Assignment: Heron of Alexandria
Heron (also Hero) was a Greek mathematician and engineer. He lived and worked in Alexandria, Egypt, around 75 A.D. During his prolific work life, Heron developed a rotary steam engine called an aeolipile, a surveying tool called a dioptra, as well as a wind organ and a fire engine.
As an engineer, he must have had the need to approximate square roots because he described an iterative method for doing so in his work Metrica. Heron's method for approximating a square root can be summarized as follows:
Suppose that x is not a perfect square and a2 is the nearest perfect square to x. For a rough estimate of the value of √x, find the value of y1 = (1/2)[a + (x/1)]. This estimate can be improved by calculating a second estimate using the first estimate y1 in place of a: y2 = (1/2)[y1 + (x/y1)].
Repeating this process several times will give more and more accurate estimates of √x.
1. a. Which perfect square is closest to 80?
b. Use Heron's method for approximating square roots to calculate the first estimate of the square root of 80. Give an exact decimal answer.
c. Use the first estimate of the square root of 80 to find a more refined second estimate. Round this second estimate to 6 decimal places.
d. Use a calculator to find the actual value of the square root of 80. List all digits shown on your calculator's display.
e. Compare the actual value from part (d) to the values of the first and second estimates. What do you notice?
f. How many iterations of this process are necessary to get an estimate that differs no more than one digit from the actual value recorded in part (d)?
2. Repeat Question 1 for finding an estimate of the square root of 30.
3. Repeat Question 1 for finding an estimate of the square root of 4572.
4. Why would this iterative method have been important to people of Heron's era? Would you say that this method is as important today? Why or why not?