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=a2+2ab+b2(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 46246^2.

Procedure:

  1. Subtract the ones digits from the number to be squared: 4640=6.46 - 40 = 6. Now you have 2 numbers - 40 and 6.
  2. Square the subtrahend from #1: 402=1600.40^2 = 1600.
  3. Multiply the product of the subtrahend and the difference from #1 by 2: 40×6×2=480.40 \times 6 \times 2 = 480.
  4. Square the difference from #1: 62=36.6^2 = 36.
  5. Add all results from #2, #3 & #4: 1600+480+36=2116. 1600 + 480 + 36 = 2116. \ _\square

How about for 3-digit numbers?

Find 7462746^2.

Procedure:

  1. Get the leftmost digit from the number to be squared and add zeroes to replace all digits to its right: 746700.746 \rightarrow 700.
  2. Subtract the result from #1 from the number to be squared: 746700=46.746 - 700 = 46. Now you have two numbers - 700 and 46.
  3. Square the subtrahend from #2: 7002=490000.700^2 = 490000.
  4. Multiply the product of the subtrahend and the difference from #2 by 2: 700×46×2=64400.700 \times 46 \times 2 = 64400.
  5. Square the difference from #1: 462=211646^2 = 2116
  6. Add all results from #3, #4 & #5: 490000+64400+2116=556516. 490000 + 64400 + 2116 = 556516. \ _\square

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

Find 174621746^2.

Procedure:

  1. Get the leftmost digit from the number to be squared and add zeroes to replace all digits to its right: 17461000.1746 \rightarrow 1000.
  2. Subtract the result from #1 from the number to be squared: 17461000=746.1746 - 1000 = 746. Now you have two numbers - 1000 and 746.
  3. Square the subtrahend from #2: 10002=1000000.1000^2 = 1000000.
  4. Multiply the product of the subtrahend and the difference from #2 by 2: 1000×746×2=1492000.1000 \times 746 \times 2 = 1492000.
  5. Square the difference from #1: 7462=556516746^2 = 556516
  6. Add all results from #3, #4 & #5: 1000000+1492000+556516=3048516. 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 12345678901234567890 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 12345678901234567890:

STEP 1: Square the number containing the 3 rightmost digits of the given number (The resulting number is 890890).

8902=8002+((800×90)×2)+902890^2 = 800^2 + \big((800 \times 90) \times 2\big) + 90^2

=640000+144000+8100=792100= 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.

78902=70002+((7000×890)×2)+89027890^2 = 7000^2 + \big((7000 \times 890) \times 2\big) + 890^2

=49000000+12460000+792100=62252100= 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 2ab2ab and add the digits of b2b^2 corresponding to the places of the zeroes in b2b^2 so one can know the rightmost digits missing in the calculator.

A. 678902=600002+((60000×7890)×2)+7890267890^2 = 60000^2 + \big((60000 \times 7890) \times 2\big) + 7890^2

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

=4609052100= 4609052100

B. 5678902=5000002+((1000000×67890)×2)+678902567890^2 = 500000^2 + \big((1000000 \times 67890) \times 2\big) + 67890^2

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

=322,499,052,100= 322,499,052,100

C. 45678902=40000002+((8000000×567890)×2)+56789024567890^2 = 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= 16,000,000,000,000 + 4,543,120,000,000 + 322,499,052,100

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

D. 345678902=300000002+((60000000×4567890)×2)+4567890234567890^2 = 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= 900,000,000,000,000 + 274,073,400,000,000 + 20,865,619,052,100

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

E. 2345678902=2000000002+((400000000×34567890)×2)+345678902234567890^2 = 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= 40,000,000,000,000,000 + 13,827,156,000,000,000 + 1,194,939,019,052,100

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

STEP 4: Square the given number.

12345678902=1,000,000,0002+((2,000,000,000×234567890)×2)+23456789021234567890^2 = 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= 1,000,000,000,000,000,000 + 469,135,780,000,000,000 + 55,022,095,019,052,100

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

Note by Adriel Padernal
3 years, 9 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Relevant article: Mental Math Tricks.

Pi Han Goh - 3 years, 9 months ago

Log in to reply

Thanks! :)

Adriel Padernal - 3 years, 9 months ago

Log in to reply

You should talk to @Michael Fuller about this. He is currently writing up this page.

Pi Han Goh - 3 years, 9 months ago

Log in to reply

@Pi Han Goh I will. Thanks again for the support!

Adriel Padernal - 3 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...