### MA180, MA185, MA190 Mathematics (Semester I)

For the course syllabus, learning outcomes and assessment details see the course web page and the continuous assessment web page.

### Homeworks

To register 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 for you!

### Tutorials

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:  MA180MA185 , MA190

#### 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 article by Fields Medallist William Thurston. He won the Fields Medal for his work in geometry.

#### CALCULUS

Calculus text:

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.

Calculus outline:

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:

A pdf record of the 2012-2013 calculus lectures will be placed below.

Lecture 1 (2012-13): Quick introduction to functions, graphs, limits and and rates of change.

Week 1 (from 2011-12)

Lecture 2 (2012-13): 25 minute introductions to functions.

Lecture 3 (2012-13): Radians, trigonometric functions, functions defined piece-wise, even and odd functions.

Lecture 4 (2012-13): Definition of a limit. Here are the photos of those who attended this lecture on 25 September 2012. Photo 1, Photo 2, Photo 3, Photo 4.

Week 2 (from 2011-12)

Lecture 5 (2012-13): Some properties of limits, and left/right hand limits. (This was not a great lecture. Apologies!)

Lecture 6 (2012-13): Continuity and the Intermediate Value Theorem.

Week 3 (from 2011-12)

Lecture 7 (2012-13): Application of the Intermediate Value Theorem. Then limits at infinity and curve sketching

Part 2 notes on rates of change (from 2011-12)

Lecture 8 (2012-13): Introduction to the derivative of a function.
Here are the photos of those who attended this lecture on 9 October 2012. Photo 1, Photo 2, Photo 3, Photo 4.

Lecture 9 (2012-13): Application of derivative to rates of change problems

Lecture 10 (2012-13): Application of derivative to maximum/minimum problems.

Lecture 11 (2012-13): Application of derivatives to curve sketching.

Lecture 12 (2012-13): Critical points: maxima and minima. Points of inflection. More on curve sketching.

Lecture 13 (2012-13): Summary of terminology: continuity, differentiability, critical point, concavity etc. Theorem: differentiability implies continuity. Rolle's Theorem. Applications.

Lecture 14 (2012-13): Mean Value Theorem followed by an introduction to logarithms.

Lecture 15 (2012-13): Logarithms "done proper".

Lecture 16 (2012-13): Inverse functions and introduction to differential Equations.

Lecture 17 (2012-13): Differential equations and cooling cups of coffee.

Here are the notes on differential equations part 3(II) from (2011-12).

Lecture 18 (2012-13): Antiderivatives

Lecture 19 (2012-13): Modelling the population of the world.

#### ALGEBRA

Algebra text:

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 Browne.

Algebra outline:

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.

Lecture 1 (2012-13): 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 lecture.

This online calculator will help with all your modular arithmetic calculations.

Lecture 1 (2011-12): Introduction to the course. Introduction to modular aritmetic. 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 partially explained!

Take a look at the Wikipedia for a good introduction to modular arithmetic.

Lecture 2 (2012-13):

Lecture 2 (2011-12): 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.

Lecture 3 (2012-13): Attacked and decoded a ciphertext 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.
Lecture 3 (2011-12):

See here for an obituary of Peter Hilton - Codebreaker and Mathematician. He worked at Bletchley Park during World War II.

Lecture 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.

Lecture 5 (2011-12): RSA public key cryptography and Euler's Phi function.

Lecture 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. Photo 1, Photo 2, Photo 3.

Lecture 6 (2012-13).

( Some may also find it useful to look at the three lectures 1, 2, 3 on RSA cryptography given in 2010-11.)

Lecture 7 (2011-12): 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 Theorem.

(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.)

Lecture 8 (2011-12): 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.

Lecture 9 (2011-12): Gave the formula for the inverse of a 2x2 matrix, and indicated its importance in affine encryption systems with message units of length 2.

Lecture 10 (2011-12): 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.

Lecture 11 (2011-12): Introduced the notion of a linear transformation and gave various examples. Also gave an example of a transformations which is not linear.

Lecture 12 (2011-12): 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.

Lecture 13 (2011-12): 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.

Lecture 14 (2011-12): 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.

Lecture 15 (2011-12): Gave the definition and some basic properties of the determinant of a 2x2 matrix.

Lecture 16 (2011-12): 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.

Lecture 17 (2011-12): 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.

Lecture 18 (2011-12): 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.

Lecture 19 (2011-12): 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 frog population.

Lecture 20 (2011-12): 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 questions (and found a slip in one of them)!