How to square large numbers quickly (part 1)
November 4th, 2007 | by Sol |I have to confess, one of my secret addictions is scouring Math books for novel approaches to solving old problems. I especially like to look for these fresh approaches in, ironically enough, old books.
Last night I was perusing a little book: “The Master System of Short Method Arithmetic and Mechanical Calculations Simplified: Methods Used by the World’s Foremost Experts” by Paul Huberich. The book was published in 1924. Page 34 has this very novel algorithm for squaring (multiplying by themselves) large numbers. In this “how to” article I describe this algorithm (in more detail than the terse explanation provided in the book, I should add) and I give a number of examples of how to apply it. I also provide suggestions on how to do the arithmetic efficiently.
Part 1 of this guide illustrates use of the algorithm for squaring 2 digit numbers.
We’ll start with a simple example.
Let’s say we want to find the square of 12. On a sheet of paper, write the the number 12 and underline it:
12
Now, write down the squares of the digits 1 and 2 underneath, adding a 0 at the beginning of any product that is not a 2-digit number. 1×1=1, 2×2=4 so we write:
12
0104
Now, double the last digit of the number we’re squaring, 2, to get 4 and multiply this product, 4, by the first digit of the number we’re squaring, 1, and we get 2×2x1=4. Write the 4 underneath the digits 0104 but one space away from the right, like this:
12
0104
4
Finally, add the last two rows of numbers together, drop the 0 at the beginning of the result, and we get our answer, 144.
12
0104
4
0144
Let’s try another example, 53. Start like this:
53
Write the squares of the digits,5×5=25 and 3×3=9, underneath like this:
53
2509
Now double the 3 in 53 and multiply the double by the 5 in 53: 3×2x5=30 and write down 30:
53
2509
30
Add the numbers in the last two rows and we get our answer, 2809.
53
2509
30
2809
We’ll complete part 1 of this guide with a final example. Let’s square 94. We start as usual:
94
Write the squares of the digits,9×9=81 and 4×4=16, underneath like this:
94
8116
Now double the 4 in 94 and multiply the double by the 9 in 94: 4×2x9=72 and write down 72:
94
8116
72
Add the numbers in the last two rows and we get our answer, 8836.
94
8116
72
8836
Practice squaring two digit numbers and you’ll soon be able to do it very quickly and, as a nice side effect, you’ll enjoy arithmetic more.
Stay tuned for part 2, which extends this approach to numbers with 3 or more digits.
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22 Responses to “How to square large numbers quickly (part 1)”
By David Rudloff on Dec 4, 2007 | Reply
I hope you will maintain the site.
By Melissa on Dec 4, 2007 | Reply
ingenious!
By DEENA on Jan 9, 2008 | Reply
I ALREADY WAITING FOR PART-II SO PLEASE MAKE IT FAST
By DEENA on Jan 9, 2008 | Reply
I AM ALREADY WAITITNG FOR PART-II I THINK YOU ARE A GREAT PERSON BECAUSE YOU ALSO WANT THE OTHERS TO IMPROVE THIER MATH SKILL
By rei on Jan 13, 2008 | Reply
Rocking!! Plz maintain this site…
By Cathryn on Jan 13, 2008 | Reply
Woah!! Sooo cool! Thanks!!
By Sol on Jan 14, 2008 | Reply
@David and Rei: Yes, I will maintain this site.
@Melissa: Thanks. I didn’t invent this approach but I’m hoping to popularize it!
@Deena: I hear you. Thanks for the compliment. I’ll try to get to Part II this week. Stay tuned.
@Cathryn: I’m glad you like it.
By Thyag on Jan 22, 2008 | Reply
That’s a super-dooper math trick!!
By Eric on Jan 22, 2008 | Reply
Trying this method on numbers larger than two digits requires a few more steps. The method I worked out seems to be working. (hopefully my formatting works)
Take your number and start just like you would with a two digit number:
for each number square it and if the result is
By Eric on Jan 22, 2008 | Reply
(sorry they restrict how much you can write)
for each number square it and if the result is less then 10, put a zero in front of it:
(4*4=16, 7*7=49, 5*5=25, 3*3=9)
3 5 7 4
—-
0 9 2 5 4 9 1 6
By Eric on Jan 22, 2008 | Reply
Now, loop through and take the first value (4) * the second value (7) * 2 again, if the result is less then 10, put a zero in front of it. Also, start by putting the result below which value you are multiplying the first value by (the second value) Then for the next values do the same (second value * third value * 2) and continue on until you have second to last value * last value * 2:
(4*7*2=56, 7*5*2=70, 5*3*2=30)
3 5 7 4
—-
3 0 7 0 5 6 x
By Eric on Jan 22, 2008 | Reply
Continue the above step but this time have the first value (4) * the third value (5) * 2 again, if the result is less then 10, put a zero in front of it. Also, start by putting the result below which value you are multiplying the first value by (the third value) Then for the next values do the same (second value * forth value * 2) and continue on until you have third to last value * last value * 2:
(4*5*2=40, 7*3*2=42)
3 5 7 4
—-
4 2 4 0 x x
By Eric on Jan 22, 2008 | Reply
Continue the above process until you reach the first value (4) * the last value (3) * 2. The result will be positioned below the last value.
(4*3*2=24)
3 5 7 4
—-
2 4 x x x
By Eric on Jan 22, 2008 | Reply
Then add up all of your results and you have the answer:
0 9 2 5 4 9 1 6
3 0 7 0 5 6 x
4 2 4 0 x x
2 4 x x x
—-
1 2 7 7 3 4 7 6
By Eric on Jan 22, 2008 | Reply
A more complex example:
x x x x x x x 4 5 8 1 2 3 5 (squared)
—-
1 6 2 5 6 4 0 1 0 4 0 9 2 5
x 4 0 8 0 1 6 0 4 1 2 3 0 x
x x 6 4 1 0 3 2 0 6 2 0 x x
x x x 0 8 2 0 4 8 1 0 x x x
x x x x 1 6 3 0 8 0 x x x x
x x x x x 2 4 5 0 x x x x x
x x x x x x 4 0 x x x x x x
—-
2 0 9 8 7 7 1 4 1 2 5 2 2 5
By mike on Mar 2, 2008 | Reply
what do the x’s mean in your problems?
By Ali Rasheed on Mar 28, 2008 | Reply
good work
By Arivoli on Apr 30, 2008 | Reply
This method do not work for squaring 987. When the 2*8*7=112, this method did not explain what should one do when the product exceeds 100?
Thanks,
Arivoli
By James on Jul 10, 2008 | Reply
Arivoli, the math works the same, you must just break up the steps like so.
9 8 7
————
8 1 6 4 4 9
x x(1)1 2 x
(1)4 4 x x x
x(1)2 6 x x
————
9 7 4 1 6 9
By James on Jul 10, 2008 | Reply
Whoops, yucky formatting. Here:
x x x 9 8 7
———–
8 1 6 4 4 9
x x 1 1 2 x
1 4 4 x x x
x 1 2 6 x x
———–
9 7 4 1 6 9
By hele on Jul 22, 2008 | Reply
this can really help me improve in my math and thinking skills.thank you for sharing your knowledge with us.