Math Fun Facts!
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.
Get a random Fun Fact!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  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
All rights reserved.

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

Products of Sums of Two Squares

Here's a nice theorem due to Fibonacci, in 1202.

Theorem. If integers N and M can each be written as the sum of two squares, so can their product!

Example: since 2=12+12 and 34=32+52, their product 68 should be expressible as the sum of two squares. In fact, 68=82+22. Is there an easy way to figure out what squares the product will be made of?

Yes! This all follows from the very cool formula:


(a2+b2) (c2+d2) = (ac+bd)2 + (ad-bc)2.

Presentation Suggestions:
Do a few more numerical examples before showing the cool formula.

The Math Behind the Fact:
The formula above can be checked trivially. But there are other ways to see why it is true. If you know some linear algebra, take a look at the Fun Fact Really Complex Matrices, and take the determinant of both sides of the matrix equation there. You will get the formula above!

Or, if you know about complex numbers, the left hand side is the squared modulus of two complex numbers and the right side is the squared modulus of their product!

To see which numbers can be written as the sum of two squares, see Sums Of Two Squares.

How to Cite this Page:
Su, Francis E., et al. "Products of Sums of Two Squares." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    number theory, sum of squares
Subjects:    number theory
Level:    Medium
Fun Fact suggested by:   Lesley Ward
Suggestions? Use this form.
4.17
 
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!