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 modular arithmetic.
Modular arithmetic works exactly as
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, is the set of numbers used to count: N = cfw_1, 2, 3, 4
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 attendance in class. Each week of tutorial
will be separated by
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
None, although a high school course in functions may be
us
0.1. SET THEORY
0.1
1
Set Theory
Sets can be very general. They need not necessarily refer to numbers or
even mathematical objects. We begin by dening exactly what we mean by
a set:
Denition 0.1.1. A set is a collection of objects, called elements.
Consid
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
None, although a high school course in functions may be us
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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 = 32 5 53
c. 10403 = 101 103
d. 1089 = 32 112
e. 11011 = 7
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 particular natural number, usually the number 1. Then,
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 back, plus
the interest on that $100: (100)(0.06) = 6. I
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t n = t1 + (n 1)d
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0.1. COMPLEX NUMBERS
EXERCISES
1. For each complex number z, nd Re(z) and Im(z).
(a) z = 1 4i
(b) z = 2 + i
(c) z = 4 + 3i
(d) z = i
(e) z = 8
2. Evaluate the sum and product of each pair of complex numbers.
(a) 1 4i and 2 + i
(b) 3 + 3i and 4 3i
(c) i an
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 back, plus
the interest on that $100: (100)(0.06) = 6. I
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 simple argument should convince
you otherwise.
Imagine
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 particular natural number, usually the number 1. Then,
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 2 is not, as x = x 2 involves a fractional power of x.
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 excellent start, and that a lot may be developed by aski
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) cfw_a, b A
(g) cfw_a, b A
(h) A
(i) cfw_ A
(j) A
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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 of the above
2. The sum of the exponents of the prime de