Question

Use the Heron of Alexandria’s method to estimate the square root of 3.

Make sure to list all the steps.

Answer 1

Heron of Alexandria's method of estimating square root is an approximation technique.

The first step is to guess a number close to the square root of the given number

Let the given number be n

1. we guess x as the number close to the square root of n

2. x new = (x +n/x)/2

we repeat this process until we are satisfied with the result according to the decimal place accuracy.

NOW SOLVING SQUARE ROOT OF 3:

since

1² = 1 and 2² =4

we guess x=1.2

1st iteration:

x new = (1.2 + 3 / 1.2) / 2

x new = 1.85 (this is our initial solution)

2nd iteration:

x new = (1.85 + 3 / 1.85) / 2

x new = 1.73581 (this is our second solution, which is closer to the actual solution upto 2 decimal places)

3rd iteration:

x new = (1.73581 + 3 / 1.73581) / 2

x new = 1.73205 (this is our third result which is accurate upto 5 decimal places)

therefore sqare root of 3 = 1.73205

For more accurate results the steps can be repeated in a similar fashion.