Estimating Square Root

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Whole numbers are straightforward, and their application, as well as equation, is hardly ever complicated. The real challenge kicks in when one needs to deal with complex numbers and decimals. Did you know there are infinitely complex numbers between two consecutive whole numbers? While this is indeed a fascinating fact, it takes mathematics to another level. 

Every student will learn the square roots of whole numbers, at least till 25 in their primary school days. So it is apparent that you can easily answer when you are asked –

√144 =?

If your answer is 12, you are right. Now let's try something new-

√146 =? 

It is not that easy, isn't it? 

Well, it is because 146's square root is not a whole number, so it is a bit challenging to find its accurate answer. In such situations estimating square root is the way to go. There are several ways to calculate the approximate values, and among them, the simplest ones are stated below. 

Approximate the Square Root to the Nearest Integer

To determine the square root to its nearest tenth, let's continue with the example stated above. 


Your foremost step will be a vague estimation, which is determining the closest roots that comes just before and after 146. In this case, it will be 

√144 < √146 < √169 

Now simplify this equation to estimate the square root to the nearest integer.

12 < √146 < 13

From here it is clear that the square root of 146 is between 12 and 13. As 146 is way closer to 144 than 169, its estimate square root integer will be 12. 

Estimating Square Root to its Nearest Tenth 

Moving forward with the same example,

12 < √146 < 13

To find the square root to its nearest tenth subtracts the lower number from the one in your question. That is –

146 – 144 = 2

Now subtract the lower number from the higher number, that is –

169 – 144 = 25

The complex part of your approximation will be 

2/25 = 0.08 

Now, round it to the nearest tenth of the decimal. For this example, it will be

0.08 ≈ 0.1

Lastly, add your estimated whole number with the decimal number. 

12 + 0.1 = 12.1

So, its estimate square root to its nearest tenth is 12.1. 

Now let's confirm the result with another square root estimation method.  

Linear Approximation of Square Root

So, it is clear that the square root of 146 is somewhere between 12 and 13. In this method, you need to draw a line and divide it into 10 equal parts, write the smaller number in its left end and the higher in its right end. It should look something like this. 

[Image will be Uploaded Soon]  

Here 2 represents 12, and 3 represents 13. Since 146 is way closer to 144 than it is to 169 that linear approximation will be 12.1. 

While it is feasible to use square root by estimation method when the number is relatively small, the real challenge comes when the number in question is significantly big. 

Approximate Square Root of Large Numbers 

For this, let's take a new example,


The first step to estimating roots for these kinds of numbers is to make groups of two starting from the rightmost digit, like 


The number of groups you get will be the number of digits in your square root. In this case, it is two. Here instead of taking the closest square root, we will take the closest squares roots in hundreds or thousands. That is-

√1,600 < √2,439 < √2,500

Now simplify this equation,

40 < √2,439 < 50

So, it is clear that the number is somewhere between 40 and 50, now since the number is significantly closer to 50, we will check the square of the number just below it,

49² = 2401, which is way closer than 2500. 

Thus, the estimation of the square root for √2,439 will be 49. 

Had the number been an almost equal distance, the numbers you need to take under consideration are from the middle.  

Even though these methods are going to help you to estimate square roots, only practice can take you places. Visit Vedantu to get an array of sample sheets and sample papers for numerous such topics and sharpen your mathematic skills. Check their live classes and tuitions to understand critical topics in a more comprehensive way. 

FAQ (Frequently Asked Questions)

1. How to Apply a Large Number Square Root Method on Numbers with an Odd Number of Digits?

Ans. If there are an odd number of digits in the number like 236 or 36548, start to make pairs from the rightmost number as usual and the last number remaining will be counted as a pair as well. Like for the number 36548, the groups will be 3 │ 65 │ 48. That means the number of digits in the square root of 36548 will be three. 

However, it is implausible to get a square root estimation question of a five-digit number as this process gets a tad bit complicated. 

2. Will a Calculator Provide an Accurate Square Root Answer?

Ans. One can use a calculator to find a non-perfect square root, as it is faster and more convenient. However, the ten-digit display of a calculator will show you a long decimal number that you will have to shorten down to the nearest tenth. For example - 

√7= 2.64575131

The shortened answer will be 2.6. But when its square is calculated, you get 6.76 which is way lesser than 7. This proves that even though calculators provide a more precise answer it is still far from accurate. 

3. How Accurate is Estimating Square Root Answers?

Ans. Keep in mind that these values are "approximate" values and not very accurate; thus, these methods are also called "estimating non-perfect square roots". For example, the estimated square root of 146 is 12.1, but when you calculate the square of 12.1, the answer is 146.41, which is close but not exactly equal to 146.

These answers are quite accurate, but they never produce the number in question when squared.