The Real Numbers

An Introduction to Real Analysis

Chapter 1

The real numbers are fundamental to real analysis, so it is natural to begin the study of real analysis by studying the key properties of the real numbers. These key properties are threefold:

  1. The algebraic properties formalize the rules of arithmetic.
  2. The order properties formalize the idea of a line of real numbers.
  3. The completeness property formalizes the idea that the real number line has no gaps.

The real numbers are also unique up to isomorphism, which formalizes the idea that any two objects having the three properties above can differ only in irrelevant ways, such as having different names. For proof, see Exercise 1-30 in Schröder (2008, 16). Because they are unique, it is justified to refer to the real numbers.

Notes

  • 1.1: The natural numbers
  • 1.1.1: Arithmetic in <span class="math inline">\(\symbb{N}\)</span>
  • 1.1.2: The natural order on <span class="math inline">\(\symbb{N}\)</span>
  • 1.1.3: Induction
  • 1.1.4: Prime numbers and unique factorization

Exercises

  • 1.1.1: Sum of first <span class="math inline">\(n\)</span> natural numbers
  • 1.1.2: Exponential versus linear growth
  • 1.1.3: Sum of powers of <span class="math inline">\(q\)</span>
  • 1.1.4: Binomial theorem

References

Schröder, Bernd S. W. 2008. Mathematical Analysis: A Concise Introduction. John Wiley & Sons.