For the course syllabus, learning
outcomes and assessment details see the course
and the continuous assessment web page
For the Science MA180 homeworks click
For the Maths & Education MA185/6 homeworks click
For the CS&IT MA190 homeworks click
for homeworks use your eight digit ID number, and choose a memorable
password for the homework system. Don't
forget your MA180/MA185/MA190/MH100 password
because I am
unable to reset it
Tutorials begin on Monday 24th September. Details can be found here
Student feedback on MA180/MA185/MA190
The student feedback data on the first four weeks of the MA180 module
is available here: MA180
WHAT IS MATHEMATICS?
I'm not too sure of the answer. But whatever it is it is possibly
something a bit diferent to what was taught in your school mathematics
classes. If you are interested in the question then you should browse this
by Fields Medallist William Thurston. He won the Fields
Medal for his work in geometry.
WHAT ARE THE EMPLOYMENT PROSPECTS
FOR A MATHs GRADUATE?
Have a look at the links
to answer this question.
The MA180 calculus lectures are based on
the textbook: "Calculus, early transcendentals " by James Stewart
(Sixth Edition). Only so much of an explanation can be achieved in
lectures, and this book can be used to reinforce (or maybe even
clarify!) explanations given in lectures. It also contains many
problems (some with fully worked solutions) on which you can practice.
Even if you drop maths in second year, this will be a handy book for
your scientific bookshelf. And if you continue with maths in second
year then you'll be able to use the book again then.
There are three sections to the module. In the first lecture we'll see
that the notion of limit is just what is needed to determine the speed
of an object. In the remaining lectures of Section I we'll:
see how to formally define a limit,; calculate a range of limits; use
limits to capture the notion of "continuous function"; give some
applications of basic results on continuity.
In Section II we'll use the notion of a limit to define the derivative
of a continuous function. We'll develop and use various basics tools
for calculation derivative. We'll consider applications to: rates of
change problems; maxima and minima problems; curve sketching.
In Section III of the module we'll study differential equations and
give some more applications.
Calculus lectures 2012-13:
record of the 2012-2013 calculus lectures will be placed below.
introduction to functions, graphs, limits and and rates of change.
1 (from 2011-12)
25 minute introductions to functions.
trigonometric functions, functions defined piece-wise, even and odd
Definition of a limit. Here are the photos of those who attended this
lecture on 25 September 2012. Photo
2 (from 2011-12)
properties of limits, and left/right hand limits. (This was not a great
and the Intermediate Value Theorem.
3 (from 2011-12)
Application of the Intermediate Value Theorem. Then
limits at infinity and curve sketching
Part 2 notes on rates of change
to the derivative of a function.
Here are the photos of those who attended this
lecture on 9 October 2012. Photo
Application of derivative to rates of change problems
Application of derivative to maximum/minimum problems.
Application of derivatives to curve sketching.
Critical points: maxima and minima. Points of inflection. More on curve sketching.
Summary of terminology: continuity, differentiability, critical point, concavity etc. Theorem: differentiability implies continuity. Rolle's Theorem. Applications.
Mean Value Theorem followed by an introduction to logarithms.
Logarithms "done proper".
Inverse functions and introduction to differential Equations.
Differential equations and cooling cups of coffee.
Here are the notes on differential equations part 3(II)
Modelling the population of the world.
I can't find a suitable book to recommend for the algebra lectures. So
you'll just have to rely on the lecture notes and on the expanded
version of the lecture notes
which has been produced by Dr Patrick
This module introduces the student to matrix algebra and systems
of equations, emphasizing that: (i) the entries of a matrix can be any
"numbers" for which we have a suitable notion of addition and
multiplication; (ii) matrix arithmetic underpins Ireland's
knowledge economy; (iii) matrix arithmetic over the "real numbers" has
a fruitful geometric interpretation. The module is divided into three
parts. Part I introduces a number system that will be new to many
students. Part II introduces matrix arithmetic over this number system,
as well as over the usual real number system. Part III develops a
geometric interpretation for matrix arithmetic and systems of equations
over the real numbers. Students will be expected to develop their
understanding of the topics through extensive calculation rather than
through formal theory.
Algebra lectures 2012-13:
The algebra lecture slides will be very
similar to last year's slides listed below. So I won't bother uploading
the algebra slides for 2012-13 unless I happen to do something
very different in a lecture.
Introduction to modular aritmetic.
This lecture had to be repeated due to a timetable mixup. So these
slides are a compilation of the original lecture and the repeat
will help with all your modular arithmetic calculations.
1 (2011-12): Introduction
to the course. Introduction to modular
The notions of multiplicative inverse n-1 and additive
inverse -n modulo m were explained. The ISBN number was explained. The
lecture was short due to a power failure, so the IBAN was only
Take a look at the Wikipedia for a good
to modular arithmetic.
Explanation of a validation test for
International Bank Accounts
Numbers (IBANs) using arithmetic mod 97. Explanation of how the mod 97
value of a large integer can be calculated by hand. Introduction to
cryptography (U-boats, Bletchley Park, secure internet pages etc.) and
an example of a simple affine cryptosystem over a 26-letter alphabet in
which message units are single letters. Some mod 26 calculations show
that the composite of the enciphering function f(n) with the
deciphering function f'(n) yields the identity function f'(f(n)) = n.
Attacked and decoded a
sent using an affine cryptosystem
with single letter message units. As part of the attack the Euclidean
algorithm was used to find the inverse of an integer n modulo m. A
careful analysis of this method for finding an inverse shows that a
number n has an inverse modulo m if and only if gcd(m,n)=1.
for an obituary of Peter Hilton - Codebreaker and Mathematician. He
worked at Bletchley Park during World War II.
4 (2012-13 & 2011-12):
A lady has a basket of eggs. If she
packs them in boxes containing
exactly 13 eggs she can pack all but 3 eggs. If she packs them in boxes
of 14 she can pack all but 6 eggs. If she packs them in boxes of 15 she
can pack all but 9 eggs. In order to determine the least possible
number of eggs she could have we used the Chinese Remainder Theorem.
(It's fun to use this theorem but a bit boring to state it. So we used
it without stating it!)
At the start of the lecture we computed the last digit of a large power
of an integer.
Take a look at the Wikepedia for a good
account of the Chinese Remainder Theorem.
5 (2011-12): RSA public key cryptography and Euler's Phi function.
6 (2011-12): RSA Cryptography and Euler's Theorem.
Lecture 5 (2012-13). Here are the photos of those who attended this
lecture on 27 September 2012.
Lecture 6 (2012-13).
Some may also find it useful to look at the three
on RSA cryptography given in 2010-11.)
Deciphered a message sent to me using the RSA system. Explained how RSA
can be used to sign/authenticate documents electronically. Stated and
proved a special case of Euler's Theorem known as Fermat's Little
(Those who'd like to see a proof of the proposition (x^e)^d = x mod n
satisfied by the integers e,d,n used in RSA cryptography could take a
look at the second part of this
lecture which I gave last year.)
Recalled the notion of an mxn matrix (m rows, n columns). Recalled
addition/subtraction of two mxn matrices, and multiplication AB of an
mxn matrix A with a nxp matrix B. Recalled the notions of: zero matrix,
identity matrix, inverse matrix. From the definition of an inverse
matrix we evaluated the inverse of a 2x2 matrix modulo 5.
Gave the formula for the inverse of a 2x2 matrix, and indicated its
importance in affine encryption systems with message units of length 2.
Used the inverse of a 2x2 matrix modulo 37 to attack an affine
cryptosystem with 2-letter message units. We finished off the attack
using a computer implementation of the cryptosystem.
Introduced the notion of a linear transformation and gave various
examples. Also gave an example of a transformations which is not
Proved that any linear transformation can be represaented by a matrix.
Stated (without proof) that any reflection in a line through the origin
is linear, as is any rotation about the origin. Noted that if S, T are
two linear transformations represented by matrices A, B, then the
composite transformation SoT: v ->S(T(v)) is represented by the
matrix product AB. Derived the matrix representing an anticlockwise
rotation through an angle theta about the origin.
Illustrated the Gauss-Jordan method for finding the inverse of a square
matrix. The method is based on three types of row operations: i) Add a
multiplie of one row to a different row; ii) interchange two rows;
(iii) multiply a row by a non-zero (i.e. invertible!!) scalar.
Explained how each of the three row operations can be represented as
pre-multiplication by a suitable matrix. Then proved that the
Gauss-Jordan method always yields the inverse of a matrix when an
inverse exists. (The proof incorporates a hidden assumption which is
true but a bit tricky to prove. Please do try to spot the hidden
assumption!) Finished off by explaining how systems of
linear equations arise in real life, and how row operations can be used
to solve such systems.
Gave the definition and some basic properties of the determinant of a
Proved that the determinant of a 2x2 matrix is equal to the area of a
parallelogram (up to sign). Then gave the definition of an eigenvalue
and eigenvector, together with examples and a method for computing
eigenvalues and vectors.
Motivated by a problem of mathematically modelling a field of breeding
rabbits, we used eigenvalues and eigenvectors to study the Fibonacci
sequence and the Golden Ratio.
Started with three equivalent definitions of the Golden Ratio, and
mentioned its role in aesthetics. The used eigenvectors to find a
closed formula for the n-th term of the Fibonacci sequence.
Motivated by a problem of mathematically modelling an island of
infected frogs, the notion of a Markov matrix and Markov process was
introduced and used to study the long term prognosis for the island's
Starting from a model of the internet as a collection of nodes, and
arrows from one node to another, we saw how the eigenvector
corresponding to the largest eigenvalue of a certain Markov matrix
gives the ranking produced by a Google search. The very important
Hamilton Cayley Theorem was squaezed in at the end of the lecture.
See this link
for a more detailed account of Google's page rank algorithm.
In Lectures 21 and 22 (2011-12) I worked through some old exam
found a slip in one of them)!