MATH135 Lecture Notes - Lecture 10: Nth Root, Cube Root, Joule
14 May 2018
School
Department
Course
Professor
MATH 135 Fall 2015: Extra Practice Set 10
These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike
assignment problems, they may discussed in depth on Piazza.
•The warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any
of them, then you should review your notes and possibly speak to an instructor before beginning the
corresponding assignment.
•The recommended problems supplement the practice gained by doing the corresponding assignment.
Some should be done as the material is learned and the rest can be left for exam preparation.
•A few more challenging extra problems are also included for students wishing to push themselves
even harder. Do not worry if you cannot solve these more difficult problems.
Warm-up Exercises
1. Compute all the fifth roots of unity and plot them in the complex plane.
2. Find all complex numbers zsolutions to z2=1+i
1−i.
3. Find a real cubic polynomial whose roots are 1 and i.
4. Divide f(x) = x3+x2+x+ 1 by g(x) = x2+ 4x+ 3 to find the quotient q(x) and remainder r(x)
that satisfy the requirements of the Division Algorithm for Polynomials (DAP).
Recommended Problems
1. Let n≥2 be an integer. Prove that
n−1
X
k=0
cos 2kπ
n=0=
n−1
X
k=0
sin 2kπ
n
2. A complex number zis called a primitive n-th root of unity if zn= 1 and zk6= 1 for all 1 ≤k≤n−1.
(a) For each n= 1,2,3,6, list all the primitive n-th roots of unity.
(b) Let zbe a primitive n-th root of unity. Prove the following statements.
i. For any k∈Z,zk= 1 if and only if n|k.
ii. For any m∈Z, if gcd(m, n) = 1, then zmis a primitive n-th root of unity.
3. Let uand vbe fixed complex numbers. Let ωbe a non-real cube root of unity. For each k∈Z,
define yk∈Cby the formula
yk=ωku+ω−kv.
(a) Compute y1,y2and y3in terms of u, v and ω.
(b) Show that yk=yk+3 for any k∈Z.
(c) Show for any k∈Z,
yk−yk+1 =ωk(1 −ω)(u−ωk−1v).
40
MATH135 Full Course Notes
Verified Note
40 documents
Document Summary
Math 135 fall 2015: extra practice set 10. These problems are for extra practice and are not to be handed. Solutions will not be posted but, unlike assignment problems, they may discussed in depth on piazza: the warm-up exercises are intended to be fairly quick and easy to solve. If you are unsure about any of them, then you should review your notes and possibly speak to an instructor before beginning the corresponding assignment: the recommended problems supplement the practice gained by doing the corresponding assignment. Some should be done as the material is learned and the rest can be left for exam preparation: a few more challenging extra problems are also included for students wishing to push themselves even harder. Do not worry if you cannot solve these more di cult problems. Warm-up exercises: compute all the fth roots of unity and plot them in the complex plane, find all complex numbers z solutions to z2 = 1+i.
