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].
