**Real analysis**is the

*rigorous version of calculus*(“analysis” is the branch of mathematics that deals with inequalities and limits). It’s an

*extension*of calculus with new concepts and techniques of proof (Bloch, 2011), filling the gaps left in an introductory calculus class (Trench, 2013). Those “gaps” are the pure math underlying the concepts of limits, derivatives and integrals. Instead of just using those formulas, you’ll be

**proving that they work**.

Essentially, you go from the “what” in calculus is true to “why” it’s true (Lebl, 2020).

## What is Taught in Real Analysis?

A typical real analysis course includes:

- Construction and properties of real numbers. For example, cardinality, algebraic and order structures of real and rational number systems.
- Introduction to logical structures and construction of proofs.
- Core calculus topics: limits, continuity, differentiation and the Riemann integrals—with a heavy emphasis on proofs.
- Sequences and series: introduced in calculus, these are heavily emphasized in real analysis.
- Introduction to differential equations.

Real analysis **heavily emphasizes proofs**. In fact, you can barely take a step in the course without axioms and postulates: every formula encountered in the class has to be proved. Instead of using formulas for derivatives and integration, you’ll be **proving that they work**.

## What Are Some Uses for Real Analysis?

Real analysis underpins many other branches of mathematics including probability theory and analytic number theory. However, it isn’t just a scholarly exercise in pedantry. Real world uses include:

- Describing physical systems with ordinary differential equations,
- Formulating optimal structures (minimums, maximums, and equilibriums),

## References

Lebl, J. (2020). Basic Analysis I. Retrieved September 20, 2020 from: https://www.jirka.org/ra/realanal.pdf

Bloch, E. (2011). The Real Numbers and RA. Springer.

Schramm, M. (2012). Introduction to RA. Dover Publications.

MCMullen, C. Course Notes. Retrieved September 20, 2020 from: http://people.math.harvard.edu/~ctm/papers/home/text/class/harvard/114/course/course.pdf

Trench, W. (2013). Introduction to RA (TRENCH_REAL_ANALYSIS).

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