Introduction
The Law of the Radicand is a simple yet powerful formula to estimate square roots of non-perfect squares. Designed to be efficient for mental math or paper calculation, this method provides high accuracy using basic arithmetic.
The Formula
Let n be a positive number. Find the largest integer a such that a² ≤ n. Then compute:
√n ≈ a + (n − a²) / (2a)
How to Use It
- Find the nearest perfect square less than or equal to
n, call its roota - Calculate the difference
d = n - a² - Apply the formula:
√n ≈ a + d / (2a)
Examples
| n | a | d | Approximation | Actual √n | Error |
|---|---|---|---|---|---|
| 66 | 8 | 2 | 8 + 2/16 = 8.125 | 8.12404 | +0.001 |
| 45 | 6 | 9 | 6 + 9/12 = 6.75 | 6.7082 | +0.042 |
| 123 | 11 | 2 | 11 + 2/22 = 11.0909 | 11.0905 | +0.0004 |
Advantages
- Easy to apply on paper or mentally
- Low error in most real-world cases
- No calculator or complex steps needed
- Teaches approximation and number sense
Possible Enhancements
For even greater accuracy, a correction term can be added:
√n ≈ a + (n − a²) / (2a) − ((n − a²)²) / (8a³)
Conclusion
The Law of the Radicand is an accessible and elegant way to estimate square roots for educational and practical purposes. With minimal effort, students and enthusiasts can perform accurate calculations and develop deeper intuition for numbers.