# Why study analysis? – Serlo

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• Introduction • What is analysis? • Why study analysis? • Propositional logic • Mathematical induction • Real numbers • Complex numbers • Supremum and infimum • Sequences • Convergence and divergence • Subsequences, Accumulation points and Cauchy sequences • Series • Convergence criteria for series • Exponential and Logarithm functions • Trigonometric and Hyperbolic functions • Continuity • Differential Calculus • Linear combinations, generators and bases • Linear mappings Why should I study real analysis? This is a legitimate question and should be answered at the beginning of every lecture and textbook on real analysis. In this chapter I want to present some reasons as an answer to this question.

## Real Analysis as the foundation of mathematics

Real analysis is an essential lecture of mathematics. Many theories such as complex analysis, functional analysis and the theory of ordinary differential equations are based on it. In addition to that numerous other modified concepts of real analysis are used in a variety of other domains of mathematics.

Therefore delving into real analysis is inevitable when one wants to study mathematics. This theory will be key to understand other fields of mathematics. You will probably need background knowledge of real analysis no matter what field of mathematics is of interest to you.

The same applies if you require mathematical knowledge as a tool for your studies or research. Whatever you need to use mathematics for, knowledge of real analysis will almost certainly help you understand your mathematical tools.

## Real Analysis as a description of nature

Most problems in natural sciences are modelled with the help of concepts covered by real analysis: How can the location and speed of a moving object be determined and forecast? How can one calculate the electric field of a system of electrically charged bodies?

The area of application of real analysis includes simple problems and extends as far as modern fields of research (eg. axiomatic quantum field theory) or quantitative modelling of biological processes. Very good knowledge of these concepts will therefore help you understand the laws of nature and formulate these. For this reason studying real analysis will be particularly helpful - whether you study or are generally interested in a natural science.

## Real Analysis in school

Analysis is one of very few lectures in mathematics already covered in school. In case you study mathematics education you will be able to use the knowledge acquired by studying real analysis when teaching. Thorough knowledge of this lecture will therefore help you with explaining mathematics to your students.

## Real Analysis as an exact mathematical theory

In contrast to analysis you got to know in school we will write out mathematical theories with immense precision. This will include us proving every theorem rigorously and defining every term with precision. As an example: in the end you will know why the derivation of xxx is xxx and what exactly a derivation is. At the same time we will justify every term to ensure you understand why certain concepts in real analysis are introduced. This will not only deepen your understanding of your knowledge of analysis already acquired in school but also enable you to solve complex problems in real analysis.

## Real Analysis as an exercise of mathematical reasoning

With analysis already having been taught in school comes the advantage of the material already seeming more or less familiar to you. This gives us the chance to concentrate on the mathematical way of working, which many students aren’t familiar with in the beginning. Real analysis is excellent to practice exact and accurate reasoning.

Mastering the mathematical way of working will always be useful to you - even when you no longer need real analysis. Math students and also those of other natural sciences are especially in demand due to being capable of this. I am confident the mathematical way of thinking will help you a lot in accomplishing your tasks and project.