## 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:

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. 1x1=1, 2x2=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 2x2x1=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,5x5=25 and 3x3=9, underneath like this:

53

2509

Now double the 3 in 53 and multiply the double by the 5 in 53: 3x2x5=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,9x9=81 and 4x4=16, underneath like this:

94

8116

Now double the 4 in 94 and multiply the double by the 9 in 94: 4x2x9=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.

shreyas kabadeJuly 12th, 2011 - 06:43

i love this trick

venkateshJuly 23rd, 2011 - 02:28

this is a very super and useful

rockycricketAugust 7th, 2011 - 08:04

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

GordonSeptember 8th, 2011 - 11:49

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).

gorakhargoshOctober 26th, 2011 - 02:36

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.

vamsiNovember 23rd, 2011 - 07:13

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

harshDecember 17th, 2011 - 01:51

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

harshDecember 17th, 2011 - 01:57

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

anonymysJanuary 22nd, 2012 - 08:11

i want help in hundreds

kushalFebruary 16th, 2012 - 05:14

This was awesome.Please post for 3 digits

Is there any such way to square root too

tevFebruary 19th, 2012 - 00:26

@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.

AnonymousFebruary 19th, 2012 - 02:05

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

Devendra singh rawatFebruary 19th, 2012 - 09:12

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.

AnonymousFebruary 19th, 2012 - 23:40

Multiplying any number

e.g

28×16=368

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

so

2 (20) (48)

2 4

2 0 8

__________

4 4 8

AnonymousFebruary 22nd, 2012 - 11:52

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)

Niasia WilliamsMarch 17th, 2012 - 04:51

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

vinayt topalApril 12th, 2012 - 00:53

aswm tricks man!!!!!

Devendra singh rawat plz let us know the trick of cube!!!!!

AnonymousApril 24th, 2012 - 14:36

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

SABA TARIQAugust 16th, 2013 - 06:54

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

vineetJanuary 26th, 2014 - 14:07

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.

123

———-

010409

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).

123

———-

010409

12

06

04

———–(Add them)

015129

———–

Calculate square of 479 in the same way.

479

————

164981

126 (9*2*7=126)

72 (9*2*4=72)

56 (7*2*4=56)

———— (Add them)

229441

————

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

vineetJanuary 26th, 2014 - 14:11

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

NaveenMarch 8th, 2015 - 08:52

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