Merced College; Don Power              

 

 

Lab -- Pythagorean Triples, Scattergrams, and Parametric Surfaces   (Mr. Power)     NAME _____________

 

1.  The Formulas.  Our first task is to find parametric formulas to generate all Pythagorean triples, i.e., sets of   integers that satisfy       the Pythagorean formula a2 + b2 = c2.

 

a.  Divide the Pythagorean formula by c2 and notice that the result is in the form x2 + y2 = 1.

 

      What are x and y in terms of a, b, and c? x = ____, y = ____ (You will use these definitions of x and y later.)

 

b.  In the equation x2 + y2 = 1, solve for x2 and factor the resulting difference of squares.

 

        Result: x2 = __________________

c.  Divide both sides by x and by (1+y).  You should get a proportion: _________ = _________

      (To check this step, cross-multiply; the result should be x2 = 1-y2)

 

d.  Both sides are equal, so we can set them equal to the same letter (use t).  Thus t:  t= ______ and t = _______

      (Notice that t must be a rational number -- we will use this fact later.)

 

e.  Clear the fractions in both equations from step d.  First: _________________ Second: ________________

 

f.  Rearrange both equations in the form Ax + By = C.  You should now have the system

g.  Solve the system of equations for x and y.  Use Cramer’s Rule.  Results:

 

 

D =

 

 

Dx =

Dy =

x =

y =

h.  Consider these equations for x and y and recall (from step d) that t is a rational number.  Call it .

      Substitute   in the equations for x and y (step g) and clear the complex fractions.  You should get:                                                                                                                               ,

j.  Go back to step a for the definitions of x and y.  Equate these definitions with the results of step h.

 

      x =             (from step a)  =                                   from step h, . . .   so if a = 2uv, what is c? _____________

 

      y =            (from step a)  =                                   from step h, . . .   so if c = v2 + u2, what is b? ___________

 

Summarize:

 

a = 2uv

 

b =

c =

 

Summary:  With these formulas, you can pick any positive integers for u and v (with v > u) and (a, b,c) will be a Pythagorean triple.

 

k.  To keep a and b positive, and to exclude excluding multiples of smaller triples, we establish these restrictions:

      (1) v>u

      (2) u and v are relatively prime (they do not have any common prime factors).

      (3) Either u or v must be odd, and the other must be even.

 

 

2.  Specific Pythagorean Triples.  In this part of the lab, we will use a spreadsheet to generate a number of Pythagorean triples and investigate a relationship among them.

 

a.  In a spreadsheet (like Excel), enter the following column headings:

 

u

v

a

b

c

 

 

 

2uv

v^2-u^2

v^2+u^2

a/c

b/c

 

b.  Enter all the possible integer selections for u and v, keeping u and v both above 0 and below 10.

        To keep a and b positive, and to exclude excluding multiples of smaller triples, make sure that:

      (1) v>u

      (2) u and v are relatively prime (they do not have any common prime factors).

      (3) Either u or v must be odd, and the other is even.

 

You should end up with between 15 and 20 pairs of numbers, starting with 1 and 2, 1 and 4, 1 and 6, ...

 

c.  Enter the appropriate formulas in the remaining columns.  For example, if the first u and v are in cells A3 and B3, respectively, then the formula in the “a” column would be =A3*B3.

 

      What you actually type in the first three lines will look like this:

 

u

v

a

b

c

 

 

 

 

2uv

v^2-u^2

v^2+u^2

x=a/c

y=b/c

1

2

=2*A3*B3

=B3^2-A3^2

=B3^2+A3^2

=C3/E3

=D3/E3

 

d.  Drag the formulas down to do the calculations for all your u’s and v’s.

 

e.  Pick a row in the bottom half of your table and demonstrate here that it really represents a Pythagorean triple:

 

 

 

 

 

f.  Use the mouse to highlight all the numbers in the a/c and b/c columns.  If you were to graph these as x’s and y’s, where would you expect them to be?  (Hint:  See Part 1a above). __________________________________

 

g.  With the numbers highlighted (from item f), select the “Chart Wizard” in Excel, choose the “XY Scatter” option (without any connecting lines), and in Step 3 of the Chart Wizard, remove the legend.  Move the resulting graph to the bottom of your worksheet.  Click on the chart area and adjust the dimensions to square up the graph.

 

h.  Are the points evenly distributed as you go around the circle? (Yes or No?) ________________________

      In what quadrants were all the points? (Why?) ___________________________________________

      Will the complete set of all Pythagorean triples complete the graph in this/these quadrant(s)? ________

            Why or why not? _______________________________

 

k.  Print the Excel worksheet.

 

 

3.  Parametric Surface Plots.  In this part of the lab, we will use a computer algebra system like Maple to graph     a surface and examine that surface.

 

a.  Use Maple assignment statements to define a, b and c, using the equations found in part 1j.  For example, a:=2*u*v;

 

b.  Use the plot3d command to graph the surface: plot3d([a,b,c],u=-10..10,v=-10..10,labels=["a","b","c"]);

 

c.  Rotate/adjust the figure as necessary to get the following views:

 


     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Orientation?     Horizontal axis: _____

 

What rotation gives you this figure?

(Hint -- Maple tells you the angles.)

 

Why does it appear that there are there two different scales for the vertical axis?
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


d.  Get the clearest view of the figure (in your opinion) and print your Maple worksheet.

 

Turn in:

-- This worksheet.

--Excel table (with scatter diagram).

--Maple worksheet.

 

 

Return to:  Merced College; Don Power               Updated 06/12/06 by Don Power