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