Additional Problems These are some problems that I recommend math majors and others interested in mathematics should try. Some combine multiple ideas in the text, some have applications that you will see in future math courses, and some are classical gems of math that may not appear in your other courses.

Homework #1

6.1: 43, 47, 55

6.2: 46, 47

A. In this problem, we will find an inverse function for y=x^3+x+1 (ie. solve this equation for x in terms of y). We'll use a clever trick of Cardano.
(a) Put x=s+t with 3st+1=0, and rewrite the given equation according to this substitution. The equation should simplify drastically. Why can we always find such s and t?
(b) Express s^3+t^3 in terms of y, and (s^3)(t^3) as a constant number. Use these facts to find a quadratic equation which has s^3 and t^3 as its two roots.
(c) Solve the quadratic equation above, and use this to solve for x. Where is this expression a continuous function of y?

Homework #2

6.4: 46, 52

6.5: 37, 42, 48

B. Use previous additional problems to show that, given any large real number M, one can find a natural number N so that 1+1/2+1/3+...+1/N is even larger than M.

Homework #3

6.6: 28

6.8: 75, 76, 85, 86

C. The functions sin(t) and cos(t) parametrise the circle: as t varies, the pair (cos(t), sin(t)) covers the unit circle x^2+y^2=1. In this problem, we will discuss a parametrisation of the circle by rational functions.
(a) For a given value of t, the line with slope t through (-1,0) meets a unique second point (x,y) on the circle. Use the equation of this line and the equation x^2+y^2=1 to solve for x and y in terms of t. Also, solve for t in terms of x and y.
The functions you found above are rational functions. It follows that if you plug in a rational number for t, x and y are rational numbers, and if you plug in a pair of rational numbers for x and y, t is a rational number. Therefore, all rational number solutions (x,y) of x^2+y^2=1 correspond to a rational number t in the parametrisation.
(b) Write t as m/n, and plug this into the expressions for x and y. By clearing denominators in x^2+y^2=1, we get an integer equation a^2+b^2=c^2. Argue that every integer triple (a,b,c) satisfying the equation a^2+b^2=c^2 either arises from (m,n) in this way, or is a multiple of such a triple.
This is a classification of all right triangles with integer sides.
(c) For a given point (x,y) on the circle, let w be the angle (in (-pi, pi), say) corresponding to it. Show that t as above is tan(w/2).

Homework #4

6.9: 53

7.2: 75, 82, 83, 84

D. Find a rational parametrisation of the hyperbola x^2-y^2=1. (see C. above)

Homework #5

7.3: 33, 34, 35

7.4: 35

Homework #7

8.2: 46

8.3: 27, 40

Homework #8

8.4: 47, 53, 56

9.1: 36, 40

E. The Cantor middle thirds set is obtained from the interval [0,1] as follows:
First remove the middle third (1/3,2/3). Second, from the two remaining intervals remove their middle thirds (1/9,2/9) and (7/9,8/9). Repeat with the four remaining intervals, and continue this process indefinitely. The leftover set is the Cantor set.
(a) Show that the remaining set has no length by using an infinite series to calculate the length of the set removed.
(b) One can write the real numbers by their base 3 expansions. Note that our decimal system says that the first digit after the decimal is how many 1/10's we can take away from the number. The next decimal digit is how many (1/10)^2's we can take away from that, the next is how many (1/10)^3's remain, and so on. The base 3 system tells how many (1/3)'s there are in the first spot, then how many (1/3)^2's, then how many (1/3)^3's, and so on. Use base 3 expansions to argue that the Cantor set has infinitely many numbers remaining in it.

Homework #9

9.3: 35-37

Homework #10

9.5: 45

Homework #11

9.8: 38, 39, 40