How to Find the Square Root of a Number Without a Calculator (Easy Methods Explained)
In today’s world, anyone can calculate the square root of a number within seconds using a calculator or a smartphone. But very few people actually know—or remember—how to calculate it without such gadgets. Sometimes, you may even face situations like exams where calculators aren’t allowed. That’s when this skill becomes truly valuable.
Learning how to find the square root of a number without a calculator is not only practical but also a great way to sharpen your math skills. In this article, we’ll explore simple and effective methods to calculate square roots step by step.
Let’s explore a few effective and time-tested methods.
1. Prime Factorisation Method (For Perfect Squares)
This method works best when the number is a perfect square (like 36, 144, or 400).
Steps:
1. Break the number down into its prime factors.
- Example: 144 = 2 × 2 × 2 × 2 × 3 × 3
2. Group the factors into pairs.
- (2 × 2), (2 × 2), (3 × 3)
3. Take one number from each pair and multiply them together.
- 2 × 2 × 3 = 12
So, √144 = 12
This method is exact and simple when dealing with perfect squares.
2. Estimation Method (For Non-Perfect Squares)
Not all numbers have neat square roots. For example, √50 or √20. In such cases, estimation helps.
Steps:
1. Find two perfect squares close to the number.
- Example: √50 lies between √49 (7²) and √64 (8²)
2. Since 50 is closer to 49, √50 will be slightly more than 7.
3. To refine, divide 50 by 7 = 7.14
4. Average 7 and 7.14 → (7 + 7.14)/2 = 7.07.
So, √50 ≈ 7.07 (which is very close to the actual 7.071).
With a little practice, you can get impressively accurate results
3. Long Division Method
This is a classical method taught in schools that gives very accurate results, step by step.
Steps (for √2025):
1. Group the digits in pairs from right to left: 20 | 25.
2. Find the largest square less than or equal to 20 → 4² = 16. Write 4.
3. Subtract: 20 – 16 = 4. Bring down 25 → 425.
4. Double the divisor (4 → 8). Now find a digit X such that (80 + X) × X ≤ 425.
- X = 5 works, since 85 × 5 = 425.
5. So the square root is 45.
This method works for perfect and non-perfect squares, though it takes some practice.
4. Memorising Common Squares
One of the easiest tricks is to memorise the squares of numbers 1–30
Example: If you know 17² = 289 and 18² = 324, then you can instantly guess √300 lies between 17 and 18.
Great for quick mental math in exams!
Do You Know?
- The symbol “√” for square root was first used in the 16th century!
- Every positive number has two square roots: one positive and one negative. For example, √25 = +5 and −5.
- You can estimate square roots mentally and often get surprisingly accurate answers without a calculator.
- The largest square number whose square root has 100 digits is:
10^{200} – 2 \cdot 10^{100} + 1
\]
—that’s a number with 200 digits!
Memorising the squares of numbers from 1 to 30 can save you valuable time in exams and mental calculations.
Frequently Asked Questions (FAQ)
Q1: Can we find square roots of non-perfect squares without a calculator?
Yes! Using methods like estimation or the long division method, you can approximate the square root to as many decimal places as needed.
Q2: What is the fastest way to find square roots without a calculator?
For perfect squares, prime factorisation or memorising squares of numbers 1–30 works fastest. For non-perfect squares, estimation is usually the quickest method.
Q3: Why should I learn to find square roots without a calculator?
It sharpens your mental math skills, improves numerical intuition, and is essential in exams or situations where calculators aren’t allowed.
Q4: Are these methods accurate?
Yes! Prime factorisation and long division give exact results for perfect squares. Estimation is very close for non-perfect squares and can be refined further for precision.
Final Thoughts
Finding the square root of a number without a calculator may feel tricky at first, but once you understand these methods, it becomes second nature. For quick results, estimation works well. For exact answers, prime factorisation or the long division method is your best friend. And if you regularly memorise squares of small numbers, you’ll have an extra edge in solving math problems quickly.
So next time you don’t have a calculator, remember—you already have the tools in your mind to find square roots with ease!