Mudd Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2007 by Francis Su
You can now subscribe to our RSS feed to get the latest Fun Facts!
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2007 by Francis Edward Su
All rights reserved.

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. Studying iterated functions is the subject of discrete 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." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Bookmark this page on: | Digg this! | Del.icio.us | Technorati | Reddit | Fark | Squidoo | Furl | Blinklist | Yahoo MyWeb | Google | Stumbleupon |

Keywords:    Newton's method, dynamical systems
Subjects:    algebra, calculus, analysis, number theory
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
3.55
 
current
rating 		Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!