MATH 203, PROBLEM SET 8
DUE IN ASHER AUEL’S MAILBOX BY 5 P.M., WEDNESDAY, APRIL 19.
1. Quotient groups
This set of problems is relevant to the Escher project. The complex exponential function
exp : C→C∗=C− {0}is defined by the power series
exp(z) = 1 + z+z2/2!+··· =
∞
X
i=0
zi
i!.
You can assume that this function satsifies
exp(z1+z2) = exp(z1)·exp(z2) and exp(iy) = cos(y) + isin(y) if y∈R.
1. Show that exp(z) = 1 if and only if z= 2πim for some integer m. (Hints: Write
z=x+iy for some x, y ∈R. You can without proof the fact that ex= 1 if and
only if x= 0, as well as standard properties of cos(y) and sin(y).)
2. Show that the map z→exp(z) is a homomorphism between the group Cunder
addition and the multiplicative group C∗of non-zero complex numbers. Let [z] =
z+ 2πiZbe the class of z∈Cin the quotient group C/2πiZ. Show that exp(z)
depends only on [z], and that the map [z]→exp(z) defines a group isomorphism
C/2πiZ→C∗. You can use without proof that every real number r > 0 equals ex
for a unique x∈R, and that every complex number αfor which |α|= 1 equals
cos(θ) + isin(θ) for a unique θin the interval 0 ≤θ < 2π.
3. Draw the set S={x+iy :x, y ∈R,0≤y < 2π}in the complex plane. Show that
Sis a set of representatives for the coset space C/2πiZ.
4. The quotient map q:C→C/2πiZsends zto [z] = z+ 2πiZ. The open subsets U
of C/2πiZare those for which q−1(U) is an open subset of C. Describe the subsets
of Swhich correspond to open subsets of C/2πiZ.
5. Draw the image under exp of the subset D={x+iy : 0 ≤x < 1,0≤y < 2π}
of S. Show that Dis a set of representatives for the quotient group C/H when
H={m+in :m, n ∈Z}.
2. Continued Fractions
6. Find all the possible simple continued fractions for the rational number x= 11/7.
7. Find the infinite simple continued fraction for √3.
8. You’ve been kidnapped by Taliban guerillas while on spring break in Pakistan. Their
leader has announced that irrational numbers offend him. He says that you will be
freed if you can tell him the rational number p/q with denominator 1 ≤q≤7 which
is the best approximation to π= 3.1415.... How would you use continued fractions
to determine this p/q?
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