Wednesday, December 25, 2013

Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW
SEASONS GREETINGS!
While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...
The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.
The extension at the bottom goes beyond what might be expected from the beginning of this exploration.
Math educators can adapt this for Algebra 1 through AP Calculus students...
THE PROBLEMS
What are the number(s) described in the following?
1. A number equals its reciprocal.
2. A number equals 25% of its reciprocal.
3.  A number equals twice its reciprocal.
4.  A number equals the opposite of its reciprocal.
5.  A number equals k times it's reciprocal. Restrictions on k? Cases?
1. 1,-1
2. 1/2,-1/2
3,  √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined
OVERVIEW and much more...
• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:
(*) New = (Old + k/Old)/2
Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]
Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could  implement this on their graphing calculator or program the algorithm themselves!