How do you calculate a square root without a calculator?

There are lots of students who use the calculator to find the square root of a number without having the slightest idea of what that calculation means. The square root of a number means to find the number that multiplied by itself will result in the number that is within the root.

How can you calculate that without using the calculator?

There are several ways of achieving the wanted result without a calculator. However, I think the easiest one consists in following this sequence including three steps:

• 1 - Make an estimate.
• 2 - Make a division.
• 3 - Calculate the mean.

First of all let’s start by making an estimate. The closer to the final result an estimate is, the less are the calculations to be performed. But the method also works with very bad estimates! For example, we intend to calculate the square root of the number `12`. Let’s suppose my estimate corresponds to `2` (it’s an awful estimate since we know the right result must be between `3` and `4` since the square of `3` is `9` and the square of `4` is `16`). Concerning the second step, let’s divide number `12` by our estimate, `12:2=6`. Then, and according to the third step, let’s calculate the mean between the last result and number `2`: `(6+2):2=4`. And now let’s repeat steps 2 and 3 till we feel fulfilled with the approach we have managed. Step 2 – the division with the new estimate: `12:4=3`. Step 3 – the mean with the last result: `(4+3):2=3.5`. Step 2 – the division with the new estimate: `12:3.5=3.43`. Step 3 – the mean with the last result: `(3.5+3.43):2=3.465`. We could keep on doing this again and again, but let’s us test this last result: `3.465xx3.465=12.006225`. We must say it is a very reasonable approach, indeed!

Could you give an example of an exercise which requires the calculation of the square root?

For sure! Let’s suppose we want to buy a square shaped plot of land and the seller tells you that its area corresponds to `1156 m^2`. We ask the seller the measure of one of the sides of the plot and he is not able to give you an answer. Having in mind that to calculate the area of a square we have to `text(side) xx text(side)`, that is the same of saying `text(side)^2`. So, all we have to do is to find what is the number that multiplied by itself equals `1156`. As we are used to having our mobiles with us, it is enough to use its calculator to find that `sqrt(1156)=34`. Instead of the mobile we can always use a sheet of paper and follow the previous explained method. Thus we get to know that each side of the plot measures `34m`.

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