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Math 350 | Advanced Calculus | Spring 2026

This study guide gives the main definitions and theorems covered in the course.
Properties of Real Numbers

Consider the following sets of numbers: \begin{align} \mathbb{N} &= \{1, 2, 3, \ldots \}\ \text{ (natural numbers)}\\ \mathbb{Z} &= \{\ldots,-3,-2,-1,0,1, 2, 3, \ldots \}\ \text{ (integers)}\\ \mathbb{Q} &= \left\{\frac{a}{b} : a,b\in \mathbb{Z}, b\neq 0 \right\}\ \text{ (rational numbers)}\\ \end{align} We see that \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}\).

There does not exist \(p\in \mathbb{Q}\) such that \(p^2=2\).

In other words, \(\sqrt{2}\notin \mathbb{Q}\). We will shortly introduce the set of real numbers, \(\mathbb{R}\), a larger set of numbers than \(\mathbb{Q}\) which contains \(\sqrt{2}\). We will define \(\mathbb{R}\) rigorously as a set satisfying eleven axioms. The first ten of these axioms are those of the following three definitions.

A field is a set \(F\) together with two binary operations, denoted by \(+\) and \(\cdot\), which satisfy the following axioms:
  1. For all \(a,b\in F\), \(a+b = b+a\) and \(a\cdot b=b\cdot a\).
  2. For all \(a,b,c\in F\), \((a+b)+c = a+(b+c)\) and \((a\cdot b)\cdot c = a\cdot (b\cdot c)\).
  3. For all \(a,b,c\in F\), \(a\cdot (b + c) = a\cdot b + a\cdot c\).
  4. There exist elements \(0,1\in F\) for which \(0+a=0\) and \(1\cdot a = a\) for all \(a\in F\).
  5. For every \(a\in F\), there exists an element \(-a\in F\) for which \(a + (-a)=0\).
  6. For every \(a\in F\), \(a\neq 0\), there exists \(a^{-1}\in F\) for which \(a\cdot a^{-1}=1\).
Let \(S\) be any set. A total order (or linear order) on \(S\) is a relation \(\lt \) on \(S\) satisfying the following two axioms:
  1. For all \(a,b\in S\), exactly one holds: \(a=b\), \(a \lt b\), or \(a\gt b\).
  2. For all \(a,b,c\in S\), if \(a\lt b\) and \(b\lt c\) then \(a\lt c\).
An ordered field is a field \(F\) together with a total order \(\lt\) satisfying the following two axioms:
  1. For all \(a,b,c\in F\), if \(a\lt b\), then \(a+c\lt b+c\).
  2. For all \(a,b,c\in F\), if \(a\lt b\) and \(c \gt 0\), then \(a\cdot c\lt b\cdot c\).