Wild About Math! Making Math fun and accessible

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.