What's The Square Root Of 70

What's the Square Root of 70? A Deep Dive into Approximation Methods

The square root of 70 isn't a whole number, nor a simple fraction. It's an irrational number, meaning its decimal representation goes on forever without repeating. This seemingly simple question opens the door to exploring various mathematical concepts and techniques used to approximate irrational numbers. This article will delve into several methods for finding an approximate value for √70, examining their accuracy, efficiency, and underlying principles.

Understanding the Problem: Why Isn't √70 a Nice Number?

Before diving into the solutions, let's understand why finding the square root of 70 is not a straightforward task. A square root of a number 'x' is a value that, when multiplied by itself, equals 'x'. For example, the square root of 9 is 3 (because 3 * 3 = 9), and the square root of 16 is 4 (because 4 * 4 = 16). These are perfect squares.

70, however, is not a perfect square. There's no whole number that, when multiplied by itself, equals 70. This means the square root of 70 will be a decimal number that extends infinitely. This characteristic makes finding an exact value impossible, forcing us to rely on approximation techniques.

Method 1: Using a Calculator

The simplest and most readily available method is using a calculator. Most calculators have a square root function (√). Simply input "70" and press the square root button. You'll get a result like 8.36660026534...

This method provides a quick and relatively accurate approximation, but it doesn't offer insight into how the answer is obtained. It's a black-box solution, perfectly acceptable for practical purposes, but not satisfying from a mathematical understanding perspective.

Method 2: The Babylonian Method (or Heron's Method)

This iterative method provides a surprisingly accurate approximation of square roots, and it's relatively easy to understand. It's based on the principle of successively refining an initial guess.

Here's how it works:

1. Make an initial guess: Start with a guess for √70. A reasonable guess might be 8, since 8 * 8 = 64, which is close to 70.

2. Improve the guess: Divide 70 by your initial guess (70 / 8 = 8.75).

3. Average the results: Average the result from step 2 and your initial guess: (8 + 8.75) / 2 = 8.375. This is a better approximation than the initial guess.

4. Iterate: Repeat steps 2 and 3, using the improved guess from step 3 as the new initial guess. Continue this process until the desired level of accuracy is achieved.

Let's perform a few iterations:

• Iteration 1: Initial guess = 8, 70/8 = 8.75, average = 8.375
• Iteration 2: Initial guess = 8.375, 70/8.375 ≈ 8.36, average ≈ 8.3675
• Iteration 3: Initial guess = 8.3675, 70/8.3675 ≈ 8.3666, average ≈ 8.36705

As you can see, with each iteration, the approximation gets closer to the actual value of √70. The Babylonian method converges relatively quickly, offering a good balance between accuracy and computational effort. This method is particularly powerful because it doesn't require advanced mathematical tools.

Method 3: Using the Binomial Theorem

The binomial theorem allows us to approximate the square root of a number by expanding a binomial expression. This method is more mathematically rigorous but also more complex.

We can rewrite √70 as √(64 + 6) = √[64(1 + 6/64)] = 8√(1 + 6/64). Now we can use the binomial theorem approximation:

√(1 + x) ≈ 1 + x/2 for small values of x.

In our case, x = 6/64 = 3/32. Therefore:

√(1 + 3/32) ≈ 1 + (3/32)/2 = 1 + 3/64

So, √70 ≈ 8 * (1 + 3/64) = 8 + 3/8 = 8.375

This method provides a reasonable approximation, but its accuracy depends heavily on the size of 'x'. Smaller values of 'x' yield better approximations. For more accuracy, more terms in the binomial expansion would be required.

Method 4: Linear Interpolation

This is a simpler approximation technique. We find two perfect squares close to 70: 64 (8²) and 81 (9²). Since 70 is closer to 64 than 81, we can approximate:

√70 is roughly between 8 and 9. 70 is 6 more than 64, and there are 7 units between 64 and 81. Thus, we can estimate:

√70 ≈ 8 + (6/7) ≈ 8.86

This method is quite rough but demonstrates a basic approach to estimation.

Method 5: Newton-Raphson Method

This is a powerful iterative method for finding successively better approximations to the roots of a real-valued function. To apply it to finding the square root of 70, we consider the function f(x) = x² - 70. The square root of 70 is the root of this function (where f(x) = 0).

The Newton-Raphson formula is:

x_(n+1) = x_n - f(x_n) / f'(x_n)

where f'(x_n) is the derivative of f(x), which is 2x_n in our case. So the iterative formula becomes:

x_(n+1) = x_n - (x_n² - 70) / (2x_n)

Starting with an initial guess (e.g., 8), we can iteratively refine the approximation:

• Iteration 1: x_1 = 8, x_2 = 8 - (64 - 70) / 16 = 8.375
• Iteration 2: x_2 = 8.375, x_3 ≈ 8.36660
• Iteration 3: x_3 ≈ 8.36660, x_4 ≈ 8.3666002653

This method converges very quickly to a highly accurate approximation.

Comparing the Methods

Each method offers a different balance between accuracy, complexity, and computational effort.

• Calculator: Easiest, most accurate for practical use, but lacks mathematical insight.
• Babylonian Method: Relatively simple, good accuracy with few iterations.
• Binomial Theorem: Requires more mathematical understanding, accuracy depends on the approximation used.
• Linear Interpolation: Simple but least accurate.
• Newton-Raphson Method: Most sophisticated, very rapid convergence to high accuracy.

The choice of method depends on the desired level of accuracy, the available tools, and the mathematical background of the user. For most practical purposes, a calculator or the Babylonian method would suffice. For a deeper understanding of numerical approximation techniques, exploring the binomial theorem or the Newton-Raphson method is invaluable.

Conclusion: The Square Root of 70 and Beyond

The seemingly simple question of finding the square root of 70 unveils a rich tapestry of mathematical concepts and approximation techniques. While we cannot obtain an exact value, the various methods discussed provide ways to achieve approximations of varying degrees of accuracy and complexity. Understanding these methods helps in appreciating the nuances of irrational numbers and the power of numerical analysis in approximating solutions to seemingly intractable problems. This understanding extends far beyond just calculating square roots; it's foundational to numerous areas within mathematics, science, and engineering. The journey to find √70 is, in itself, a valuable learning experience.