# REAL ANALYSIS 1 - 2024/5

Module code: MAT1032

## Module Overview

This module is an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module leads, among other things, to a deeper understanding of what it means for a sequence or series to converge. Tools such as convergence tests are presented and their validity proved, and the rigorous use of definitions and logic play a central role. This course lays the foundations for the Level 5 module in Real Analysis 2 (MAT2004) in particular, and, more generally, underpins other modules where a culture of rigorous proof exists.

### Module provider

Mathematics & Physics

### Module Leader

BEVAN Jonathan (Maths & Phys)

## Overall student workload

Independent Learning Hours: 42

Lecture Hours: 44

Seminar Hours: 5

Guided Learning: 15

Captured Content: 44

Semester 1

None.

## Module content

Indicative content includes:

• Irrational, algebraic and transcendental numbers.

• The axioms of real numbers. Denseness of rational and irrational numbers. Maximum, minimum, supremum and infimum of sets, sequences and functions. The triangle inequality.  Simple estimates.

• Natural induction, set notation, cardinalities of sets (in particular the rationals and reals)

• Axiom of Completeness, and its consequences for the existence of limits. Role of quantifiers in stating and verifying mathematical definitions.

• Sequences: convergence, and other properties. Boundedness, Cauchy sequences, subsequences and the Theorem of Bolzano-Weierstrass.

• Infinite series, convergence and absolute convergence. Convergence tests, including proofs.  Power series, radius and region of convergence.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 25
Examination Exam (2 hours) 75

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding, interpretation and manipulation of mathematical statements.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through logical proofs or counterexamples to unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One in-semester test taken during the semester, worth 25% of the module mark, corresponds to Learning Outcomes 1, 2, 3.

• A synoptic examination (2 hours), worth 75% of the module mark, corresponds to Learning Outcomes 1, 2, 3.

Formative assessment

There are three formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback

Individual written feedback is provided to students for formative unassessed courseworks. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from all three courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback during lectures and seminars.

## Module aims

• Introduce students to quantifiers, logical statements, countability, suprema, maps, sequences and series.
• Enable students to determine limits of sequences, to determine countability, and to test and prove the convergence of series and sequences.
• Facilitate students' understanding of analysis through applications of various techniques in frequently encountered problems.

## Learning outcomes

 Attributes Developed 001 Students will understand the real numbers, their axioms and the role of completeness in the existence of limits and solutions to equations. KC 002 Students will be able to interpret and apply quantifiers in mathematical statements, and quote and apply basic theorems in analysis. KCT 003 Students will be able to calculate limits of sequences and (power) series, and prove/disprove convergence using the definitions. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Give a detailed introduction to real numbers, sequences, series and convergence and ensure experience in the methods used to interpret, understand and solve problems in analysis

The learning and teaching methods include:

• Four one-hour lectures per week for eleven weeks, with typeset notes to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.

• Five seminars for guided discussion of solutions to problem sheets (provided to students in advance for completion to reinforce their understanding and guide their learning).

• Three unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students the opportunity to review parts of lectures that they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MAT1032

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience, and Sustainability. This module is designed to allow students to develop  knowledge, skills, and capabilities in the following areas:

• Digital Capabilities: The SurreyLearn page for MAT1032 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies and facilitates collaborative learning and information sharing.

• Employability: The module MAT1032 equips students with skills which significantly enhance their employability. Students gain mathematical proficiency, which hones critical thinking and problem-solving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

• Global and Cultural Capabilities: Students enrolled in MAT1032 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

• Resourcefulness and Resilience: MAT1032 is a module which demands a rigorous approach to Real Analysis, to which students will learn to adapt. They will gain skills in analysing problems and lateral thinking. Students will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.