What will I study in this module?

In the first semester the module covers

  • Elementary number theory (8 lectures),
  • Matrix arithmetic (8 lectures),
  • Eigenvalues and vectors for 2x2 matrices (8 lectures),
  • Limits and continuity (8 lectures),
  • The derivative of a function (8 lectures),
  • The anti-derivative of a function (8 lectures),
  • and explains how these six topics can be applied to

  • cryptography,
  • geometry/computer graphics,
  • Google page rank,
  • existence of antipodal positions on the Earth's surface with equal air pressure,
  • rates of change problems and maximization problems,
  • prediction of how the world population of humans will grow.

How will the module be assessed?

In the first semester there will be:

  • 40% for quizzes.
    In the last week of the semester there will be two 1-hour quizzes, together worth 40%. Each quiz will consist of three questions. They will be modelled on the past exam paper available here.
  • 40% for homeworks.
    The homework problems are worth 40%. They will be delivered as six online problem sheets which will be made available here. Submission deadlines are strict. There are about 12 questions per problem sheet and to score 100% on the Semester I CA component you need to submit 60 or more correct answers.
  • 20% for group work.
    Students will submit two small-group projects, each worth 10%. Details.

MA180 students

The first semester score will be averaged with the second semester score to obtain an overall score for the 15 credit MA180 module. There will be six more homework sheets in Semester II and students must pass the 12 homeworks on average in order to be eliible to pass the MA180 module.

MA185 students:

The first semester score will be returned as the score for the MA185 Analysis and Algebra module, and will also count for 50\% of the MA187 Mathematical Skills module.

MA190 students:

The first semester score is equal in weight to the second semester analysis and calculus module (though second semester algebra is returned as part of he CT102 module).

Live lectures, on-campus workshops, textbooks, and contact

  • Lecturer: Prof Graham Ellis. (He can be contacted at graham.ellis@nuigalway.ie. Mathematical questions should normally be addressed to your workshop tutor and not to him. Mathematical questions can also be addressed to the people in SUMS.)
  • Lectures: Students can participate in live Zoom lectures at 1pm Mondays, 10am Tuesdays, 10am Wednesdays and 10am Thursdays. The link will be posted on Blackboard and circulated by email to registered students prior to the lectures.

    Lectures will be recorded and the videos will be posted below. So if you happen to miss a live lecture then you can always watch it dead. But any contributions from students during the lectures will not be recorded.
  • Workshops: On-campus workshops begin on Monday 28 September. Details can be found here.
  • Recomended text: The lecture notes and web links (below) and continuous assessment problems contain all material necessary for this module.

    The algebra lectures are based on the e-textbook
    A Short Introduction to University Algebra, by Graham Ellis
    which will be made available on Blackboard.

    The calculus lectures are based on the textbook:
    Calculus, early transcendentals by James Stewart (Sixth Edition)
    which can be accessed online via this library link. 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.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard Mathematics MA180 & MA185 & MA190 (Semester I) pages. Blackboard will also be used for announcements, quizzes, second semester exams, and for posting grades.

Homework sheets and Deadlines

The online homework sheets are available here.

Deadlines

Friday 16 October, 5pm First Okuson homework sheet
Friday 30 October, 5pm Second Okuson homework sheet
Friday 06 November, 5pm First small-group project
Friday 13 November, 5pm Third Okuson homework sheet
Friday 27 November, 5pm Fourth Okuson homework sheet
Wednesday 16 December, 10am

Thursday 03 December, 10am

First quiz
Friday 11 December, 5pm Second small-group project
Thursday 17 December, 10am Second quiz
Friday 04 December, 5pm

Friday 18 December, 5pm

Fifth Okuson homework sheet
Friday 18 December, 5pm

Friday 22 January, 5pm

Sixth Okuson homework sheet

Recorded lectures and notes

Lecture Notes

click number to view notes
Lecture Videos, Summaries and Other Material
Intro

25 September 2020: slides and prerecorded video

This is a talk about first year Mathematics and Mathematical Studies (Science) given as part of Orientation Week.
Lecture 1
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Maria Chudnovsky.

Gave an informal introduction to modular arithmetic, and included an application to the ISBN book number.

For an alternative introduction to modular arithmetic take a look at this Youtube clip. Then take a look at this clip, this clip and this clip.
Lecture 2
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Sebastian Schreiber.

Explained Euclid's algorithm for finding the greatest common divisor of two numbers, and used it to find the inverse of some number n modulo m. An application of modular arithmetic to IBAN bank numbers was explained.

Take a look at this clip for another example of using the Euclidean algorithm to find the inverse of a number in modular arithmetic.

For more background on modular arithmetic take a look at the wikipedia page here.


Lecture 3
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Dusa McDuff.

Explained the basic ideas underlying cryptography. Discussed the Enigma machine and an affine cryptosystem on single letter message units.

For more background on the Enigma machine take a look at the wikipedia page here.
For more background on affine cryptosystems take a look at the wikipedia page here.
Lecture 4
Video of lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Peter Scholz.

Deciphered an enciphered message sent from Agent 007.
Lecture 5
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Edward Frenkel.

Explained the Chinese Remainder Theorem.

For more background on the Chinese Remainder Theorem take a look at the wikipedia page here. Also, take a look at this youtube explanation which uses easily calculated numbers.

By the way, the puzzle used to introduce the Chinese Remainder Theorem is due to Brahmagupta who was born 598 AD (see here), so please don't attribute its old-fashioned stereotypes to me. 🙂
Lecture 6
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Sylvia Serfaty.

Introduced Euler's phi (or totient) function. Also gave the definition of a public key cryptosystem

For more background on Euler's phi function take a look at the wikipedia page here.
Lecture 7
Video of lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Terence Tao.

Stated and illustrated Euler's Theorem. Then stated and proved a special case known as Fermat's little theorem.

For more background on Euler's Theorem take a look at the wikipedia page here.
For more background on Fermat's little heorem take a look at the wikipedia page here.
Lecture 8
Post-recorded podcast of the lecture. (I forgot to press "record" at the start of the live lecture, so this is a recording of a repeat run given without any audience. )
Handwritten notes from the lecture. (The handwritten notes from the podcast are available here.)
Waiting room video: interview with Olga Paris-Romaskevich.

Explained the RSA public key cryptosystem.

For more background on the RSA cryptosystem take a look at the wikipedia page here.

It is worth considering, with the benefit of hindesight, the following quote

"both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate [number theory], and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean."

from G.H. Hardy's A Mathematician's Apology. This short book is well worth a read and is available online here.
Lecture 9
Video of lecture.
Handwritten notes from lecture.
Waiting room video: interview with Curtis McMullen.

We considered a stone being dropped from the top of the Leaning Tower of Pisa. We assumed that the distance at time t is 4.9t2 (something physicists tell us should be true). We used the formula y= 4.9t2 to begin a discussion of functions. We then investigated the question: what is the speed of the stone at time t=2 seconds?

The lecture touched on the notions of "continuous function", "limit" and "derivative" (="speed"). The remaining 23 lectures in Semester I will provide more details on these three fundamental notions.

The lecture also included reference to a function which was shown by Karl Weierstrass to be everywhere continuous (no breaks in its graph) and nowhere differentiable (no point on its graph has a well-defined tangent). The function was a lesson to physicists for the need for rigour.

For a more detailed summary of what calculus is all about, read the section "A Preview of Calculus" in Stewart, pages 2-9,

For more details on the Weiertrass function see here.
Lecture 10
Video of lecture.
Handwritten notes from lecture.
Waiting room video: interview with Karen Uhlenbeck.

We recalled that a function f:D--->C consists of a domain D, a codomain D and a rule for assigning precisely one element of the codomain to each element of the domain. When the domain and codomain are not explicitly specified then we just take D to be the largest subset of the reals for which the "function rule" makes sense, and we just take C to be the set of all real numbers. We recalled that functions can be represented by graphs and we studied some examples. During the examples we met concepts such as "horizonal asymptote", "vertical asymptote", "x-intercept", "y-intercept", "range of a function". For more on the basics of functions read Stewart, Section 1.1.
Lecture 11
Video of the lecture.
Handwritten notes from the lecture.
Waiting room interview: interview with Roger Penrose.

Introduced the concept of a limit of a function f(x) as x tends to some number c. Gave some examples too. For more details on the notion of a limit read Section 2.2 in Stewart. For more details on the formal definition of a limit presented at the end of the lecture, read Section 2.4 in Stewart.
Lecture 12
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: interview with Jim Simons.

Used Weierstrass's definition of a limit to prove that the limit, as x---> 1, of (x6-1)/(x-1) is equal to 6.

Then stated a proposition listing some properties of limits. I didn't bother deriving/proving this proposition from Weierstrass's definition. Instead of giving a proof, I illustrated how to apply the proposition to establish/prove certain limits.

For more details on the formal definition of a limit, read Section 2.4 in Stewart.
Lecture 13
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: Lecture by Cédric Villani

Introduced and illustrated the Sandwich Lemma. In particular, I showed that
limx->0Sin(x)/x = 1 .
Lecture 14
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: Numberphile on the Poincaré Conjecture.

Introduced the notion of a continuous function. The final example of a continuous function was f(x) = 1/x . For more details on the notion of continuity see Section 2.5 in Stewart.
Lecture 15
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: Numberphile on a flaw in the Enigma machine.

Stated the Intermediate Value Theorem. Then used it to approximate the solutions to some polynomial equations. Also used it to `prove' that on any great circle on the Earth there exists a pair of opposite points with equal atmospheric pressure.

For more details on the notion of continuity, and on the Intermediate Value Theorem, see Section 2.5 in Stewart.
Lecture 16
Video of lecture.
Handwritten notes from the lecture.
Waiting room video:Marcus du Sautoy on Gödel's incompleteness of arithmetic

Introduced left-hand and right-hand limits. Stated a proposition which relates left-hand limits and right-hand limits to the usual notion of a limit at a point x=a. Gave some examples to illustrate these notions/relationships.

Gave some examples of "limits at infinity" and related them to asymptotes. Then introduced the most important definition of this semester.

Finished with a limit involving trigonometric functions.

See Section 2.2 of Stewart for more on left-hand and right-hand limits. See Section 2.6 of Stewart for more on limits at infinity and asymptotes. See Section 3.3 of Stewart for more on limits of trigonometric functions.
Lecture 17
Video of the lecture.
Handwritten notes of the lecture.
Waiting room video: video about William Rowan Hamilton.

Introduced the notion of a matrix and the operations of addition, subtraction and multiplication.

For more background on matrix addition look at the wikipedia page here.

For more background on matrix multiplication look at the wikipedia page here.
Lecture 18
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: intervew with Peter Sarnak.

Introduced the concepts of: scalar multiplication for matrices, an nxn identity matrix, the inverse of an nxn matrix.

Derived a formula for the inverse of a 2x2 matrix.

Introduced the matrix affine cryptosystem.
Lecture 19
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: lecture about Emmy Noether.

Deciphered a ciphertext from KARLA, produced from an affine matrix cryptosystem.
Lecture 20
Video of the lecture.
Hndwritten notes of the lecture.
Waiting room video: commentary on G.H. Hardy's A Mathematician's Apology

Introduced the concept of a linear transformation of the plane. Showed that reflection in a line through the origin is a linear transormation.

For more background on linear transformations take a look at the Open Corseware notes from MIT here.
Lecture 21
Video of the lecture.
Handwritten notes of the lecture.
Waiting room video: Numberphile on knot theory.

Explained why every linear transformation of the plane can be represented by a 2x2 matrix.

Stated that: (i) reflections in a line through the origin of the plane are linear transformation; (ii) rotations about the origin of the plane are linear transformations; (iii) the composite of any two linear transformations of the plane is linear.

Stated a theorem which asserts that composition of transformations corresponds to multiplication of matrices. Matrix multiplication has been invented just so that this theorem is true.

Ended the lecture showing how rotation of the plane about the origin can be represented as matrix multiplication, and hence is linear.
Lecture 22
Video of the lecture.
Handwritten notes of the lecture.
Waiting room video: Numberphile on Fermat's Last Theorem.

Then recalled how to multiply two matrices, and recalled what is meant by the inverse of a matrix.

Explained how the inverse of an n×n matrix can be useful for solving a system of n linear equations (=no powers or products of unknown variable) in n unknowns.

Ended with an explanation of how to use row operations to find the inverse of an n×n matrix.
Lecture 23
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: Numberphile on the Riemann hypothesis.

Began with a review of row operations and the method for inverting a matrix. Then explained why the Gauss-Jordan method for finding the inverse of a matrix works.

Gave an example to illustrate that row operations can be used to solve systems of linear equations arising from "real life" problems.

For more background on systems of linear equations take a look at the wikipedia page here.
Lecture 24
Video of the lecture.
Handwritten notes of the lecture.
Waiting room video:lecture on Sofie Germain and Karl Friedrich Gauss.

Explained gaussian elimination as an algorithm for finding a solution to a system of n linear equations in n unknowns.

Returned to calculus and gave the definition of the derivative of a function.
Lecture 25
Video of the lecture.
Handwritten notes of the lecture.
Waiting room video: Ashkay Venkatesh talking about primes and knots.

Began with an explanation of why the derivative f'(x) of a function f(x) represents the slope to the tangent of the curve y=f(x) at the point x.

Then explained the rules for differentiating: (1) a sum of functions, (ii) a scalar product of a function, (iii) a product of two functions, (iv) a quotient of functions, (v) a composite of functions (the Chain Rule).
Lecture 26
Video of lecture.
Handwritten notes from the lecture.

Used the Chain Rule to solve two problems, one aboutthe rate at which an aeroplane flies away from a beacon, the second about the rate at which a lighthouse beams travels along the shore.
Lecture 27
Video of the lecture.
Handwritten notes of the lecture.

Discussed max/min applications and solved one max/min problem about laying a cable from a lighthous to a point on the shore.

See Section 4.7 in Stewart for further examples of max/min problems.
Lecture 28
Video of the lecture.
Handwritten notes from the lecture

Explained the details of the two 60-minute end of semester quizzes at 10am on Wednesday 16 December and Thursday 17 December. The quizzes will be at 10am on Thursday 3 December and 10am on Thursday 17 December. Then answered Q1 and Q2(b) from last years en of semester test.
Lecture 29
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: How do brains count?

Used the derivative and the second derivative to help sketch the curve of a function. Talked about a curve being
"concave up" -- on the intervals where the acceleration is positive.
"concave down" -- on the intervals where the acceleration is negative
and having
"points of inflection" -- a point where concavity changes
"critical points" -- points where the derivative is zero or not defined.

See Section 4.3 in Stewart for more information on "concave up" and "concave down" functions.
Lecture 30
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: the fundamental theorem of algebra.

Summarized terminology on: continuity, differentiability, critical points, points of inflection. Worked through an example on determining conditions for a piecewise defined function to be differentiable. See Section 2.8 in Stewart for more on the fact that differentiability implies continuity.
Lecture 31
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: Numberphile on Euclid's big problem.

Explained the Mean Value Theorem (and the special case known as Rolle's Theorem). Gave an application to establishing that a certain cubic polynomial has exactly one root.

Finished off by recalling the intuitive understanding of logarithms and exponents that is taught as part of the Leaving Certificate syllabus. This level of understanding will suffice for your science practical labs. But there are issues, such as what do we mean by raising a number to an irrational power, that I'll address in subsequent calculus lectures.

CORRECTION: It is in fact known that (√2)√2 is irrational. This result was proved by Rodion Kuzmin in 1930. See this Wikipedia page for more details.
Lecture 32
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: Numberphile on the Mandelbrot set.

Explained that the first class quiz will be at 10am on Thursday 03 December, and that it will be an online Okuson quiz covering the material normally cover by Q1, Q2, Q4 on previous end of semester tests/exams. The quiz will open at 10am; it is a 50 minute quiz; you will be asked to upload 10 answers. If you can answer Q1, Q2, Q4 on previous years' exam papers then you will fly the quiz.

Then finished off answering Q1, Q2, Q3(a) and Q4 from last year's end of semester test.
Lecture 33
Video of the lecture.
Handwritten notes from the lecture.
Waiting room video: the power of mathematical visualization.

Introduced the determinant of a 2×2 matrix. Showed that the determinant of a matrix is equal to +/- the area of the parallelogram formed from its two columns. Also showed that det(AB)=det(A)det(B).
Lecture 34

Lecture 35

Lecture 36

Lecture 37

Lecture 38

Lecture 39

Lecture 40

Lecture 41

Lecture 42

Lecture 43

Lecture 44

Lecture 45

Lecture 46

Lecture 47

Lecture 48

Supplementary material

CLICKER OPINION POLLING may be used in some lectures.

ONLINE CALCULATOR FOR CLOCK ARITHMETIC

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

WHAT IS MATHEMATICS?

I'm not too sure of the answer. But whatever it is it is possibly something a bit larger than 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. You could also take a look at the lovely little book A Mathematician's Apology by G.H. Hardy which is available online here.

WHAT ARE THE EMPLOYMENT PROSPECTS FOR A MATHS GRADUATE?

Have a look at this link to answer this question.

PRACTICE HOMEWORK PROBLEMS

A few students have asked for more problems, similar to the homework problems, to practice on. I'll place some here after each homework has closed:

ADVANCED READING SUGGESTIONS

If you are finding the pace of lectures too slow and want to browse some advanced textbooks that cover advanced topics related to material in the lectures then you might take a look at

STUDENT FEEDBACK

I'll place student feedback here.

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