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How to Calculate the Square Root of 3 with Accuracy and Simplicity

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Calculating the square root of 3 can be a little challenging, especially without a calculator. However, by using simple methods such as the Babylonian method (also known as Heron's method) and the binomial theorem, you can approximate the value with remarkable accuracy. In this article, we will explore these methods step-by-step.

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The Babylonian Method (Heron's Method)

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The Babylonian method, also known as Heron's method, is an iterative process that provides a useful and efficient way to approximate square roots. Follow these simple steps to calculate the square root of 3:

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Steps for the Babylonian Method

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Choose an Initial Guess

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Your initial guess can be any number close to the expected square root. For the square root of 3, a good starting point is 1.5.

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Apply the Formula

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Use the formula: (x_{n 1} frac{x_n frac{S}{x_n}}{2}), where (S) is the number whose square root you want to find (3 in this case).

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Iterate

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Repeat the calculation until you reach the desired accuracy.

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Example Calculation

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Initial Guess:

(x_0 1.5)

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First Iteration:

(x_1 frac{1.5 frac{3}{1.5}}{2} frac{1.5 2}{2} 1.75)

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Second Iteration:

(x_2 frac{1.75 frac{3}{1.75}}{2} approx frac{1.75 1.7143}{2} approx 1.73215)

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Third Iteration:

(x_3 frac{1.73215 frac{3}{1.73215}}{2} approx frac{1.73215 1.73205}{2} approx 1.73205)

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After a few iterations, you can see that (sqrt{3}) is converging to approximately 1.73205, which is a good approximation of (sqrt{3}).

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The Binomial Theorem Method

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For an alternative approach, you can leverage the binomial theorem to approximate (sqrt{3}). Here’s how:

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Manipulating the Number

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Manipulate the number 3 to facilitate the binomial theorem:

r r r Express 3 as a fraction: (sqrt{3} sqrt{frac{48}{16}} sqrt{frac{49 - 1}{16}} frac{7}{4}sqrt{1 - frac{1}{49}})r r r

Binomial Approximation

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The binomial theorem approximation is given by:

r r [sqrt{3} frac{7}{4} left(1 - frac{1}{98} - frac{1}{2!98^2} - frac{1 cdot 3}{3!98^3} - frac{1 cdot 3 cdot 5}{4!98^4} - cdots right)]r r

If you calculate up to 2 terms in the parenthesis, you get: 1.732_{142} cdots, which is 3 digits accurate under the decimal point.

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If you calculate up to 3 terms, you get: 1.73205_{174} cdots, which is 5 digits accurate under the decimal point.

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If you calculate up to 4 terms, you get: 1.7320808_{195} cdots, which is 7 digits accurate under the decimal point.

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Every additional term in the calculation improves the accuracy by almost two decimal places.

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Conclusion

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The square root of 3 is approximately 1.732. By using either the Babylonian method or the binomial theorem, you can achieve this approximation with remarkable precision. These methods not only provide a practical way to calculate square roots but also demonstrate the beauty and elegance of mathematical theorems.

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