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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: \(1600 + 480 + 36 = 2116. \ _\square\)

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.

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

\(= 1,000,000,000,000,000,000 + 469,135,780,000,000,000 + 55,022,095,019,052,100\)

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

Note by Adriel Padernal
9 months, 3 weeks ago

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Relevant article: Mental Math Tricks. Pi Han Goh · 9 months, 3 weeks ago

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@Pi Han Goh Thanks! :) Adriel Padernal · 9 months, 3 weeks ago

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@Adriel Padernal You should talk to @Michael Fuller about this. He is currently writing up this page. Pi Han Goh · 9 months, 3 weeks ago

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@Pi Han Goh I will. Thanks again for the support! Adriel Padernal · 9 months, 3 weeks ago

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