Understanding the Square Root of 1/3: A Detailed Exploration

The square root of 1/3 is a mathematical concept that often appears in various fields, from basic arithmetic to advanced calculus. In this article, we will explore this specific square root, provide a clear answer, and discuss methods to approximate its value without a calculator.

The Exact Value

The square root of 1/3 can be expressed as follows:

The square root of 1/3 is (1/sqrt{3} sqrt{3}/3) which is approximately 0.577. This can also be written as (sqrt{frac{1}{3}} frac{sqrt{frac{10000}{3}}}{100}).

A Better Approximation Method

For a more precise approximation, let's follow a step-by-step method without the use of a calculator:

Step 1: Simplify the Expression

Start with the fraction: (sqrt{frac{1}{3}} frac{sqrt{frac{10000}{3}}}{100}) Next, approximate (frac{10000}{3} approx 3333). The closest integer to the square root of 3333 is 58, as (58^2 3364).

Step 2: Refine the Approximation

Let (sqrt{3333} 58 Delta) where (Delta) is a small value. Then, (58^2 2 cdot 58 cdot Delta Delta^2 3364). Since (Delta^2) is very small, we can approximate it as 0, so (58^2 2 cdot 58 cdot Delta approx 3364). Solving for (Delta), we get (Delta approx -frac{31}{116} approx -0.27). Finally, (sqrt{frac{1}{3}} approx frac{58 - 0.27}{100} approx 0.5773).

Exact Value vs. Approximation

Using a calculator, the exact value of (sqrt{frac{1}{3}}) is 0.57735026919. The error in our approximation is very small, typically less than 0.0001.

Conclusion

The square root of 1/3 is an important mathematical concept that can be accurately calculated through different methods. By understanding the exact value and refining our approximation techniques, we can achieve great precision in calculations.