Chapter 1
Numbers
1.1
Prime Numbers
We start by dening collections of numbers, or sets, which we will be using throughout the book. The natural numbers, also known as the counting
numbers, denoted N,
McGill University
Department of Mathematics and Statistics
MATH180 The Art of Mathematics
COURSE OUTLINE
Instructor:
Dr. Sidney Trudeau
Office: BH 1127
e-mail: [email protected]
Prerequisites
Non
0.1. MODULAR ARITHMETIC
0.1
1
Modular Arithmetic
If it is 11 oclock now, what time will it be in 3 hours? If you answered 2,
then you are suggesting that 11 + 3 = 2. Welcome to the wonderful world
of
MATH 180 - Summary Notes
These notes are intended to provide a quick, concise reference to the course material, but should not be
considered as a sufficient replacement of the textbook and/or attendan
EXERCISES
1. Evaluate the following sums:
M8
(80
:3
H
a
(b)
3
H
0
M8
cfw_H 1% E1 124
(C)
M8
3
H
(1)
M8
3
H
00
1
2
M8
(6)
3
H
m
2
Mn + 2)
M8
(0
n1
H
2. Write the following rational numbers as a fractio
EXERCISES
1. Determine whether the following are prime or composite numbers. If
composite, decompose these into a product of primes as guaranteed
by the Fundamental Theorem of Arithmetic:
(a) 2011
(b)
0. 1. SET IIIEORY l5
EXERCISES
1. Let A 2 cfw_(1, b, (:7 cfw_a, b Determine whether the following are true or
false.
(a (17334: E A
(b cfw_ml, (3 6 A
cfw_mine C A
cfw_(1,1) E: A
2. List all the subset
0. 1. SET IIIEORY l5
EXERCISES
1. Let A 2 cfw_(1, b, (:7 cfw_a, b Determine whether the following are true or
false.
(a (17334: E A
(b cfw_ml, (3 6 A
cfw_mine C A
cfw_(1,1) E: A
2. List all the subset
0.1. COMPLEX NUMBERS
EXERCISES
14 For each complex number 2, nd Recfw_z) and 1111(2).
(a) 3:1,12
(b) zz~2+z
(C) z:1+31
(1)2:2
cfw_e :28
'2 Evaluate the sum and product of each pair of complex 11u111he
EXERCISES
1. Find an integer between 0 and 5 congruent to 1362 modulo 6.
2. Find an integer between 0 and 16 congruent to 1362 modulo 17.
3. Find an integer between 0 and 9 congruent to 1362 modulo 10
CHAPTER I. FUNCTIONS
EXERCISES
1. Evaluate the degrees of the following polynomials:
(a) A6$3+4x+2
(b310x5Ax4+x3Ax2+ch1
(c)3:17A1
(d) 1315+;623A2
2, For each of the following polynomials, verify that
1
Math 180 Midterm Version 1. Please answer all questions on the scantron
provided.
1. If you decompose 40320 into a product of primes, the sum of the
exponents is
(a) 8
(b) 9
(c) 10
(d) 11
(e) none o
McGill University
Department of Mathematics and Statistics
MATH180 The Art of Mathematics
COURSE OUTLINE
Instructor:
Dr. Sidney Trudeau
Office: BH 1127
e-mail: [email protected]
Prerequisites
Non
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0.1. SOLUTIONS TO ODD NUMBERED EXERCISES
0.1
1
Solutions to Odd Numbered Exercises
Primes
1a. 2011 is prime. This can be veried by checking that all primes less
than 43 do not divide 2011.
b. 2385 = 3
0.1. PROOF BY INDUCTION
0.1
1
Proof by Induction
Induction is a method of proof oftentimes used to prove statements concerning natural numbers. The idea is the following: You prove the statement
for a
0.1. FINANCIAL MATHEMATICS
0.1
1
Financial Mathematics
If you deposit $100 in the bank for one year, and if the interest rate in eect
is 6%, then, in one year, you will have $106. That is, your $100 b
MGCR 271: Assignment #3
Winter 2017
Due date: March 16th at the beginning of class. (please print, do not email). No late
submissions accepted.
Instructions:
Please print this booklet and answer all
6
EXERCISES
1. Evaluate the following sums:
(a)
(b)
(c)
(d)
(e)
(f)
1
2n
n=4
n=0
n=1
n=8
1
4n
1
n
1
10n
1
n
2
n=2
n=1
2
n(n + 2)
2. Write the following rational numbers as a fraction
p
of integers:
0.1. SET THEORY
15
EXERCISES
1. Let A = cfw_a, b, c, cfw_a, b. Determine whether the following are true or
false.
(a) a, b, c A
(b) cfw_a, b, c A
(c) cfw_a, b, c A
(d) cfw_a, b A
(e) cfw_a, b A
(f) cf
0.1. COMPLEX NUMBERS
0.1
1
Complex Numbers
There is always some apprehension when it comes to complex numbers. Peo
ple usually know i = 1 but feel they are missing a lot. We will see that
this is an e
Chapter 1
Functions
1.1
Polynomials
You have all seen polynomials before. They are simply expressions involving
whole powers of x. For example, 2x + 1 and 3x4 + 16x3 are polynomials
1
whereas 5x2 + x
0.1. PROOF BY INDUCTION
0.1
1
Proof by Induction
Induction is a method of proof oftentimes used to prove statements concerning natural numbers. The idea is the following: You prove the statement
for a
0.1. SERIES
0.1
1
Series
The term series refers to an innite sum. The ancient Greeks did not
believe you could add innitely many positive numbers together and get
anything less than innity. However, a
MGCR 271: Assignment #4
Winter 2017
Due date: April 4th at the beginning of class (please print, do not email). No late submissions
will be accepted.
Instructions:
Please print this booklet and answe
MGCR 271: Assignment #2
Winter 2017
Due date: February 14th at the beginning of class. (please print, do not email). No late
submissions accepted.
Instructions:
Please print this booklet and answer a
MGCR 271: Assignment #3
Winter 2017
Due date: March 16th at the beginning of class (please print, do not email). No late submissions will be accepted.
Instructions:
Please print this booklet and answ
MGCR 271: Assignment #2
Winter 2017
Due date: February 14th at the beginning of class. (please print, do not email). No late
submissions accepted.
Instructions:
Please print this booklet and answer a