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

... find the approximate value of √5. In order to determine √5, let ...

5 ÷ G1 = 2.5
5 ÷ G2 = 2.222
5 ÷ G3 = 2.236

That means √5 is approximately 2.236.

Answer 2

The method developed by Babylonians to find the square root of numbers which were non perfect Squares is,

The given number is √5.

As, 2 < √5 < 3

Now, take the number between 2 and 3, which is smaller.

So, $G_{1}=2$

$\frac{G_{1}}{2}=\frac{5}{2}=2.5$

Mean of 2 and 2.5 is

$G_{2}=\frac{2+2.5}{2}=2.25$

So, $G_{3}=\frac{5}{G_{2}}=\frac{5}{2.25}=2.23$

2.23 × 2.25≠5

Now,

$\frac{5}{G_{3}}=\frac{5}{2.23}=2.24=G_{3}$

Which , is required proof.

As, $(G_{3})^2=2.23 \times 2.24=5{\text{approx}}$

Hence proved.