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How to square large numbers quickly (part 1)

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:


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. 1x1=1, 2x2=4 so we write:


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 2x2x1=4. Write the 4 underneath the digits 0104 but one space away from the right, like this:


Finally, add the last two rows of numbers together, drop the 0 at the beginning of the result, and we get our answer, 144.


Let's try another example, 53. Start like this:


Write the squares of the digits,5x5=25 and 3x3=9, underneath like this:


Now double the 3 in 53 and multiply the double by the 5 in 53: 3x2x5=30 and write down 30:


Add the numbers in the last two rows and we get our answer, 2809.


We'll complete part 1 of this guide with a final example. Let's square 94. We start as usual:


Write the squares of the digits,9x9=81 and 4x4=16, underneath like this:


Now double the 4 in 94 and multiply the double by the 9 in 94: 4x2x9=72 and write down 72:


Add the numbers in the last two rows and we get our answer, 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.

Comments (72) Trackbacks (1)
  1. Unbelievable, My father gave me a copy of this book years ago and he’s left this planet over twenty-five years ago.
    My book has no front or back cover, only the name on the pages. I have been using this book over my whole life.
    Anyway, I am quoting from the book for something I’m writing and decided to see if I could find out who wrote it. I assumed the book was out of print and I would need to do an all out search for a rare book.
    WHAM, there it is. Now, I’m not sure which is better, the book or the internet.

  2. i love this trick

  3. this is a very super and useful

  4. very cool!
    i saw this method of carry the multyplying by 11.
    11 x 17 = 187; put the last digit in units column ie 7, add the two no’s 1+7 in tens column, and place the first no. in hundreds column, so for 11×27 = 297.
    If digits add to more than 10 then carry the ten to the first digit ie 11×38 = 418, 8 goes in ones column, 8+3=11, one from the 11 gous in tens column, carry the 1 to the 3, 3+1=4
    the method continues when multiplying 11x three digit number.
    11x 124 = 1364
    place 4 in units coumn, add the 4 and 2, place in tens column, add the 1 and 2 place in hundreds column, place the 1 in thousands column.
    As before if the numbers add to more than 10 carry the one; 11x 467 = 5137

  5. I was studying squaring numbers, and had to quickly do squares 1-20. I realize quick math is in SATs. I just am sitting here during homeroom and am excited to learn this; I didn’t pick it up from the math teacher (as usual).

  6. This is the ancient Indian Vedic method of squaring large numbers by using duplexes. =)
    Much of what Arthur Benjamin teaches is also derived from Vedic math. I’ve realized because I’m learning both systems.

  7. Nice..!and great..! Post for 3 digit numbers also

  8. this is really very helpful to me . I want that every one should see

  9. i also want to tell something see below;

    every no. like 111,222,333,444,etc. are all multiples of 37 and add these no. like 1+1+1:2+2+2:3+3+3:4+4+4:etc. respectively no. we get3;6;9;12 etc. if we multiply these no. like 37*3:37*6:37*9:37*12:etc we get 111:222;333;444

  10. i want help in hundreds

  11. This was awesome.Please post for 3 digits
    Is there any such way to square root too

  12. @anonymys It is possible to do it with hundreds, and indeed any number, but it gets more complicated.
    Note that the above is actually just a special case of (a + b)^2, where ‘a’ is the units part and ‘b’ is the tens. i.e. if you want to square 53, a = 3, b = 50.
    (a+b)^2 expands to a^2 + 2ab + b^2 = 3^2 + 2*3*50 + 50^2 = 9 + 300 + 2500 = 2809.
    (note that 50^2 = 5^2 * 10^2 = 25 * 100 = 2500 : 10^x is just a 1 with ‘x’ zeros after it. In the above example, it’s done by shifting the digits without showing the zeroes).
    If you want to do hundreds as well, you’ll have to use a third term ‘c’ to represent hundreds. This gives (a+b+c)^2, which expands to a^2 + 2ab + 2ac + b^2 + 2bc + c^2
    e.g. 153^2 = (3 + 50 + 100)^2 = 3^2 + 2*3*50 + 2*3*100 + 50^2 + 2*50*100 + 100^2 = 9 + 300 + 600 + 2500 + 10000 + 10000 = 23409.
    So it’s doable, but realistically, you’re probably better off grabbing a calculator.

  13. its good but in equal time i can get the answer by multipying the number.

  14. I discovered 3 rules. With the help of these rules u can find out the cube of any no between 1-100 within 7-15 seconds.

  15. Multiplying any number

    write 2 of 28 as it is, multiply 2 with 6 add 8 (2×6+8), multiply 8 with 6 (8×6)
    2 (20) (48)

    2 4
    2 0 8
    4 4 8

  16. i have got a better trick…example 13..

    1st :- take square of 10 = 100
    2nd:- take square of unit digit of the no, here 3 is the unit = 9
    3rd :- add 1+2 step = 109
    4th :- Multipy unit digit by 20 = 60
    5th :- Add 3 + 4 the step = 169 (final answer)..

    You can use this trick for any no from (11-19)

  17. To Devendra singh rawat : what were these 3 rules that you were talking about I really could used that

  18. aswm tricks man!!!!!
    Devendra singh rawat plz let us know the trick of cube!!!!!

  19. This is basic algebra : (x+y)*(x+y) = x*x + 2*x*y +y*y
    12: x=10, y=2
    53: x=50, y=3
    94: x=90, y=4
    Interesting but ‘reducing’ the two multiplications + one addition of traditional long multiplication to four multiplications and three additions is not a step in the direction of efficiency

    For three digit numbers the inefficiency gets worse
    (x+y+z)(x+y+z) = x*x + y*y +z*z + 2*x*y + 2*x*z +2*y*z

    e.g. 123: x= 100, y=20, z=3

  20. What a fantastic manner to find the square of the any number.lovely

  21. Here’s a trick to calculate square of 3-digit number.It’s a little bit more complicated than 2-digit.

    Calculate square of 123 ?
    1> Step 1 is similar to the 2-digit.
    Calculate square of individual digits and keep them in order as said for 2-digit no.
    2>Fundamental for step 2 is similar to 2-digit,but applies 3-times(see below).
    a)double unit digit and multiply it with 10’s digit(i.e 2 in this case). Place it leaving 1 space from right(same as 2-digit).[calculated as 12]
    b)Again double unit digit and multiply it with 100th digit(i.e 1 in this case). Place it leaving 2 space from right.[calculated as 6]
    c)double 10’s digit(i.e 2 in this case) and multiply it with 100th digit(i.e 1 in this case). Place it leaving 3 space from right[calculated as 4]. Finally add all of them(see below).
    ———–(Add them)

    Calculate square of 479 in the same way.
    126 (9*2*7=126)
    72 (9*2*4=72)
    56 (7*2*4=56)
    ———— (Add them)

    Sometimes I make things messy. I hope it will help you.

  22. The above post for 3-digit doesn’t reflect proper places for step-2.I hope you understand where to place them.

  23. folks I want to know how to compute square of many digits 4,5,6,7……in simple & tricky way. can any1 pls explain?

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