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