

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.
√146
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,
√2439
The first step to estimating roots for these kinds of numbers is to make groups of two starting from the rightmost digit, like
24│39
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.
FAQs on Estimating Square Root
1. What is the main idea behind estimating the square root of a number?
The main idea behind estimating a square root is to find an approximate value, not an exact one. This is especially useful for numbers that are not perfect squares. The method involves identifying the two perfect squares that your number lies between. The square root of your number will then be between the square roots of those two perfect squares, allowing you to make an educated guess about its value.
2. How can you estimate the square root of a non-perfect square, for example, 85?
To estimate the square root of 85, you follow these steps:
- Step 1: Find the two closest perfect squares. 85 lies between 81 (which is 9²) and 100 (which is 10²).
- Step 2: This means the square root of 85, or √85, must be between 9 and 10.
- Step 3: Observe which perfect square 85 is closer to. Since 85 is much closer to 81 than to 100, the square root will be much closer to 9 than to 10.
- Step 4: You can estimate a decimal. A good first estimate would be 9.2. To check, 9.2 x 9.2 = 84.64, which is very close to 85. This makes 9.2 a good estimation for √85.
3. Why do we learn to estimate square roots when calculators can give a precise answer instantly?
Learning to estimate square roots is a crucial skill for several reasons, even with calculators available. It strengthens your number sense and mental maths abilities. This skill helps you quickly check if a calculator's answer is reasonable, which is vital in exams to avoid errors. Furthermore, it builds a foundational understanding of how numbers relate to each other, a concept essential for more advanced topics in algebra and calculus.
4. How accurate is the estimation method, and is it always reliable?
The accuracy of estimation depends on how close the number is to a perfect square. The method provides an approximate value, not an exact one. For instance, our estimate for √85 was 9.2, but 9.2² is 84.64, not exactly 85. While it is highly reliable for getting a close value quickly (e.g., for multiple-choice questions or checking calculations), it is not a substitute for methods like long division when a high degree of precision is required.
5. What is the first step in estimating the square root of any number as per the Class 8 NCERT syllabus?
As per the standard method taught in the CBSE/NCERT curriculum for Class 8, the very first step to estimate the square root of a non-perfect square is to identify the two consecutive whole numbers between which the square root lies. This is done by finding the two perfect squares immediately below and above the given number.
6. Is estimating a square root the same as using the long division method?
No, they are different methods with different purposes. Estimation uses logical reasoning and known perfect squares to find a quick, approximate answer. The long division method is a step-by-step algorithm used to calculate a more precise, digit-by-digit value of the square root. Estimation is about 'ballparking', while long division is about 'calculating'.
7. In what real-world situations is estimating a square root more useful than finding the exact value?
Estimating is more practical in many real-world scenarios where quick, approximate answers are sufficient. For example:
- In basic physics, to quickly estimate the time it takes for an object to fall.
- In carpentry or design, to get a rough idea of the diagonal length of a square or rectangular piece.
- When budgeting or shopping, to quickly assess space or dimensions without needing precise figures.
- To simply verify the plausibility of a result from a complex calculation or a computer program.

















