## MATH101 Supplementary Algebra 1

First Semester | Also available: Second Semester Summer School |

This 9-point half-paper covers methods and applications of algebra. It consists of the algebra component of MATH 160. Note that to go on to MATH 170, you will need both MATH 101 and MATH 102 - or their equivalent, MATH 160.

### ***** Not available in 2019! *****

### Paper details

This algebra paper is the natural continuation of Year 13 Mathematics.

You will learn about three-dimensional vectors and their many uses (such as in geometry, computer graphics, surveying and even calculus). The vector representation of lines, planes and projections leads naturally to the discussion of linear systems of equations. The basic properties of matrices are studied together with some applications. Complex numbers and polynomials complete this section of the course.

### Introduction

Algebra and calculus form the basic tools used to produce most mathematical frameworks for modelling quantifiable phenomena. For example, to model the movement of an object through space we need first to create an algebraic structure in which to specify where our object is, and then we can study how that position changes with time (i.e. its movement) using calculus.

Many other problems arising in areas such as Economics or Chemistry, can be examined in a mathematical way using the same basic ideas. For example, we may need to minimize 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 minimization are based on a mixture of algebra and calculus theories.

This course aims to develop these tools and techniques both for use in other subjects and in preparation for further study of Mathematics.

### Paper details

This paper is the natural continuation of Year 13 Mathematics.

In the algebra half, you will learn about three-dimensional vectors and their many uses (such as in geometry, computer graphics, surveying and even calculus). The vector representation of lines, planes and projections leads naturally to the discussion of linear systems of equations and matrices. The basic properties of matrices are studied together with some applications. Complex numbers and polynomials complete this section of the course.

### Potential students

MATH 160 is intended both for those with a main interest in studying Mathematics and/or Statistics, and those whose interest in Mathematics is mainly to support other areas of study. These areas might include the physical, health and biological sciences, computer and information science, engineering, surveying, architecture, economics and finance, and philosophy of science. An understanding of basic algebraic and differential and integral techniques is of benefit to all students exposed to the analysis of processes, whether involving one or several variables.

### Prerequisites

Formally, MATH 160 has no prerequisites other than “sufficient achievement in NCEA Level 3 Calculus”. However, we strongly suggest that if you have not passed the externally assessed NCEA Calculus papers “Apply differentiation methods in solving problems” (AS91578) and “Apply integration methods in solving problems” (AS91579), then you should consider taking MATH 151 before attempting MATH 160. If you have high achievement (mostly Excellences and Merits) in NCEA Level 3 Calculus, please see a maths advisor for consideration of direct entry to MATH 170. The placement tool can help you decide which paper is appropriate for you.

### Main topics

Algebra:

- Vectors; linear and planar geometry and applications
- Solving linear systems
- Matrices and applications
- Complex numbers
- Polynomials and their roots.

Algebra and Calculus notes will be made available online during the semester.

**Recommended text for Calculus**: *Calculus* by James Stewart (Truncated edition) or the full *Calculus* (metric edition 8) by James Stewart are available from the University Book Shop; if you are planning on taking MATH 170, you should consider getting the full version.

### Useful references

Several standard texts are suitable for reference. For example:

*Elementary Vector Algebra*by A.M. MacBeath*Algebra, Geometry and Trigonometry*by M.V. Sweet

### Lecturer (Semester 1)

Professor David Bryant, room 232A

### Lectures (Semester 1)

Algebra: Mon, Wed and alternate Fridays, 10 am

### Office Hours

These are hours where you can drop in to see us in our offices without making an appointment first.

David: TBA

At other times, please make an appointment by talking to us in person, or by e-mail or phone.

### Tutorials

You are required to attend one tutorial per week, and they contribute to your final grade. You will be assigned a tutorial time before the beginning of the semester, and changes to these times can be arranged with your tutor. Note that you will only have to complete the Algebra questions in each tutorial.

### Internal Assessment

There are five marked assignments which make up your assignment mark A.

After most lectures there will be a online quiz which must be completed before the beginning of the next lecture. The top 80% of your quiz grades will be used to determine your quiz mark (Q).

You can check your marks by clicking on the Resources link at the top of this page.

### Terms Requirement

**You have to fulfill the terms requirement in order to be allowed to sit the final exam.**

In this paper, to pass “terms” you need to:

- Attend at least 5 out of the 10 tutorials
- Complete at least 10 out of the 20 quizzes.

### Exam format

The 3-hour final exam consists of multiple-choice and long-answer questions.

### Calculators

In the exam, you may use any calculator from List A (Scientific Calculators) of the University of Otago's approved calculators; these are Casio FX82, Casio FX100, Sharp EL531, Casio FX570 and Casio FX95.

### Final mark

### Sample problem

The aircraft’s flightpath goes through coordinates (1,2,0) and (23,-19,3). The top of the hill is at (18,-13,2).

How close does the aircraft get to the top of the hill? Vectors make this an easy calculation.

### J Willard Gibbs...

..., 1839-1903, was a pioneer in vector analysis. His family lived in Connecticut and Gibbs became Professor of Mathematical Physics at Yale in 1871 — rather surprisingly before he had published any work! He made major contributions to thermodynamics, the electromagnetic theory of light and statistical mechanics.

### Sample problem

In a certain city, commuters go to work by car or bus. A study shows that from each year to the next year 20% of car users change to travelling by bus, while 15% of bus users change to travelling by car. What percentage of commuters travel by car, once things have settled down?