Math 170: Ideas in Mathematics
Homework 2
Due at recitation section, Feb 12-14, 2008

Proofs by contradiction (reduction to the absurd)
(1) Prove that there is no pair of strictly positive integers (x,y) such that x²-y²=1.
Hint: Factor x²-y².

(2) Prove that if a is a rational number a=p/q, and b is a number which is not rational, then a+b cannot be rational.
Hint: Start by assuming that a+b=r/s is a rational number. What can you conclude about b?

Proofs by induction
(3) Define a sequence a₀, a₁, a₂,... by the formula a_n+1 = 2a_n - a_n². Show by induction that for all n=0,1,2,..., we have a_n=1 - (1 - a₀)²ⁿ.

(4) Prove using induction that 6^(2n-1)+1 is divisible by 7.

Fundamental theorem of arithmetic

(5) Recall the result proved in class, that for any integers p and q greater than 1 with p < q, there exists an integer e and an integer r with 0 ≤ r < p such that q = ep + r. We saw that the integer r is the remainder of the division of q by p. Can you explain what e is in terms of p and q (without using r)? Do you know a formula for this quantity?

(6) The proof of the fact that there is an infinite number of primes used numbers of the form p₁p₂...p_n+1 where each p_i is a prime number. Is such a number always prime?

Conic sections

(7) Identify each of the following quadratic equations in two variables as an ellipse, a hyperbola or a parabola:
3x² + 2xy - 5y² + 3x - 11y = 2.
x² + xy + 4y² + 2x - 6y = 0.
3x² + 6xy + 3y² - x - 2y = 7.

(8) Take the ellipse equation x²/4 + y² = 1.
Plot four points in the plane corresponding to the two points of the ellipse with x-coordinate equal to zero and the two points of the ellipse with y-coordinate equal to zero. From this, sketch the ellipse. What are the positions of the two foci?

(9) Consider the quadratic equation -5x² + 2xy + 3y²=0. Its discriminant is positive, so it seems to be a hyperbola. But is it? Can you sketch some points on it and see what it looks like? Can you factor the expression and explain your graph?

(10) How many ellipses are there having foci located at the points (0,-1) and (0,1)? What does the family of such ellipses look like? Can you make a sketch?