Estimating Square Roots
Not every number is a perfect square. Numbers like 2, 3, 5, 7, 10, and 50 do not have whole-number square roots. Their square roots are irrational numbers — decimals that go on forever without repeating.
In many practical situations, we need an approximate value of a square root without using a calculator. This is where estimation comes in.
The key idea is to find the two consecutive perfect squares between which the given number lies. This tells us between which two whole numbers the square root falls. We can then refine the estimate to get a closer approximation.
Estimation of square roots is an important skill in Class 8 NCERT Maths. It builds number sense and prepares students for working with irrational numbers in Class 9.
What is Estimating Square Roots?
Definition: Estimating a square root means finding an approximate value of √n when n is not a perfect square.
Key concepts:
- A perfect square is a number whose square root is a whole number (e.g., 1, 4, 9, 16, 25, ...).
- A non-perfect square is a number whose square root is not a whole number (e.g., 2, 3, 5, 7, 10, ...).
- The square root of a non-perfect square is an irrational number.
- Estimation gives a rational approximation (decimal) that is close to the actual value.
Estimating Square Roots Formula
Step-by-step method to estimate √n:
Step 1: Find the two consecutive perfect squares between which n lies.
If a² < n < b², then a < √n < b
Step 2: Determine how close n is to a² and b².
Estimate ≈ a + (n − a²) / (b² − a²)
Where:
- a = the smaller whole number (√ of the smaller perfect square)
- b = a + 1
- n = the number whose square root is being estimated
- This is linear interpolation between a and b.
Derivation and Proof
Why does the estimation formula work?
Step 1: We know that a < √n < b where a and b are consecutive integers.
Step 2: The function f(x) = x² is approximately linear over the short interval [a, b].
Step 3: Using linear interpolation:
- √n ≈ a + (n − a²) / (b² − a²)
Step 4: Since b = a + 1:
- b² − a² = (a+1)² − a² = 2a + 1
So the formula becomes:
√n ≈ a + (n − a²) / (2a + 1)
This gives a good approximation, typically accurate to one decimal place. For higher accuracy, the long division method or successive approximation (Newton’s method) is used.
Types and Properties
Estimation problems can be classified as follows:
1. Finding between which two integers √n lies:
- Identify the perfect squares on either side of n.
2. Estimating to one decimal place:
- Use the interpolation formula for a decimal approximation.
3. Trial and improvement:
- Try values like 5.1, 5.2, ... and square them to narrow down the answer.
4. Verifying an estimate:
- Square the estimated value and check how close it is to n.
5. Real-world problems:
- Finding the side of a square field given its area (non-perfect square).
Solved Examples
Example 1: Example 1: Between which two integers?
Problem: Between which two consecutive integers does √50 lie?
Solution:
- 7² = 49 and 8² = 64
- 49 < 50 < 64
- So 7 < √50 < 8
Answer: √50 lies between 7 and 8.
Example 2: Example 2: Estimation using interpolation
Problem: Estimate √50 to one decimal place.
Solution:
Step 1: 7² = 49, 8² = 64. So a = 7.
Step 2: Using the formula:
- √50 ≈ 7 + (50 − 49) / (64 − 49)
- = 7 + 1/15
- = 7 + 0.067
- ≈ 7.1
Verification: 7.1² = 50.41 (close to 50) ✓
Answer: √50 ≈ 7.1.
Example 3: Example 3: Estimating √200
Problem: Estimate √200.
Solution:
- 14² = 196, 15² = 225
- √200 ≈ 14 + (200 − 196) / (225 − 196)
- = 14 + 4/29
- = 14 + 0.14
- ≈ 14.1
Verification: 14.1² = 198.81; 14.14² = 199.94 ✓
Answer: √200 ≈ 14.1.
Example 4: Example 4: Estimating √3
Problem: Estimate √3.
Solution:
- 1² = 1, 2² = 4
- √3 ≈ 1 + (3 − 1) / (4 − 1) = 1 + 2/3 = 1.67
Verification: 1.67² = 2.79 (not very close). Try 1.73: 1.73² = 2.99 ✓
Note: The interpolation gives a rough estimate. For √3, the actual value is approximately 1.732.
Answer: √3 ≈ 1.7.
Example 5: Example 5: Estimating √85
Problem: Estimate √85 to one decimal place.
Solution:
- 9² = 81, 10² = 100
- √85 ≈ 9 + (85 − 81) / (100 − 81)
- = 9 + 4/19
- = 9 + 0.21
- ≈ 9.2
Verification: 9.2² = 84.64 (close to 85) ✓
Answer: √85 ≈ 9.2.
Example 6: Example 6: Trial and improvement
Problem: Find √40 by trial and improvement.
Solution:
- 6² = 36, 7² = 49. So 6 < √40 < 7.
- Try 6.3: 6.3² = 39.69 (too small)
- Try 6.4: 6.4² = 40.96 (too large)
- Try 6.32: 6.32² = 39.94 (close but small)
- Try 6.33: 6.33² = 40.07 (just above 40)
Answer: √40 ≈ 6.32.
Example 7: Example 7: Square field problem
Problem: A square garden has an area of 300 m². Estimate the length of each side.
Solution:
- Side = √300
- 17² = 289, 18² = 324
- √300 ≈ 17 + (300 − 289) / (324 − 289) = 17 + 11/35 = 17 + 0.31 ≈ 17.3
Verification: 17.3² = 299.29 (close to 300) ✓
Answer: Each side is approximately 17.3 m.
Example 8: Example 8: Estimating √150
Problem: Estimate √150.
Solution:
- 12² = 144, 13² = 169
- √150 ≈ 12 + (150 − 144) / (169 − 144) = 12 + 6/25 = 12 + 0.24 = 12.24
Verification: 12.24² = 149.82 (very close) ✓
Answer: √150 ≈ 12.2.
Example 9: Example 9: Between which integers does √700 lie?
Problem: Between which two integers does √700 lie?
Solution:
- 26² = 676 and 27² = 729
- 676 < 700 < 729
- So 26 < √700 < 27
Answer: √700 lies between 26 and 27.
Example 10: Example 10: Estimating √10
Problem: Estimate √10 to one decimal place.
Solution:
- 3² = 9, 4² = 16
- √10 ≈ 3 + (10 − 9) / (16 − 9) = 3 + 1/7 = 3 + 0.14 ≈ 3.1
Verification: 3.16² = 9.99 ≈ 10 ✓
Answer: √10 ≈ 3.2 (actual value ≈ 3.162).
Real-World Applications
Geometry: Finding the side of a square when its area is known but not a perfect square. For example, a square of area 72 cm² has side √72 ≈ 8.49 cm.
Distance Calculation: The distance formula uses square roots: d = √[(x₂−x₁)² + (y₂−y₁)²]. If the sum inside is not a perfect square, estimation gives an approximate distance.
Construction: Engineers and architects estimate square roots when calculating diagonal measurements of rooms or plots.
Physics: Many physics formulas involve square roots (e.g., time of free fall: t = √(2h/g)). Estimation gives quick approximate answers.
Mental Maths: Quickly estimating square roots helps in competitive exams and everyday mental calculations.
Key Points to Remember
- If n is not a perfect square, √n is an irrational number.
- Find the two consecutive perfect squares between which n lies: a² < n < (a+1)².
- Then a < √n < a + 1.
- Interpolation formula: √n ≈ a + (n − a²) / (2a + 1).
- Always verify your estimate by squaring it.
- The closer n is to a², the closer √n is to a.
- Trial and improvement gives more accurate results but takes more steps.
- For exact values, use the long division method.
- Perfect squares between 1 and 400: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400.
- Memorising squares up to 20² = 400 helps in faster estimation.
Practice Problems
- Between which two consecutive integers does √75 lie?
- Estimate √30 to one decimal place using the interpolation method.
- Estimate √250 and verify by squaring your answer.
- Use trial and improvement to estimate √60 correct to one decimal place.
- A square plot has area 500 m². Estimate the length of each side.
- Estimate √7 and compare with the actual value 2.646.
- Between which two integers does √1000 lie?
- Estimate √175 to one decimal place.
Frequently Asked Questions
Q1. What does it mean to estimate a square root?
It means finding an approximate decimal value of √n when n is not a perfect square. For example, √50 ≈ 7.07.
Q2. Why can't we find the exact square root of non-perfect squares?
The square root of a non-perfect square is an irrational number — it has infinitely many decimal places that never repeat. We can only write approximate values.
Q3. What is the interpolation formula for estimating square roots?
If a² < n < (a+1)², then √n ≈ a + (n − a²) / (2a + 1). This gives a good one-decimal-place estimate.
Q4. How accurate is the estimation method?
The linear interpolation method is usually accurate to within 0.1 or 0.2 of the actual value. For more accuracy, use trial and improvement or the long division method.
Q5. How do I verify my estimate?
Square your estimated value. If the result is close to the original number n, your estimate is good. For example, if you estimate √50 ≈ 7.1, check: 7.1² = 50.41 ≈ 50. Good estimate.
Q6. What are perfect squares between 100 and 200?
100 (10²), 121 (11²), 144 (12²), 169 (13²), and 196 (14²).
Q7. Is √2 rational or irrational?
√2 is irrational. Its value is approximately 1.414 and the decimal never terminates or repeats.
Q8. Can estimation be used in exams?
Yes. Many competitive exams and board questions ask you to estimate square roots or find between which integers a square root lies. The interpolation method is fast and acceptable.
Q9. What is trial and improvement?
Try successive decimal values (e.g., 6.1, 6.2, 6.3...), square them, and narrow down to the value closest to n. It is slower but more accurate than interpolation.
Q10. How does this connect to Class 9 topics?
In Class 9, you study irrational numbers and their representation on the number line. Estimation of square roots is the foundation for understanding and locating irrational numbers.
