Squaring Numbers: Another Method

Math Tricks is an app on Google Play with lots of tricks and shortcuts to specific math problems. In this app I have discovered that not only is the following method applicable to squaring any 2-digit number but also it is applicable to squaring any real number.

See other tricks @ Mental Math Tricks.

How?

From the FOIL (First, Outer, Inner, Last) method, we can derive that

$(a + b)^2 = a^2 + 2ab + b^2$

(This is obviously the law of squaring a binomial.)

...which means we can square any real number with any number of digits!

How it works for 2-digit numbers

Find $46^2$.

Procedure:

1. Subtract the ones digits from the number to be squared: $46 - 40 = 6.$ Now you have 2 numbers - 40 and 6.
2. Square the subtrahend from #1: $40^2 = 1600.$
3. Multiply the product of the subtrahend and the difference from #1 by 2: $40 \times 6 \times 2 = 480.$
4. Square the difference from #1: $6^2 = 36.$
5. Add all results from #2, #3 & #4: $1600 + 480 + 36 = 2116. \ _\square$

How about for 3-digit numbers?

Find $746^2$.

Procedure:

1. Get the leftmost digit from the number to be squared and add zeroes to replace all digits to its right: $746 \rightarrow 700.$
2. Subtract the result from #1 from the number to be squared: $746 - 700 = 46.$ Now you have two numbers - 700 and 46.
3. Square the subtrahend from #2: $700^2 = 490000.$
4. Multiply the product of the subtrahend and the difference from #2 by 2: $700 \times 46 \times 2 = 64400.$
5. Square the difference from #1: $46^2 = 2116$
6. Add all results from #3, #4 & #5: $490000 + 64400 + 2116 = 556516. \ _\square$

How about for numbers with digits more than 3? Previous procedure.

Find $1746^2$.

Procedure:

1. Get the leftmost digit from the number to be squared and add zeroes to replace all digits to its right: $1746 \rightarrow 1000.$
2. Subtract the result from #1 from the number to be squared: $1746 - 1000 = 746.$ Now you have two numbers - 1000 and 746.
3. Square the subtrahend from #2: $1000^2 = 1000000.$
4. Multiply the product of the subtrahend and the difference from #2 by 2: $1000 \times 746 \times 2 = 1492000.$
5. Square the difference from #1: $746^2 = 556516$
6. Add all results from #3, #4 & #5: $1000000 + 1492000 + 556516 = 3048516. \ _\square$

Advantage

The one and only significant advantage I see for this method is that it gives us a more accurate result for squaring numbers like those with 10 digits (and of course with any number of digits) like the number $1234567890$ using only a scientific calculator (not a computer calculator; this method is most useful if it can only display 10 digits or less).

Example (difficult)

To square a number like $1234567890$:

STEP 1: Square the number containing the 3 rightmost digits of the given number (The resulting number is $890$).

$890^2 = 800^2 + \big((800 \times 90) \times 2\big) + 90^2$

$= 640000 + 144000 + 8100 = 792100$

STEP 2: Place the nearest digit positioned to the left of the leftmost digit of the resulting number to the left of the resulting number, and square the new resulting number.

$7890^2 = 7000^2 + \big((7000 \times 890) \times 2\big) + 890^2$

$= 49000000 + 12460000 + 792100 = 62252100$

STEP 3: Repeat STEP 2. For squaring numbers with 7 or more digits, one should count how many zeroes there are in $2ab$ and add the digits of $b^2$ corresponding to the places of the zeroes in $b^2$ so one can know the rightmost digits missing in the calculator.

A. $67890^2 = 60000^2 + \big((60000 \times 7890) \times 2\big) + 7890^2$

$= 3,600,000,000 + 946800000 + 62252100$

$= 4609052100$

B. $567890^2 = 500000^2 + \big((1000000 \times 67890) \times 2\big) + 67890^2$

$= 250,000,000,000 + 67,890,000,000 + 4609052100$

$= 322,499,052,100$

C. $4567890^2 = 4000000^2 + \big((8000000 \times 567890) \times 2\big) + 567890^2$

$= 16,000,000,000,000 + 4,543,120,000,000 + 322,499,052,100$

$= 20,865,619,052,100$

D. $34567890^2 = 30000000^2 + \big((60000000 \times 4567890) \times 2\big) + 4567890^2$

$= 900,000,000,000,000 + 274,073,400,000,000 + 20,865,619,052,100$

$= 1,194,939,019,052,100$

E. $234567890^2 = 200000000^2 + \big((400000000 \times 34567890) \times 2\big) + 34567890^2$

$= 40,000,000,000,000,000 + 13,827,156,000,000,000 + 1,194,939,019,052,100$

$= 55,022,095,019,052,100$

STEP 4: Square the given number.

$1234567890^2 = 1,000,000,000^2 + \big((2,000,000,000 \times 234567890) \times 2\big) + 234567890^2$

$= 1,000,000,000,000,000,000 + 469,135,780,000,000,000 + 55,022,095,019,052,100$

$= 1,524,157,875,019,052,100 \ _\square$