2.17. EXERCISES 39

(b) Show that if β is any nonzero irrational number, and N is a positive integer,there exists 0 < n ≤ N and an integer m such that

∣∣β − mn

∣∣ < 1nN ≤

1n2 . Hint:

You might consider β − [β ]≡ α where [β ] is the integer no larger than β whichis as large as possible.

(c) Next notice that from the proof, the same will hold for any β a positive number.Hint: In the proof, if there is a repeat in the list of numbers, then you wouldhave an exact approximation. Otherwise, the pigeon hole principle applies asbefore. Now explain why nothing changes if you only assume β is a nonzeroreal number.

15. This problem outlines another way to see that rational numbers are dense in R. Pickx ∈ R. Explain why there exists ml , the smallest integer such that 2−lml ≥ x sox ∈ (2−l (ml−1) ,2−lml ]. Now note that 2−lml is rational and closer to x than 2−l .

16. You have a rectangle R having length 4 and height 3. There are six points in R. Oneis at the center. Show that two of them are as close as

√5. You might use pigeon

hole principle.

17. Do the same problem without assuming one point is at the center. Hint: Considerthe pictures. If not, then by pigeon hole principle, there is exactly one point in eachof the six rectangles in first two pictures.

18. Suppose r (λ ) = a(λ )p(λ )m where a(λ ) is a polynomial and p(λ ) is an irreducible poly-

nomial meaning that the only polynomials dividing p(λ ) are numbers and scalarmultiples of p(λ ). That is, you can’t factor it any further. Show that r (λ ) is of theform

r (λ ) = q(λ )+m

∑k=1

bk (λ )

p(λ )k , where degree of bk (λ )< degree of p(λ )

19. ↑Suppose you have a rational function a(λ )b(λ ) .

(a) Show it is of the form p(λ )+ n(λ )∏

mi=1 pi(λ )

mi where {p1 (λ ) , · · · , pm (λ )} are rela-

tively prime and the degree of n(λ ) is less than the degree of ∏mi=1 pi (λ )

mi .

(b) Using Proposition 2.14.6 and the division algorithm for polynomials, show thatthe original rational function is of the form

p̂(λ )+m

∑i=1

mi

∑k=1

nki (λ )

pi (λ )k

where the degree of nki (λ ) is less than the degree of pi (λ ) and p̂(λ ) is somepolynomial.