Squaring 3 Digit Numbers

This is a general method of squaring 3 digit numbers which will work for any three digit number. Now I will admit, it had me stumped the first time I read through this method in the book and I had to work through a couple of examples to work out what you were supposed to do.

Before we get started if you haven't already, I suggest you have a look at Squaring 2 Digit Numbers as we make use of this method as part of squaring 3 digit numbers.

Next we should make sure we all understand the terms being used.
A three digit number is made up of three digits, a hundreds digit, a tens digit and a units digit.
When I refer to the first digit I am referring to the hundreds digit, the second digit is the tens digit and the third, or last digit is the units digit.

The three general steps to this method are:

 

Don't worry if that sounds a little confusing we will go though the process later in more detail.

It was because of the difficulty I had in working out what was being done with this process that I present it a little differently than the book. It is still the same but organised a little different making it, in my opinion, a little easier to follow.

For the following examples I will use my modified layout for the first two then I will do an example following the layout in the book. Please note these layouts are trying to show you clearly what you are doing in the calculation but you do not have to write all of it out. In the final example I will show you how you do the calculation simply writing down the answer.

Example 1 : 678²

To illustrate what we are doing we will use squaring 678 as our example.

Step 1

To begin our solution we will use the method for squaring 2 digit numbers on the tens and units digits, the 7 and 8. In this step we treat the 78 as a two digit number and we ignore the 6.

a) square the units digit.
The 8 is our units digit so we square that giving us a result of 64.

We write down the 4 and carry the 6.

b) multiply the two digits and double.
We multiply the 7 and 8 together then double the result. We get 112 from this to which we add the carry from a for a total of 118.

We write 8 and carry the 11.

c) square the tens digit.
   The 7 is our tens digit and 7 squared is 49. We add 11, the carry  from b, to the 49 giving us 60.

We write the 60 in front of the 84 we already have giving us our initial four digit number.

Remember, if the result is less than four digits then put leading a zero to make it four digits.

The tens and units of this answer, the 84 are, in fact, the final result for these digits. The remaining two steps will only affect the hundreds and up.

Step 2

We now do our "open cross-product" by multiplying the first and last digits in our three digit number, the 6 and the 8. We then double the result.

The trick here is you add the result of this step to the first two digits of the result of step 1, which is why step 1 must have a four digit result. Or, to put it another way the units of the result of this step is put in the hundreds column.

There are two ways you can proceed here, add the numbers up in each step and get a progressive total or simply write the result of each step in its correct place then column-wise add up everything at the end. For clarity I will take the second option and write everything in its place. If you were mentally doing this then the progressive total approach would be better.

Step 3

We now use the method for squaring 2 digit numbers on the hundreds and tens units, the 6 and the 7, but this time we will omit the squaring of the units. Again we treat the 67 here like a two digit number and ignore the 8.

a) multiply the two digits and double.
    We multiply the 6 and the 7 together then double the result.
We write the 84 so that the units digit, the 4, is under the tens digit, the 9, from step 2.

b) square the tens digit of our two digit number.
    The 6 is the tens digit of our two digit number so we square that giving us 36.

We put the 36 so that the units digit, the 6, is under the tens digit, the 8 from part a.

We now have all the numbers we need and they are all lined up correctly so all we need to do is to add them up column-wise. If we start on the right and work left we only have a 4 in the first column.

In the next column we only have an 8.

In the third column we only have a 6 and a zero so the sum is 6.

In the fourth column we have 6, 9 and 4, adding these together we get 19 so we write 9 and carry 1.

In column 5 we have 8 and 6 to which we add the 1 from the carry for a sum of 15. We write 5 and carry the 1.

In the last column we just have 3 to which we add the 1 we carried, for a sum of 4.

We now have our result of 459,684 as the square of 678.

As I said before you don't have to write everything out as I have done, but you could if you feel more comfortable doing so until you get more familiar with the method. See the fourth example as to how you can just write down the answer.

Example 2 : 123²

We will look at another example but this time instead of just showing the process I will try to add extra explanation of why results are placed where they are so you can understand why the process works.

We will look at squaring 123.

Step 1

To begin our solution we will use the method for squaring 2 digit numbers on the tens and units digits, the 2 and 3. In this step we treat the 23 as a separate two digit number and we ignore the 1.

a) square the units digit.
The 3 is our units digit so we square that giving us a result of 9.

We write down the 9.

This is the only time we are multiplying the actual units digit of our three digit number with another units digit. In this case we are multiplying the 3 by itself giving us 9. The 9 will be the units digit of our final answer.

All the following steps will be using the tens and or hundreds digit of our three digit number so these will only affect the tens digits or higher in answer in the following steps.

b) multiply the two digits and double.
We multiply the 2 and 3 together then double the result.

We write 2 and carry the 1.

Like the 9 for the units digit, the 2 is the final value for the tens digit of our answer. Every step from no on involves multiplying units that the minimum value is in the hundreds or higher.

c) square the tens digit.
   The 2 is our tens digit and 2 squared is 4. We add 1, the carry  from b, to the 4 giving us 5.

We put the 05 in front of the 29 we already have giving us our initial four digit number.

As the result is less than four digits we put a leading zero to make it four digits.

Multiplying two tens digits will result in a result that the units are in the hundreds. the hundreds are the third digit from the right. We still have another step that will affect the hundreds digit so the first two digits in our four digit number, the 05, are not the final values for our result.

Step 2

We now do our "open cross-product" by multiplying the first and last digits in our three digit number, the 1 and the 3. We then double the result.

We just multiplied the hundreds digit with the units digit of our three digit number so the lowest unit of the result must be a hundreds digit. As the hundreds are the third digit from the right in a number that is why we line up the lowest digit of the 06, the 6, under the third column of the answer.

You could add the following zeros and put 0600 and lining everything up on the right but this is unnecessary. This is speed math so we leave out any unnecessary steps such as filling in zeros.

Step 3

We now use the method for squaring 2 digit numbers on the hundreds and tens units, the 1 and the 2, but this time we will omit the squaring of the units. Again we treat the 12 here like a separate two digit number and ignore the 3.

Why omit the squaring of the units here? Because we have already done it! The units in this case is the 2, if you look back to part c) in step 1 you will see we already squared the 2 when we we treated it as the tens digit of the 23 the first two digit number we looked at.

At this point we have done the final step that involves the hundreds digits. If we added up our numbers so are we would have 1129, the 129 are the last three digits of our final answer. The 1 in the thousands is not final as there is another step that will give us another thousands digit to add.

a) multiply the two digits and double.
    We multiply the 1 and the 2 together then double the result.

We write the 04 so that the units digit of the answer, the 4, is under the tens digit, the 0, from step 2.

The positioning of the 4 in the fourth column from the right puts it in the thousands column. Why? The reason is that we just multiplied the 1, the hundreds digit, with the 2, the tens digit of our three digit number. A hundred multiplied by a ten will give us a thousand hence we line the units of the result of this step up with the thousands column of our answer.

This is the final step that involves the thousands digit so adding the digits above we have 05129 and the 5129 are the last four digits of our final answer.

b) square the tens digit of our two digit number.
    The 1 is the tens digit of our two digit number so we square that giving us 01.

We put the 01 so that the units digit, the 1, is under the tens digit, the 0 from part a.

Looking at what we just did, we multiplied the 1, the hundreds digit by itself another hundreds digit. A hundred times a hundred gives us ten thousand and ten thousand is the fifth column from the right so that is why we line up the units digit of this steps result in the fifth column.

It should be easy to keep track of where to put the result of each step as after lining up the hundreds place in step 2, each step after that we simply move one column to the left.

In the previous step we saw that we had a running total of 05129 and we are adding the 01 so that the 1 is lined up with the 0 in 05129 giving us 015129 which is actually our final answer.

We now have all the numbers we need and they are all lined up correctly. If we now add up each column. The first column on the right just has a 9.

The second column just has a 2

The third column we have 5 and 6 giving us 11, we write 1 and carry 1.

The fourth column we have 0, 0 and 4 and the 1 we carried giving us 5.

The fifth column we have 0 and 1 so we put 1 here.

The last column is just a zero so we can ignore it and there is our answer: 15,129

Once you get the hang of this process it is very easy to do and I find it quite fast.

Example 3 : 457²

This time we will follow the layout mentioned in the book, the calculations are the same the only difference is in laying out the equation.

We will look at squaring 457.

Step 1

As always we start by using the method for squaring 2 digit numbers on the tens and units digits, the 5 and 7 and we ignore the 4.

a) square the units digit.
The 7 is our units digit so we square that giving us a result of 49.

We will show the 49 on the right hand side under the 7.

From this 49 we really only need to write down the 9 as part of our answer. The 4 is carried to the next step.

b) multiply the two digits and double.
We multiply the 5 and the 7,which is 35 then we double it giving 70.

We will show the 70 to the left of the 49, below the 5.

Adding the 4 carried from step 1 to the 70 we get 74, so we can write down the 4 as the tens digit of our final answer.

c) square the tens digit.
We square the 5 which gives us 25.

We will show the 25 to the left of our two previous results.

We have the 7 from the earlier calculation that we carry over to the 25 which gives us 32. We will put the 32 in grey as this is not the final values for these units in the final answer.

Step 2

We now do our "open cross-product" of the first and third digit, in this case the 4 and 7.

Multiplying the 4 and 7 then doubling we get 56.

The 56 is added directly below the 32 we got from the previous step.

Adding the 32 and 56 we get 88.

The 8 in the hundreds place is now the final value for this digit in our final answer.

Step 3

We now use the 4 and 5, the first and second digit of our number for the remaining steps.

a) multiply the two digits and double.
    We multiply the 4 and 5 and double the result,  this gives us 40.

We put the 40 under the leftmost 8, the thousands column.

b) square the tens digit of our two digit number.
    We square the 4 which is 16.

We put the 16 to the left of the 40 from the previous step.

Now we can add the 8 and the 0 from the 40 which is 8. We put down the 8 and we carry the 4 to the 16. 16 plus the 4 is 20. Writing down the  20 we get our final result of 208849.

 

Example 4 : 439²

This example will be light on explanation as we are concentrating on how you would actually write down the answer as you go through the calculation.

We will square 439.

Step 1

We take the 39 as our first two digit number and square it.

a) square the units digit.

We square the 9 giving us 81.

We write down the 1 and carry the 8.

b) multiply the two digits and double.

We multiply the 3 and the 9 which is 27  and doubling it gives 54. We then add the carry from the previous step giving us 62.

We write the 2 and carry the 6.

c) square the tens digit.

We now square the 3 giving us 9 then we add the 6 from the carry giving us 15.

Here we do not write it down but instead we cross out the 6 from the previous carry and write the 15 as the new carry. This adding to the carry is required here as we have one more calculation to do that involves the hundreds digit of the answer.

Step 2

We do the open cross product using the 4 and the 9.

The 4 multiplied by 9 gives us 36 which we double to get 72. Now we add the 15 we carried in the previous step to give 87.

We write down the 7 and carry the 8.

Step 3

We now use the second of our two digit numbers this time with the 4 and 3.

a) multiply the two digits and double.

We multiply the 4 and 3 together then double the result, this gives us 24. We add the 8 carried from the previous step to get 32.

We write down the 2 and carry the 3.

b) square the tens digit of our two digit number.

Our final task is to square the 4 which gives us 16. We add the 3 carried over to get 19

We write the 19 down and we have our answer of 192,721.

As you can see this is an efficient way to square a three digit number and with just a little practice it is easy to simply write down the answer as you go along.

To help you practice you can download a worksheet from here.

 

 
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