Additional Problems 2
Math 311, Spring 2009
Note: problems with (**) are harder ones.
(A-7) Prove that if b > 0, then there exist only finitely many positive integers nsuch that 0 < n ≤b. Hence
the set {n:n∈N, n ≤b}is a finite subset of N, and it has a maximum element if it is nonempty.
(A-8) It is known that C={a+bi :a, b ∈R}(where i2=−1) is a field. Prove that Cis not an ordered
field, that is, one cannot define an order relation on Cwhich satisfies (O1)-(O5). (hint: prove by
contradiction: if there is an order <, then either 0 < i or i < 0. · · · )
(A-9) The floor function is defined by
bxc= max {n∈Z|n≤x}.
And the fractional part function is {x}=x− bxc. For all x, 0 ≤ {x}<1.For example, b2.3c= 2,
{2.3}= 0.3; and b−2.3c=−3, {−2.3}= 0.7. Prove that if ais an irrational number, then
(a) For any ε > 0, there exist m, n ∈Nsuch that |na −m|< ε;
(b) For any a, b satisfying 0 <a<b<1, there exists n∈Nsuch that {na} ∈ (a, b).
Homework 2: due Feb 5 (Thursday) 5pm
Required problems: 4.1-4.4(b,i,n,v), 4.6, 4.8, 4.14a, 4.15, 5.1-5.2(c,d), 5.5, A-7
Optional problems: 6.4, 6.6(**, 2 points), A-8, A-9
General Rule for Homework (apply to all homework assignment, unless other specified):
1. Each homework assignment is 10 points. 5 points for completion, 4 points for correctness, and 1
point as award. Extra points are possible.
2. Each homework assignment has two parts: required problems and optional problems. If you solve all
required problems correctly, then you get 9 points. If you also solve noptional problems correctly,
then you get 9 + npoints.
