## MA180-I & MA185 & MA190-I ALGEBRA and CALCULUS (SEMESTER I) LECTURER: GRAHAM ELLIS

### SYLLABUS & LEARNING OUTCOMES

For the course syllabus, learning outcomes and assessment details see the course web page and the continuous assessment web page. At the end of this module you'll be able to tackle questions such as those on the Semester I 2013/14 exam paper. (At the end of the corresponding Semester II module you'll be able to takle questions such as those on the Semester II 2013/14 exam paper.) All past exam papers are available from the library website.

### FORTNIGHTLY HOMEWORKS

The continuous assessment in Semester I consists of six fortnightly algebra/calculus homeworks of equal weight.

The continuous assessment counts 40% towards the module assessment for MA180-I and MA190-I. To pass MA180 or MA190 students must pass the year's continuous assessment consisting of 12 algebra/calculus homeworks.

For MA185 students the Semester I homeworks count 50% towards module MA187. The remaining 50% of module MA187 is based on six homeworks in Semester II.

Please click here to access the MA180, MA185 and MA190 homework sheets. (Due to a technical computer problem caused by a lightening strike this link will not work before Wednesday 6 September.) The first homework will be due on 29th September. Late submissions will not be graded.

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 password because I am unable to reset it for you!

### WEEKLY WORKSHOPS

Workshops begin on Monday 18th September. Details can be found here.

### 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.

### STUDENT FEEDBACK

I'll place student feedback here.

### ALGEBRAMATERIAL

Algebra text:
Algebra & Geometry: An introduction to University Mathematics by Mark V.Lawson.
A pre-publication pdf version of this text is available on blackboard. This version is for private use only and the pdf version must not be made available on the internet.

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.

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

Algebra lectures 2017-18:

The algebra lecture slides will be uploaded to the web after each lecture and links to the slides will be given below. A brief outline of each lecture will be added/modified below shortly after each lecture.

### 1

Lecture 1:
Introduction to modular arithmetic. An application to the ISBN book number was explained.

For another introduction to modular arithmetic take a look at this Youtube clip. Then take a look at this clip, this clip and this clip

### 2

Lecture 2:
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.

### 3

Lecture 3:
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.

### 4

Lecture 4:
Deciphered an enciphered message sent from Agent 007.

### 5

Lecture 5:
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,

### 6

Lecture 6:
Introduced Euler's phi (or totient) function.

For more background on Euler's phi function take a look at the wikipedia page here.

### 7

Lecture 7:
Began with the quote

"both Gauss and less 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.

Then explained the RSA public key cryptosystem.

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

### 8

Lecture 8:
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.

Took attendance at todays lectures. Here are the photos: photo1, photo2, photo3.

### 9

Lecture 9:
Introduced the notion of a matrix and the operations of addition, subtraction and multiplication.

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

Take a look at this clip for examples of matrix multiplication.

### 10

Lecture 10 :
Explained the notion of an affine matrix cryptosystem.

### 11

Lecture 11:
Deciphered a ciphertext obtained from an affine matrix cryptosystem. In the process I got lots of practice of matrix multiplication.

### 12

Lecture 12:
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.

### 13

Lecture 13:
Explained why every linear transformation of the plane can be represented by a 2x2 matrix. Stated a theorem that asserts that composition of transformations corresponds to multiplication of matrices. Matrix multiplication has been invented just so that this theorem is true.

I didn't get around to deriving the matrix representing rotation through an angle theta about the origin. See the slides of a previous year's lecture for this important derivation.

Took attendance at todays lectures. Here are the photos: photo1, photo2.

### 14

Lecture 14:
Illustrated the Gauss-Jordan method for inverting a matrix. The method uses a sequence of row operations.

### 15

Lecture 15:
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.

### 16

Lecture 16:
Defined the determinant and adjoint of a 2x2 matrix. Gave a formula for the inverse of a 2x2 matrix in terms of its determinant and adjoint. Explained that the determinant of a 2x2 matrix is equal to the area of a certain parallelogram up to sign.

### 17

Lecture 17:
Proved that the determinant of a 2x2 matrix is equal to the area of a certain parallelogram up to sign. Then introduced and illustrated the notions of eigenvector and eigenvalue of a matrix.

### 18

Lecture 18:
Explained how eigenvectors are involved in Google's page rank algorithm. (I intensionally over simplified the explanation. In particular, the importance In of a page is determined from the full network of pages on the internet and not just those [8 in my explanation] containg the given searched words.)

More details on the page rank algorithm can be found here.

Also stated and illustrated the important Hamilton-Cayley Theorem.

### 19

Lecture 19:
Explained how to find eigenvalues of a 2x2 matrix using the characteristic equation. Explained how to find eigenvectors for the given eigenvalues.

Derived the recurrence relation Fn = Fn-1 + Fn-2 for the number of rabbits in a field after n months, based on some assumptions about rabbit breeding.

### 20

Lecture 20:
Talked about various occurences of the Golden Ratio.

### 21

Lecture 21:
Explained how to express a suitable 2x2 matrix A in the form A=T-1 D T where D is diagonal. Here "suitable" means that A must have two eigenvectors such that the matrix T containing the two eigenvectors as columns is invertible.
Used the above expression to find a formula for the terms Fn in the Fibonacci sequence.

### 22

Lecture 22:
Used eigenvalues and eigenvectors to study a diseased population of frogs.

### 23

Lecture 23:
Did some revision.

### 24

Lecture 24:
Did a bit more revision.

### CALCULUS MATERIAL

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 2017-18:

The calculus lecture slides will be uploaded to the web after each lecture and links to the slides will be given below. A brief outline of each lecture will be added/modified below shortly after each lecture.

### 1

Lecture 1:
We considered a stone being dropped from the top of the Eiffel Tower. 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. A function assigns one output to each input. We then asked the question: what is the speed of the stone at time t=2 seconds? To answer this we used the notion a limit.

### 2

Lecture 2:
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".

### 3

Lecture 3:
Introduced the concept of a limit of a function f(x) as x tends to some number c. Gave some examples too.

### 4

Lecture 4:
This lecture was a hotchpotch of basic material: x-intercepts and y-intercepts; definition of a radian; definition of cos(x), sin(x) and tan(x); examples of functions defined piecewise.

### 5

Lecture 5:
Introduced notation for composite functions. Described what it means for a function to be even or odd. Described the absolute value function and noted that it is even. Gave a proposition about the limit of a sum of functions, the limit of a scalar multiple of a function, the limit of a product of functions, and the limit of a quotient of functions. Ended with details on the "Sandwich Lemma" and illustrated how it could be used to determine a limit.

### 6

Lecture 6:
Introduced left-henad limits and right-hand limits. Explained that the limit of f(x) exists at a if and only if the left-hand and right-hand limits exist at x and are equal. Gave two informal and one formal definition of what it means for a function f(x) to be continuous at a point x=a.

### 7

Lecture 7:
Stated the Intermediate Value Theorem. 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.

### 8

Lecture 8:
Gave some examples of "limits at infinity". Then introduced the most important definition of this semester: the definition of the derivative of a function.

### 9

Lecture 9:
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).

### 10

Lecture 10 :
Discusssed "rates of change applications" and solved two problems.

### 11

Lecture 11:
Discusssed "max/min applications" and solved one problem.

### 12

Lecture 12:
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.

### 13

Lecture 13:
Critical points: maxima and minima. Points of inflection. More on curve sketching. (This lecture was given by John Burns. The attached slides were the basis of the lecture and not the actual slides produced during the lecture.)

### 14

Lecture 14:
Summary of terminology: continuity, differentiability, critical point, concavity etc. Theorem: differentiability implies continuity. Rolle's Theorem. Applications. (This lecture was given by Goetz Pfeiffer. The attached slides were the basis of the lecture and not the actual slides produced during the lecture.)

### 15

Lecture 15:
Stated the Mean Value Theorem. Then recalled the idea of a logarithm as the "inverse" to taking exponents. We would like to think of loga y = x as meaning y = ax. A serious difficulty with taking this as the definition of a logrithm is that we are not too sure (yet in the module) what we mean by ay when y is irrational. Nevertheless, this not so solid definition does suggests that loga(x) should be a real valued function, with domain the positive real numbers, satisfying the following two basic properties:

(i) loga(uv) = loga u + log_a v

(ii) logaun = nlogau .

In the next lecture we'll give a better definition of the logarithm function.

### 16

Lecture 16:
Defined the natural logarithm ln(x) (often written as loge(x) ) as the area under the curve y=1/t from t=1 to t=x for x≥1. For 0<x<1 defined ln(x) to be the negative of the area under the curve y=1/t from t=1 to t=x. Then showed that ln(x) has the properties required of a logarithm.

### 17

Lecture 17:
Explained that an injective function f:D-->R has an associated inverse function f,sup>-1:C-->D where C=f(D) is the range of f. Then gave a formula for the derivative of the inverse function. The function exp(x) or ex was introduced as the inverse of the natural logarithm function. The function f(x)=3x can be rigorously defined as the inverse to the function log3(x) = ln(x)/ln(3) . Ended the lecture by talking briefly about differential equations and their solutions.

### 18

Lecture 18:
Spent the whole lecture discussing my cooling cup of coffee. Mentioned Newton's Law of Cooling and expressed it as a differential equation. Solved this differential equation to describe the changing temperature of my coffee.

### 19

Lecture 19:
Introduced anti-derivatives and discussed the Malthusian Law as a model of world population growth.

### 20

Lecture 20:
Discussed the Logistic Equation as a model for world population growth.

Lecture 21:

Lecture 22:

Lecture 23:

Lecture 24: