Question

The ancient Babylonians developed a method for calculating nonperfect squares by 1700 BCE. Complete the statements to demonstrate how to use this method to find the approximate value of . In order to determine , let G1 = 2, a number whose square is close to 5. 5 ÷ G1 = , which is not equal to G1, so further action is necessary. Average 2 and G1 to find G2 = 2.25. 5 ÷ G2 ≈ (rounded to the nearest thousandth), which is not equal to G2, so further action is necessary. Average 2.25 and G2 to find G3 = 2.236. 5 ÷ G3 ≈ (rounded to the nearest thousandth), which is equal to G3. That means is approximately 2.236.

Answer 1

Step-by-step explanation:

The ancient Babylonians developed a method for calculating nonperfect squares by 1700 BCE. To use this method to find the approximate value of a nonperfect square, such as √5, follow these steps:

Let G1 be a number whose square is close to the given number, in this case, 2, since 2^2 = 4 is close to 5.

Divide the given number by G1, which gives 5 ÷ 2 = 2.5.

Average G1 and the result from step 2 to find G2 = (2 + 2.5) ÷ 2 = 2.25.

Divide the given number by G2, which gives 5 ÷ 2.25 ≈ 2.222.

Average G2 and the result from step 4 to find G3 = (2.25 + 2.222) ÷ 2 = 2.236.

Divide the given number by G3, which gives 5 ÷ 2.236 ≈ 2.236, which is equal to G3.

The number obtained in step 6 is the approximate value of the square root of the given number.

Therefore, using this method, the approximate value of √5 is 2.236.