calculus-2014-05-21

DRAFT c○ 2014 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse
Before rotation
After a 45 degree rotation
in the clockwise direction.
Figure 7.5. A rotation of 45 ◦ in the clockwise direction.
Before reflection
through the dashed line
After a reflection through
through the dashed line.
Figure 7.6. A Reflection.
10. Show that the natural numbers, N, and M = {1, 2, 3, 4, 6, 7, 8, 9, . . . } (the natural numbers minus
the number 5) are equidecomposable.
Hint: Break both N and M into two pieces such that one pair of pieces from each
are identical and the other pair of pieces are congruent by a shift to infinity (i.e. a
translation).
11. Why might people find your answer to Investigation 10 surprising? Explain.
Our next example, showing that a circle is equidecomposable to a circle minus a point, is similar
to Investigation 10 but since it is done on a circle, this adds a layer of complexity.
12. In Figure 7.7 is a circle of radius 1. Cut a piece of string whose length is equal to the radius, then
beginning at P 0 , mark off a point P 1 that is 1 unit (the length of the string) along the circle
away from P in the clockwise direction.
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