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Homework
Due Monday, January 26
Below is a list of some of the complicated questions that students sometimes ask when
you introduce new concepts or revisit old ones. I want you to briefly answer each one of
these questions (three to five lines) to the best of your ability, as you might to a student.
Many of these questions will resurface in class (ideally through your discussion). You
also will want to answer some of these questions as part of your portfolio
1. Why can’t we divide by 0? If we just define
1   and be done with it?
0
1  i , why don’t we define
2. What do we mean when we divide a number by
2 or  ?
3. My high school geometry teacher said that you can’t trisect an angle with
straightedge and compass. Why not? How can you show that you can’t do
something?
4. If you add an infinite number of positive quantities, do you have to get infinity?
5. How can you add up things that get infinitely small and still get infinity? I don’t

1
understand why  should be infinite.
i 1 n
6. We can construct an equilateral triangle, a square, and a regular hexagon, can we
construct a regular pentagon or a regular 7-gon?
7. What are the complex numbers? Do they really exist?
8. Is there a point on the number line just to the right of 0?
9. Why is a negative times a negative a positive?
10. Why do you have to reverse the inequality sign when you multiply both sides by a
negative number?
11. Are there more even or odd integers? Explain.
12. How do we know that the Pythagorean Theorem is true?
13. They say that the sum of the angles of a triangle is always 180 degrees, but I just
drew a triangle and measured the angles and only got 178 degrees, why?
14. Are there more irrational numbers or rational numbers? How do we know?
15. What is an ellipse? How do you draw one?
16. What is the number e? Where does it come from?
17. You say that  is about 3.14 and is the same for all circles, yet when we measured
it in class, some people got 3 and others got 3.5, why should it always be the
same?
18. How do you know that the decimal for  never repeats or ends? How does
anyone find so many digits for it?
19. How do you calculate ln(x) or e x ?
20. I have a four function calculator at home, and when I divided 1 by 9, I get
.11111111, but when I add it to itself 9 times I get .99999999 not 1, so
(1  9)  9  1 . Is this rule just made up or is my calculator lying? Even if my
calculator is lying, isn’t 1  9  . 1 , so . 1  9  . 9 and again it isn’t 1. Doesn’t this
mean that multiplication doesn’t work for infinite decimals?
21. I added .7  . 8  1. 6 5  1.666...5 . This is OK isn’t it?
22. Is an integral really the area under the curve? I always thought that it was just an
approximation of the area. In fact, what do we mean by the area of a curved
shape?