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The ancient Babylonians developed a method for calculating non-perfect squares by 1700 BCE. Complete the statements to demonstrate how to use this method to find the approximate value of [tex]\sqrt{5}[/tex].

1. In order to determine [tex]\sqrt{5}[/tex], let [tex]G_1 = 2[/tex], a number whose square is close to 5.
2. [tex]5 \div G_1 = \square[/tex], which is not equal to [tex]G_1[/tex], so further action is necessary.
3. Average 2 and [tex]5 \div G_1[/tex] to find [tex]G_2 = 2.25[/tex].
4. [tex]5 \div G_2 \approx \square[/tex] (rounded to the nearest thousandth), which is not equal to [tex]G_2[/tex], so further action is necessary.
5. Average 2.25 and [tex]5 \div G_2[/tex] to find [tex]G_3 = 2.236[/tex].
6. [tex]5 \div G_3 \approx \square[/tex] (rounded to the nearest thousandth), which is equal to [tex]G_3[/tex]. That means [tex]\sqrt{5}[/tex] is approximately 2.236.



Answer :

GinnyAnswer
Let's go through the steps to approximate [tex]\(\sqrt{5}\)[/tex] using the ancient Babylonian method, also known as Heron's method or the method of successive approximations.

1. Initial Approximation [tex]\(G_1\)[/tex]:
- We start with [tex]\(G_1 = 2\)[/tex], a number whose square is close to 5.

2. First Division:
- Calculate [tex]\(5 \div G_1\)[/tex]:
- [tex]\(5 \div 2 = 2.5\)[/tex]
- This is not equal to [tex]\(G_1\)[/tex], so further action is necessary.

3. Second Approximation [tex]\(G_2\)[/tex]:
- To find [tex]\(G_2\)[/tex], average 2 and [tex]\(5 \div G_1\)[/tex]:
- [tex]\(G_2 = \frac{2 + 2.5}{2} = 2.25\)[/tex]

4. Second Division:
- Calculate [tex]\(5 \div G_2\)[/tex] and round to the nearest thousandth:
- [tex]\(5 \div 2.25 \approx 2.222\)[/tex]
- This is still not equal to [tex]\(G_2\)[/tex], so further action is necessary.

5. Third Approximation [tex]\(G_3\)[/tex]:
- To find [tex]\(G_3\)[/tex], average 2.25 and [tex]\(5 \div G_2\)[/tex]:
- [tex]\(G_3 = \frac{2.25 + 2.222}{2} \approx 2.236\)[/tex]

6. Third Division:
- Calculate [tex]\(5 \div G_3\)[/tex] and round to the nearest thousandth:
- [tex]\(5 \div 2.236 \approx 2.236\)[/tex]
- This is now approximately equal to [tex]\(G_3\)[/tex].

Therefore, [tex]\(\sqrt{5}\)[/tex] is approximately [tex]\(2.236\)[/tex].

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