# MATH140 Fundamentals of Modern Mathematics 2

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Continuation from MATH 130 with an emphasis on mathematical thinking and formalisation and their importance for applications of mathematics.

The techniques covered in this paper form the basic tools used to produce mathematical frameworks for modelling quantifiable phenomena. For example, to model the movement of an object through space, we begin with an algebraic structure in which to specify where our object is, and then study how that position changes with time using methods developed in calculus. Many other problems arising in areas such as Economics or Chemistry can be examined mathematically using the same basic principles. For example, we may need to minimise a manufacturing cost, or the time for a chemical reaction to take place, or the effects of river pollution; in each case the techniques used for the minimisation are based on a mixture of tools relying on both algebra and calculus.
This paper aims to develop proficiency with algebra and calculus, both for use in other subjects and in preparation for further study of Mathematics. MATH 140 is the natural continuation of MATH 130, and provides a strong mathematical background to support other subjects as well as forming a necessary prerequisite for progression to 200-level Mathematics.

Paper title Fundamentals of Modern Mathematics 2 MATH140 Mathematics 0.1500 18 points Semester 2 (On campus) \$929.55 Tuition Fees for international students are elsewhere on this website.
Prerequisite
MATH 130 or MATH 160
Restriction
MATH 170
Schedule C
Arts and Music, Science
Eligibility

This paper should appeal to a wide variety of students, including Mathematics and Statistics majors or those studying Computer Science, Physics, Chemistry, Surveying, Biological Sciences, Genetics or other disciplines with a quantitative component requiring competent manipulation of mathematical formulae and interpretation of mathematical representations of systems.

Contact

math140@maths.otago.ac.nz

Teaching staff

Paper Structure
1. Truth and falsehood
We cover key ideas and skills relating to logic and mathematical thinking, proofs, formal arguments and fallacies. Concepts are developed using examples from number theory, cryptography, and propositional logic.
2. Complex numbers
The emphasis will be on understanding the connection between Cartesian and polar representations of complex numbers, the geometric viewpoint of complex multiplication and their central importance to dynamic and physical systems.
3. Matrices and subspaces
We explore and extend the theory and geometry of matrices and linear algebra started in MATH 130. We show how matrices are used to understand systems of equations and subspaces, introducing rank, dimensions and bases. Eigenvalues, eigenvectors and determinants are introduced and linked with concrete applications.
4. Calculus
We extend ideas from calculus introduced in MATH 130. The toolkit is expanded significantly. Important special functions are discussed in context. We explore the Taylor approximation of a function, with key examples and applications. We reintroduce differential equations and link them with ideas from integration. We examine ways these ideas generalise to higher dimensions, revisiting partial derivatives, the gradient, and encountering some of the challenges of high dimensional integrals.
Teaching Arrangements

Four lectures per week.
Weekly tutorials, one hour in length.
Assessment is 55% internal, 45% final exam.

Textbooks

Scholarship, Lifelong Learning, Information Literacy, Critical Thinking, Communication

Learning Outcomes

Students who successfully complete the paper will:

• Appreciate mathematics as a modern discipline and learn what mathematicians and mathematical scientists do
• Gain increasing fluency with the processes of abstraction and generalization in the mathematical sciences
• Begin to recognize and develop mathematical arguments and apply logic and mathematical rigor
• Manipulate mathematical expressions, derive new expressions from others and develop skills for explaining and communicating mathematical arguments

## Timetable

### Semester 2

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
Blackboard

#### Lecture

Stream Days Times Weeks
Attend
A1 Monday 12:00-12:50 28-34, 36-41
Tuesday 12:00-12:50 28-34, 36-41
Wednesday 12:00-12:50 28-34, 36-41
Thursday 12:00-12:50 28-34, 36-41

#### Tutorial

Stream Days Times Weeks
Attend one stream from
A1 Thursday 13:00-13:50 29-34, 36-41
A2 Thursday 15:00-15:50 29-34, 36-41
A3 Friday 10:00-10:50 29-34, 36-41
A4 Friday 12:00-12:50 29-34, 36-41
A5 Friday 15:00-15:50 29-34, 36-41