When looking up methods for calculating square roots without a calculator some methods are more complicated than others. The method we will look at here is simple enough to do in your head, which is what is was designed for. When looking for the square roots of larger numbers we build on top of this method and, admittedly, it does get more complicated as the numbers get larger and you probably have to resort to writing down parts as you calculate but the method builds on itself and the steps are not really complicated.

As part of our calculation we need to know the squares of the single digit numbers, ie from 1 to 9. Take a little time if necessary to familiarize yourself with the squares of these numbers. These values are:

We will look at find the square root of three or four digit numbers because they both give a two digit answer.

## A Three Digit Example

Letâ€™s find the square root of 729

### Step 1

The first step is, starting from the right hand side, put a slash after every second digit as you move to the left.

In this case, we put the slash between the 2 and the 7, when moving left.

For the three-digit and four-digit numbers we only have the one slash, indicating we have a two digit answer because the single slash breaks the number into two groups and the number of groups tells us the number of digits in the square root. This step also gives us the first number we we use to calculate the first digit of our answer.

### Step 2

Going from left to right, the first group of numbers on the left of the first slash is either a single digit or two digits. In this case, the only digit before the first slash is the 7. To find the first digit of the answer we need to work out the largest digit which when squared is less than 7. Or, more generally,

- The first digit of the answer is always the largest single digit, which when squared is less than the value of the first group of digits.

2^{2} = 4

3^{2} = 9

As 4 is less than 7 but 9 is greater than 7 this means the first digit of our answer is 2.

### Step 3

We square the first digit of our answer and then subtract this from the first group of digits in the number.

2 squared is 4 so we subtract this 4 from the 7 which gives us 3.

### Step 4

Up to now everything we have done is exact, that is, you follow the steps and it works but now we come to a part of the process where things get a little bit fuzzy, especially when there are odd numbers involved. Odd numbers present us with a choice, three choices actually and we don't know which choice is the correct one. All we can do is make a choice and try, knowing that if we are wrong we need to backtrack to here, make a different choice and continue on again.

To explain lets continue with the example.

In this step we take the result of the subtraction in the previous step, halve it then add a zero on the end (basically halving and multiplying it by ten).

The result of the subtraction in the previous step gave us 3, which is an odd number so halving it to an integer can give us either 1 or 2. Adding the zero on the end means our choices are 10 or 20 but we can also make a third choice by splitting the difference and selecting 15. Often taking the middle choice works but not every time

For this example we will take the middle choice and use 15.

We divide the 15 by the first digit of the answer, the 2. 15 divided by 2 gives us 7 so this becomes the second digit of our answer. However, this is just a tentative answer, we may have made the wrong choice so we need to confirm the result.

At this point we have our two digits of the answer, although the second digit is not yet final and there may also be a remainder.

The next two steps are required to find the remainder and also confirm that the last digit of the answer is correct or not.

### Step 5

In this step we will do a procedure similar to what was done when squaring numbers. Actually we doing the calculation for squaring the answer but we omit the final step of squaring the tens digit as we already used that when finding the first digit of the answer.

We will write this step down although you should be able to do this and the next steps mentally.

We will calculate a little table of numbers under our answer by doing the following:

1 - square the units digit of the answer, in this case the 7.

7 x 7 = 49

The 9 we put in the third column of our little table under the 7 of our answer. We will carry the 4.

2 - Multiply the tens digit and the units digit then double the result.

2 x 7 = 14

14 x 2 = 28

We add the 4 carried over from step 1 to the 28 giving us 32, we write the 3 in column one of our little table and the 2 in column two.

### Step 6

We subtract the first digit of our three digit number, the 3, from the result of our subtraction in step 3, which was also 3.

3 - 3 = 0

We will put the 0 up under the 2 of 729 as shown in the following image.

### Last Step

We bring down the last two digits of the 729, the 2 and the 9 next to the 0.

We then subtract the remaining two digits of our three digit number from this 029.

The result is 0 so we have no remainder and so we have also confirmed that the 5 is the correct digit and that the square root of 729 is 27 with no remainder.

### The Alternatives

Instead of looking at new examples lets go back to step 4 of our example and choose the other alternatives and see what happens if we chose 10 or 20 instead of the 15.

#### If we chose 20

If in Step 4 we took half of the 3 as 2 then added added a zero we get 20, dividing the 20 by the first digit of the answer, the 2, we get 10 which is impossible as we need a single digit. However, in this case we can try 9 instead.

Whenever we get a number higher than 9 when we calculate the next digit of the answer we always need to reduce it to 9 or possibly lower.

##### Step 5

We calculate our little table under our answer by doing the following:

1) - square the units digit of the answer, in this case the 9.

9 x 9 = 81

The 1 we write into the third column of the table. We will carry the 8.

2) - Multiply the tens digit and the units digit then double the result.

2 x 9 = 18

18 x 2 = 36

We add the 8 carried over from step 1 to the 36 giving us 44, we write the fours into columns one and two of our little table.

We subtract the first digit of our three digit number, the 4, from the result of our subtraction in step 3, which was 3.

3 - 4 = -1

We have a problem, we cannot have negative numbers, so the 9 is too high.

We need to go back to step 4 and make another choice or the alternative is to reduce the 9 to 8 and recalculate our little table. There is no hard and fast rule here, how you choose to proceed is up to you, all I can do is give you the guidelines to follow.

#### If we chose 10

If in Step 4 we took half of the 3 as 1 then added added a zero we get 10, dividing the 10 by the first digit of the answer, the 2, we get 5.

##### Step 5

We calculate our little table under our answer by doing the following:

1) - square the units digit of the answer, in this case the 5.

5 x 5 = 25

The 5 is the units result for our three digit number. We will carry the 2.

2) - Multiply the tens digit and the units digit then double the result.

2 x 5 = 10

10 x 2 = 20

We add the 2 carried over from step 1 to the 20 giving us 22, we write the 2's into column one and two of our little table.

We subtract the number in column one of our table, the 2, from the result of our subtraction in step 3, which was 3.

3 - 2 = 1

We will put the 1 up under 729 as shown in the following image.

##### Last Step

We bring down the last two digits of the 729, the 2 and the 9 next to the 1.

We then subtract the remaining two digits of our three digit number from this 129.

The result is 104 so we have a remainder of 104. This is a problem because with square roots the rule is:

The remainder must be no larger than twice the answer

The reason for this rule is that for any two whole numbers whose difference is 1, the difference in their squared values is always twice the value of the lower number plus 1.

Some examples

6 and 7

6^{2} = 36

7^{2} = 49

The difference in the squared values is: 49 - 36 = 13

The value of 6 doubled plus one is: 6 x 2 + 1 = 13

28 and 29

28^{2} = 784

29^{2} = 841

The difference in the squared values is: 841 - 784 = 57

The value of 28 doubled plus one is: 28 x 2 + 1 = 57

So if the remainder is more than twice the answer then the answer is too low.

As our answer is 25 and the remainder is 104 and 25 x 2 + 1 = 51, this means our answer should be at least 26 or more.

We need to go back to step 4 and make another choice or change the last digit of our answer from 5 to 6 or 7 and try the calculation again. Again, there is no hard and fast rule as which way is better as long as you eventually get to the correct answer.

When in doubt, on average, choosing the middle choice will get you to the answer faster but not always.

## A Four Digit Example

We will now find the square root of a four digit number

### Step 1

Starting from the right hand side, put a slash after every second digit as you move to the left.

In this case, we put the slash between the 1 and the 9, when moving left.

### Step 2

To find the first digit of the answer we need to work out the largest digit which when squared is less than 29.

5^{2} = 25 which is less than 29

6^{2} = 36 which is more than 29

So the first digit of our answer is 5.

### Step 3

We square the first digit of our answer and then subtract this from the first group of digits in the number.

5 squared is 25 so we subtract this 25 from the 29 which gives us 4.

### Step 4

In this step we take the result of the subtraction in the previous step, the 4, halve it then add a zero which gives us 20.

half of 4 is 2

adding a zero to the 2 we get 20

We divide the 20 by the first digit of the answer, the 5. 20 divided by 5 gives us 4 so this becomes the second digit of our answer. However, this is just a tentative answer, we need to confirm the result.

### Step 5

We calculate our little table under our answer by doing the following:

1) - square the units digit of the answer, in this case the 4.

4 x 4 = 16

We write the 6 in the third column of our little table. We will carry the 1.

2) - Multiply the tens digit and the units digit then double the result.

5 x 4 = 20

20 x 2 = 40

We add the 1 carried over from step 1 to the 40 giving us 41, we write the 4 into column one and the 0 into column two.

### Step 6

We subtract the first digit of our three digit number, the 4, from the result of our subtraction in step 3, which was also 4.

4 - 4 = 0

We will put the 0 up under 2918 as shown in the following image.

### Last Step

We bring down the last two digits of the 2918, the 1 and the 8 next to the 0.

We then subtract the remaining two digits of our three digit number from this 018.

The result is 2 so we have a remainder of 2 and so we have also confirmed that the 4 is the correct digit and that the square root of 2918 is 54 with 2 remainder.

For the free members there are plenty of square roots worksheets you can download and use which are separated into different difficulty levels

When working out "half" of an odd number probably the easiest way to think of it is this way.

Subtract 1 from the odd number and halve the result, this will give you the "lower half".

Add 1 to the odd number and halve the result, this will give you the "higher half".

To get the third choice after you have added the zero to the lower and higher halves you subtract the lower number from the higher number, divide it by two then add the result to the lower number.

For Example for 3

3 - 1 = 2, half of 2 is 1, this is our "lower half", adding a zero we get 10.

3 + 1 = 4, half of 4 is 2, this is our "higher half", adding a zero we get 20.

20 - 10 = 10, half of this is 5.

10 + 5 = 15 which is our middle choice.