hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed or follow us on Twitter.              The Math Fun Facts App!

List All : List Recent : List Popular
About Math Fun Facts / How to Use
Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su

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

# Chords of a Unit Circle Figure 1

Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all other points on the circle, as in Figure 1. What is the product of lengths of those chords?

Surprise: for N points, the product is just N. Pretty cool, huh?

Presentation Suggestions:
Do simple cases first and draw pictures: for N=2, there is just one chord (the diameter of the circle, length 2). For N=3, there are two chords (both of length Sqrt, product=3). For N=4, there are 3 chords (one diameter of length 2, two of length Sqrt, product=4). Have people conjecture the answer for themselves!

The Math Behind the Fact:
If you know about complex numbers, you can prove this by letting the points of the circle be given by roots of unity, and letting z be any point (not necessarily the unit disc) and looking at the product of lengths of the chords from z to all N points on the circle. Divide this expression by (z-1), and take the limit as z goes to 1, and voila!, you will get N.

If you liked this Fun Fact, now see what happens with Chords of an Ellipse.

Su, Francis E., et al. "Chords of a Unit Circle." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Keywords:    complex numbers, chords, unit circle
Subjects:    algebra, geometry, calculus, analysis
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
4.31
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!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity Want another Math Fun Fact?

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