Approximating Square Roots Using Long Division Method: A Fun and Effective Approach

Approximating square roots can be a valuable mental exercise, especially for those who grew up without calculators. While many rely on modern technologies to find square roots, learning to approximate them using the long division method is a skill that enhances mathematical fluency and problem-solving abilities. This article delves into the method, providing examples and insights into how to efficiently approximate square roots mentally.

Introduction

The long division method for approximating square roots is a technique that helps you find the square root of a number to a sufficient degree of accuracy. This method was particularly useful in the pre-calculator era, but it remains a valuable tool for those who appreciate the satisfaction of mental calculations. In this guide, we will explore how to apply the long division method, using examples from the perspective of a retired chemistry teacher.

Step-by-Step Guide to Approximating Square Roots

Example 1: Approximating the Square Root of 123

Consider the number 123. To approximate its square root, we first recognize that 11 squared is 121, which is less than 123. The next perfect square, 12 squared, is 144, which is more than 123. This means that the square root of 123 lies between 11 and 12.

We can then calculate how far 123 is from 121 and adjust our estimate accordingly. Since 123 is 2 more than 121, we know that the square root of 123 is 2/23 of the way from 11 to 12. This fraction can be approximated as follows:

2/23 is approximately 0.087, which would give an answer of 11.087. However, as a retired chemistry teacher, you might find 2/23 slightly smaller than 2/20 (which is 0.1). To make it smaller, you might think of it as 0.09. Therefore, you would estimate the square root of 123 as 11.09.

The actual calculator answer is 11.0905, confirming that our approximation is quite close.

Example 2: Approximating the Square Root of 875

For the number 875, the nearest perfect squares are 841 (29 squared) and 900 (30 squared). By calculating the distance from 841 to 875 (which is 34), we find that:

34/59 can be approximated to 34/60 (0.567), which simplifies to 0.57 when divided by 6. Since 34/59 is slightly larger than 34/60, we add a small adjustment to our estimate and consider 0.58. Therefore, the square root of 875 is approximately 29.58.

The actual calculator answer is 29.5804, showing that our approximation is accurate.

Example 3: Approximating the Square Root of 0.72

For 0.72, we first convert it to the exponent form: 72 x 10^-2. Then, we find the square root of 72, which lies between 8^2 (64) and 9^2 (81). When we calculate how far 72 is from 64 (8/17 of the way), we approximate it as follows:

8/17 is approximately 0.47, which can be simplified to 0.48 (8/16). The square root of 72 is approximately 8.4. The square root of 10^-2 (0.01) is 0.1, so the square root of 0.72 is 8.4 x 10^-1 or 0.84.

The actual calculator answer is 0.848, making our approximation highly accurate.

Conclusion

Approximating square roots using the long division method is a powerful mental exercise that enhances your mathematical skills. By practicing this method, you can quickly and accurately estimate square roots, which is particularly helpful in scientific and engineering contexts where precise measurements are crucial.

Sharpen your skills, have fun, and enjoy the process of mental calculation!