Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Last year, on the television programme Antiques Roadshow... work out the approximate profit.

Archimedes and Numerical Roots

Why do this problem?

This problem offers students the opportunity to engage with and make sense of a numerical method for finding roots.

Possible approach

"How could I find the square root of three if I didn't have a calculator?" Collect together students' suggestions - it is likely that various methods of trial and improvement will be suggested, as well as the observation that the value will be between 1 and 2.

"Trial and improvement takes time. Here is another numerical method for finding roots." Introduce the algorithm for finding a new approximation. Give students some time to experiment with the method to get a feel for it and to observe how it converges to $\sqrt{3}$.
"Can you adapt the method to find roots of other numbers? Can you explain why it works?"
Again, give the students time to explore these two questions.
Finally, bring the class together so they can share their ideas and explanations.

Key questions

What does the method $(\frac{(\frac{3}{n} + n)}{2})$ calculate, if $n$ is an approximation to $\sqrt{3}$?
How can you change the method to work out other square roots?
How does the equation $n = (\frac{(\frac{3}{n} + n)}{2})$ help you to make sense of why the method works?