Here’s a very quick way to generate the square root of N. Let A_{0}=N. Then generate a sequence of numbers A_{1}, A_{2}, A_{3}, etc. (on your calculator, for instance) by using the formula:

A_{k+1} = 1/2 ( A_{k} + (N/A_{k}) ).

This will give a sequence that converges very quickly to the square root of N. In fact, it converges so quickly, that it generally doubles the number of correct digits after each step!

This formula arises as a result of using Newton’s method. Can you figure out how?

**Presentation Suggestions:**

Draw a picture, if it is helpful, of how Newton’s method works. Challenge them to explore what happens if you start off with different values of A_{0}.

**The Math Behind the Fact:**

Repeatedly applying a function over and over is called *iteration*. Iterated functions are studied in *dynamical systems*. Newton’s method is one example of how iteration can be very useful.

**How to Cite this Page:**

Su, Francis E., et al. “Quick Square Roots.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by**:

Francis Su