Research Methods in Mathematics
Homework 4 solutions
T. PERUTZ
(1) Solution. (a) Since x2 = 2, we have (p/q)2 = 2, so p2 = 2q2 . By denition,
an integer is even if it is twice another integer. Since it is twice q2 , the integer
p2 is even.
(b) The square
Research Methods in Mathematics
Lecture 9: Functions and limits
T. PERUTZ
Functions and limits
Last time, we introduced the system R of real numbers. A sightly informal denition
of function goes as follows. A function f is a rule which assigns to certain
Research Methods in Mathematics, Lecture 10
Examples of limits
T. PERUTZ
In the last lecture we gave the denition of a limit. Heres a reminder: if f is a function
whose domain includes all real numbers close to a , except perhaps a itself, we say
that f (
Research Methods in Mathematics
Homework 1 solutions
T. PERUTZ
(1) (a) In each row, there is exactly one square on the diagonal. Therefore, the
number of diagonal squares equals the number of rows, that is, (n + 1).
(b) The chessboard has (n + 1)2 squares
Research Methods in Mathematics
Homework 3
T. PERUTZ
(1) Solution. The denition says that a fraction is represented by an ordered pair
of natural numbers (p, q), where (p, q) represents the same fraction as (np, nq)
for all natural numbers n . Suppose we
Research Methods in Mathematics
Homework 2 solutions
T. PERUTZ
Due at the beginning of class, Thursday September 8.
(1) Solution. (a) Let P(n) be the statement that
1 + 2 + + n = n(n + 1)/2.
Note: P(n) is not a number, nor is it an algebraic expression. I
Research Methods in Mathematics
Lecture 8: Real numbers
T. PERUTZ
Completeness
The system of rational numbers, Q , is an example of what is called a eld. That means
We have a rule for adding rational numbers.
Addition is associative and
commutative. Ther
Research Methods in Mathematics
Lecture 7: Least upper bounds
T. PERUTZ
In this lecture, well be working with rational numbers. The system of all rational
numbers is denoted by Q (for quotientan old-fashioned name for a fraction).
Notation. Well use the s
Freshman Research Initiative:
Research Methods in Mathematics
T. PERUTZ
2
2.1
Counting and induction
Number systems
In science nothing capable of proof ought to be accepted without proof.
Though this demand seems entirely reasonable, I cannot regard it as
Research Methods in Mathematics, Lecture 1
T. PERUTZ
Part I: From counting to calculus
The ghosts of departed quantities?
In a 1734 pamphlet entitled The Analyst, the philosopher George Berkeley argued that
innitesimal calculus, the subject that had been
Research Methods in Mathematics
Lecture 3: Induction is inescapable.
T. PERUTZ
More on the axioms for natural numbers
Of the axioms explained in the last lecture, the principle of induction seems much more
subtle than the rest. Heres a consequence of thos
Research Methods in Mathematics
Lecture 4: Addition and multiplication. Fractions
T. PERUTZ
Addition
The denition of addition of natural numbers is done recursively. We want to dene
n + m . We think of this as a function fn of m , with n xed: we want to s
Research Methods in Mathematics
Lecture 6: Rational numbers; upper bounds
T. PERUTZ
The integers
Whilst it is always possible to add two natural numbers, it is not always possible to
subtract. For this, we need to introduce the system of integers, in whic
Research Methods in Mathematics
Lecture 5: Inequalities
T. PERUTZ
Comparing natural numbers
If m and n are natural numbers, we write m > n (or n < m ) to mean that there is some
natural number p such that m = n + p .
Theorem 1 Let m and n be any two natur