## MATH1055 Numbers and Vectors

### 10 creditsClass Size: 220

Module manager: Dr Oleg Chalykh
Email: o.chalykh@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2021/22

### Pre-requisite qualifications

Grade B in A-level Mathematics or equivalent.

### This module is mutually exclusive with

 MATH1005 Core Mathematics MATH1010 Mathematics 1 MATH1012 Mathematics 2 MATH1026 Sets, Sequences and Series

This module is approved as a discovery module

### Module summary

This module introduces students to three outstandingly influential developments from 19th century mathematics: - complex numbers- vectors- and the rigorous notion of limit. Complex numbers are the natural setting for much pure and applied mathematics, and vectors provide the natural language to describe mechanics, gravitation and electromagnetism, while the rigorous notion of limit is fundamental to calculus. Along the way, students will go beyond the straightforward calculation and problem solving skills emphasized in A-level Mathematics, and learn to formulate rigorous mathematical proofs.

### Objectives

On completion of this module, students should be able to:

a) perform algebraic calculations with complex numbers and solve simple equations for a complex variable;
b) determine whether simple sequences and series converge;
c) perform calculations with vectors, write down the equations of lines, planes and spheres in vector language, and, conversely, describe the geometry of the solution sets of simple vector equations;
d) construct rigorous mathematical proofs of simple propositions, including proofs by mathematical induction.

### Syllabus

1. Proof by induction.
2. Complex numbers: modulus, argument; de Moivre's Theorem; geometry of the complex plane; complex roots.
3. Sequences: definition of convergence; algebra of limits; squeeze rule; monotone convergence theorem (statement only).
4. Series: definition of convergence; divergence test, comparison tests, ratio test.
5. Vector geometry: parallelogram law; scalar product, norm; vector product.
triple product; equations of lines, planes and spheres.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 11 1.00 11.00 Tutorial 5 1.00 5.00 Private study hours 84.00 Total Contact hours 16.00 Total hours (100hr per 10 credits) 100.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular example sheets.

!!! In order to pass the module, students must pass the examination. !!!

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated