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# 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: $$1600 + 480 + 36 = 2116. \ _\square$$

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: $$1000000 + 1492000 + 556516 = 3048516. \ _\square$$

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.

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

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

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

11 months, 2 weeks ago

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