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This enrichment mathematics workshop is aimed at students studying, or about to study mathematics, physics or philosophy at A-level or International Baccalaureate Diploma and other pre-university courses.

Common to all E.T.A. Mathematics Enrichment Workshops is the objective to expose curious students (aspiring to study mathematics, science, computing or engineering based disciplines at university), to a richer mathematical experience than that found within the walls of the generic A-level, IB Diploma,French bac, Abitur classroom.

In particular, THE EIGHT BENEFITS from attending one of our unique Mathematics Enrichment Workshops are:

1) Mathematics Olympiads

2) University Mathematics Degree application interviews

3) STEP (Sixth Term Examination Paper) Mathematics applications to the University of Cambridge and the University of Warwick and the MAT (Mathematics Admissions Test) for the University of Oxford.

4) IB Diploma Math SL/HL Explorations and Mathematics Extended Essays.

5) Mathematics and Further Mathematics A-level

6) IB Diploma Mathematics HL

7) Cambridge Pre-U Mathematics and Pre-U Further Mathematics

8) Cambridge International Mathematics AS-Level and Mathematics and Further Mathematics A-level.

This workshop uses a variety of intuitive visual proofs, interspersed with a tantalising sprinkling of mathematical analysis and technology to examine the beautiful and contradictory world of “the infinite” from a variety of perspectives, and highlights key developments in the history of mathematics centred on the concept of “infinity”. In so doing, this workshop exposes the student to the branches of analysis, number theory, set theory and transfinite arithmetic.

As part of the prerequisites for the workshop, registered students will be given access to the A Hitchhiker's Guide To Infinity: Preliminary Unit, with instructions to complete a set selection of exercises from this Unit before the commencement of the workshop.

This workshop is broken down into the following sections:

MOTIVATION: Paradoxes of Infinity

- Achilles and the Tortoise paradox
- Bolzano's paradoxes of infinite series

SECTION 1: Infinite Classes

SECTION 2: Hierarchies of Infinity

SECTION 3: Hilbert's Grand Hotel paradox

CONCLUSION: Continuum Hypothesis

This workshop begins with a seduction of mind-bending curiosities and paradoxes involving the mathematical notion of “infinity”. Specifically, Zeno's paradox of Achilles and the Tortoise is used as a motivation to introduce an analytical treatment of oscillating, divergent and convergent infinite series. This analytically focused workshop then proceeds to utilizes subtle mathematical tools to closely examine the nature of the different infinities of the natural numbers, integers, rational numbers and the real numbers systems. These different hierarchies of infinity are then further investigated by using such concepts as cardinality, countable v's uncountable, the number continuum and the bounding of irrational numbers by rational numbers, and transfinite arithmetic. Set Theory methods are then used to consider Hilbert's thought experiment of allocating an infinite number of guests into a hotel with an infinite number of rooms. The workshop culminates with Cantors concept of 1-1 correspondence of sets with proper subsets of themselves. The workshop concludes by examining the implications of the Continuum Hypothesis, and an update on its status!

This online workshop is delivered through an **interactive whiteboard**. This is a very powerful platform with the following features: file sharing, session recording, breakout rooms, polling, screen sharing, pan-and-zoom,limitless pages, maths tools and symbols editor, live chat, etc. In advance of the workshop, students will get access to a **User Guide** on how to use the main functions of the learning whiteboard.

Shortly before, during, and for one month after the workshop, you will have access to your own, secure **portfolio-database dashboard** record with a link to the course delivery board. There is no software to install. Students and parents can:

- submit tasks and download tutor set tasks
- access the dashboard from within the Europe Tutorial Academy website
- access tutor feedback and recommendations
- keep track of pre- and post-workshop notices

This enrichment workshop would normally be delivered in-person, but during Corona restrictions we are offering this workshop online.

Ian Tame

Senior Tutor