How to Square a Ten Digit Number in Your Head - Part 1

 A ten digit number is a billion of some sort.  3,141,592,653 is an example. This post, the first of three, will show how to square this number in your head.

This isn't a Rain Man thing, where you have to have extraordinary powers, but a technique which simplifies a very large complex task into many small, easy tasks. The math involved has no step more difficult than multiplying single digit numbers together, doubling them, and adding four similar products. For example: 3x3 (doubled) + 1x5 (doubled) + 4x6 (doubled) + 1x2 (doubled) + 5x9 (doubled). This is the longest calculation involved in the squaring of the above number.

The difficulty in the task is not the arithmetic, but the load on the memory. It does take some time to develop the ability to visualize and remember the numbers in your head.

This post will go through the technique up to 5 digit numbers, and provide all the necessary building blocks. The second post will go though the example with a five digit number,  and the third with a ten digit number, the one above, which are the first ten digits of pi, as you no doubt noticed.

Let's start with a two by two example: 31x31.

The technique involves cross multiplying, going from left two right, meaning first the hundred's digit, then the ten's, then the units. 3x3 is 9, for the hundred's. 3x1 (doubled) is the ten's, for 6. 1x1 = 1 gives the unit. The final answer is 961.

The doubling occurs because cross multiplying when squaring always involves multiplying the same two numbers twice.

Let's go to squaring a 3 digit number, 314. The ten thousand's digit is 3x3. The thousand's is 3x1 twice. Up to here, it's the same as the 2 digit number 31, with 96 as the first two digits, in this case 96,000. Whenever I hit where the comma separates the number, I make a special effort to remember that number. That is, I want to remember 96 for the thousands, set that aside, and come back to it after finishing up the three numbers to the right of the comma.

96 is the year Clinton ran against Dole. It's the year the Packers won the Super Bowl with Farve. Both numbers are divisible by 3.  These are a couple of ways to remember 96. Use whatever feels most comfortable. Later on I'll discuss a more robust method to remember many numbers.

For the hundred's digit we have 3x4 (doubled), plus 1 squared, which is 25. For the ten's, 1x4 (doubled). For the units, 4 squared, which is 16. Added together is 2500 + 80 + 16. Always go left to right, hundreds then tens then units, as this is much easier than trying to go from right to left. This gives 2580 + 16 for 2916. Fishing the 96 thousand from our memory, we add the 2596 to give 98,596 for the final answer.

When doing this in my head, I picture it like this:

3    4

   1

Wherever there are two numbers on a line, they are multiplied together and doubled. A single number on a line is squared. Thus to get the hundred's digit (the most involved calculation for squaring a 3 digit number) it's 3x4 (doubled) plus 1 squared, for 25.

Before going on to squaring 4 digit numbers, practice with some 3 digit numbers. With a little practice, it should only take a few minutes to square 3 digit numbers in your head. The first few times might take ten, fifteen, or even twenty minutes, as the difficult will be remembering the numbers, but with some training, picturing and remembering the numbers will become a lot easier.

Going on to 3,141 squared.

The first digit on the left will be millions, which is 3x3 = 9 million. Since the next number is separated by a comma, a special effort is made to remember the 9. I think of a baseball team.

The next three digits will be hundred thousands, ten thousands, and thousands.

The hundred thousand's digit is 3x1 (doubled), which is 6. The ten thousand's digit is 25, the exact same calculation as for the hundred's digit for the three digit number.

The thousand's digit, the one with the most calculations, I picture as follows:

3 1

1 4

This is a bit ambiguous, since the 1 appears twice, but think of these numbers as a ring going in counter clockwise motion, meaning the 1 to the right of the 3 represents the last digit, not the second one, which is below the 3.

The calculation is 3x1(doubled) + 1x4(doubled), which is 14. A useful trick is whenever you have the same number on two lines, such as the 1, you can add the same number together (1+1=2) and the other two numbers (3+4=7), and multiply them together. This also yields 14, and is easier.

Now we add 6 (hundreds), 25 (tens), and 14 (units), to get 600 + 250 + 14 = 864. This is easy to remember as it's 3 consecutive descending even numbers. When we get to squaring 5 digit numbers, we'll look at a more robust way to handle this.

Although we're actually dealing with hundred thousands, ten thousands, and thousands, it doesn't make any difference for the calculation, so think of it as hundreds, tens, and units to simplify. 

Notice we're applying the exact same technique as we go from triplet (the three numbers separated by commas) to triplet.  It's just a matter of remembering the triplets, and juxtaposing them.

The first triplet is the single number 9. The second is 864.

Finally we have 3 digits for the hundred's place (141), 2 digits for the ten's (41) and 1 for the units (1).

1    1

   4

1x1 (doubled) + 4 squared is 2 + 16 = 18. That's the hundreds. 4x1 (doubled) is the tens for 80. 1 squared is 1 for the units, which gives 1800 +  80 + 1 = 1881.

881 is easy to remember because 8 divided by 8 is 1. Use your preferred method to remember 881. Also remember there is a 1 in front of it to add to the thousand's triple we had from before, which was 864.

The three triplets are 9, 895 (864+1), and 881, giving as the final answer 9,897,881.

Being able to square 4 digit numbers is no mean feat. I practiced this quite a lot before moving to squaring 5 digit numbers, and recommend the same for the reader. As you can see, the calculations are all quite easy, but there's a lot of numbers to keep in your head and remember, which will take the mind some time to get acclimated to.

The next post will look at squaring 5 digit numbers.