Real Analysis

Real analysis is the rigorous study of the real numbers as mathematical objects, and it is more broadly the rigorous, proof-based version of calculus. Where calculus focuses on computing derivatives and integrals, analysis focuses on proving properties about them. For example, calculus students assume that the mean value theorem is true and then use it to solve computational problems, while analysis students prove the mean value theorem in the first place.

Each math subject must tether itself to a rock of assumptions, called axioms, from which other results derive. Real analysis begins with an axiomatic definition of the real numbers. From this definition, familiar arithmetic properties can be derived. The concept of distance is then formalized into the concept of a metric, and the different ways in which real numbers can be "close" to one another forms the basis of metric topology. From there, sequences and series of real numbers are studied, which naturally entail discussions of limits, which describe the behavior of processes tending towards zero or infinity. Calculus graduates will recall that limits then lead to derivatives, and derivatives to integrals. "Nice" functions are formalized with a discussion of continuity along the way.

As with many topics in mathematics, a basic understanding of set theory is required. Many introductory books on analysis, topology, and abstract algebra include primers on set theory, and readers here are directed to reference the Naive Set Theory topic. Accordingly, familiarity with concepts such as set algebra, order relations, and functions is assumed.

Real analysis is one of the most crucial topics in advanced mathematics because it serves as a nexus for many other branches of mathematics. Subjects such as complex analysis, probability, and differential geometry are built on top of the real numbers. Likewise, whole subjects such as general topology and many parts of linear algebra are made up of abstractions of more concrete concepts originally built on the real numbers. Put simply, a heck of a lot of math works with real numbers, so getting familiar with them is a darn good use of time.

In universities, the real analysis course sometimes doubles a torture device for professors to wield over undergraduates, whereby the professor "introduces" innocent fledgling math majors to "real math" by way of intimidation and psychological terror. The familiarity of the subject matter paradoxically compounds the difficulty, as students have to not only learn how to write proofs, but also learn which facts in the sea of "math we just know works" depend on others yet unproved so far in the course, and which are fair fodder for whatever that week's problem set of proofs. This website aims to avoid all that.

Conquering an introductory book on real analysis is a right of passage and a mark of distinction for all aspiring mathematicians.