SP1, SP2 and SP3), to be added to the PSL, to get the square of a four-digit number.

General Rule No. 3

The sub-totals are always less than one, the number of the sub-products


Sub-Totals

A ‘sub-total’ is simply, a temporary sum, in between, each time you add a sub-product. In squaring a two-digit number (such as, 23²), there is no sub-total because you directly add SP1 to the PSL. A sub-total is optional; you can either include or ignore it. But personally, I highly recommend you to practice including it because it is important in our study of Square Edging.

Three-Digit SSQ (3D.SSQ)

In 3.D SSQ, the same rules you learned in 2D.SSQ still work. But of course, there are new added features.

Given Problem:

743² = ?

Step 1:Create a PSL

This is the easiest part. Simply write down the index squares of 7, 4 and 3.

743² = 49’16’09

Step 2:Solve the first sub-product SP1

First Sub-Product (SP1)

By the rules in dealing with sub-products;

Rule 1: Multiply the digits 4 and 3

4x3 = 12

Rule 2: Double the product (Optional, if you wish, you can skip this one)

12 x 2 = 24

Rule 3: Take note, the last digit of SP1 must be in the tens decimal place.

4x3x2 = 24 ← PS1

Step 3: Get the sub-total by adding the first sub-product to the PSL

Sub-Total (ST1)

Take note, this sub-total is just simply like a stop-over. It is not yet our final destination.

......743² = 49’16’09 ← PSL
.+4x3x2 = .......2'4 ← SP1
................. 49’18’49 ← ST1

Step 4: Solve the second sub-product


Second Sub-Product (SP2)

Notice that here in 3D.SSQ, the two added features are the sub-total ST1 and this second sub-product (SP2). In dealing with SP2, GIVE EXTRA CARE, in cross multiplying the digits.

Again, by the rules dealing with sub-products using the R.A.R. Multiplication Pattern, the all digits to the right of 7 are 4 and 3. We should not consider 4 and 3 as individual and separate digits. The A in R.A.R. stands for the word “All”. So don’t forget this – consider ALL to the right as ONE GROUP. So the digits 4 and 3, as a group, will become 43. You must read it as forty three.

Rule 1: Multiply the reference digit 7 by 43

7 x 43 = 301 Correct

7 x 4 x 3 = 83 Incorrect

Rule 2: Just to remind you, DTP, “always double the product”

301 x 2 = 602

Rule: The last digit 2 of 602 should be in the hundreds place

7x43x2 = 602 ← SP2

Step 5: As a final step, get the total sum by adding the second sub-product (SP2) to the sub-total (ST1)

.................. 49’18’49 ← ST1

...+7x43x2 = 6'02 ← SP2 (provided that 2 of 602 aligned to 8 of the ST1)

.................. 55’20’49 ← TSum

Summary:

......743² = 49’16’09 ← PSL

........ +4x3x2 = 2'4 ← SP1

.................. 49’18’49 ← ST1

...+7x43x2 = 6'02 ← SP2

.................. 55’20’49 ← TSum