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From the Fun Fact files, here is a Fun Fact at the Medium level:

Quick Square Roots

Here's a very quick way to generate the square root of N. Let A0=N. Then generate a sequence of numbers A1, A2, A3, etc. (on your calculator, for instance) by using the formula:

Ak+1 = 1/2 ( Ak + (N/Ak) ).

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 A0.

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. <>.

Keywords:    Newton's method, dynamical systems
Subjects:    algebra, calculus, analysis, number theory
Level:    Medium
Fun Fact suggested by:   Francis Su
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