What is analysis?

At its heart, mathematical analysis is the study of the continuum: the idea of a continuous, unbroken number line and the consequences of this idea. In this course, we shall introduce ideas from analysis which should allow you to convincingly answer the above questions (and much, much more!). In some cases, we shall provide explicitly answers, and in other cases we shall leave the questions open for you to think about on your own and to discuss with other students.¹¹ 1 We do not claim to completely answer all these questions: one of the joys of mathematics is that there is always scope to delve deeper, and it is possible that you shall return to and revise your understanding of some of these questions in later courses.

To understand simpler number systems such as $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ we only require algebra. These number systems arise simply by considering what happens when we add, subtract, multiply and divide. Real numbers $\mathbb{R}$ are different in this regard, and much more complicated. To understand $\mathbb{R}$ we need not only ideas from algebra, but also new ideas from analysis. The most fundamental new idea is called the completeness axiom, and is discussed in Chapter 1. This is the fundamental property which distinguishes the real numbers from the rational numbers. It makes precise the idea that the real numbers form a continuum. As such, all the major results in the course stem from the completeness axiom.

In Chapters 2 and 3 we shall study sequences, which are just ordered lists of real numbers such as $(1/n)_{n\in\mathbb{N}}=(1,1/2,1/3,\dots)$. This quickly leads to limits, another fundamental concept in analysis. You briefly encountered sequences and limits in IMU (for instance, in the context of geometric series), without going into the details of how they are rigorously defined. You may also be aware of identities such as

This expresses the idea that, for large $n$, the terms of the sequence $1/n$ get closer and closer to $0$. If you haven’t seen this idea before, then that’s perfectly fine: we shall introduce these concepts from the ground up. Using limits, we can make start to make sense of decimal expansions: taking the sequence $(3,3.1,3.14,3.141,\dots)$ of truncated decimal approximations to $\pi$, the idea is that the terms of this sequence get closer and closer to $\pi$ and so the ‘limit’ of this sequence equals $\pi$.

Limits are the bread-and-butter of analysis: they provide the right framework for studying the continuum. They crop up everywhere, all over mathematics, and a major goal of this course is to understand what they are and how to use them to solve problems.