Chapter 2 - The Real Number Line
Section 2.1 - The Algebraic and Order Properties of Real Numbers
Proposition. 2.1.1: \(\mathbb{R}\) is a field, with zero element \(0\) and identity \(1\).
Definition. The rational numbers \(\mathbb{Q}\) is the field of fractions of the natural numbers \(\mathbb{N}\).
Theorem. 2.1.4: There does not exist a rational number \(r\) such that \(r^2 = 2\).
Definition. An ordered field is a field \(F\) together with subset \(F^+\) such that
- \(F+\) is closed under addition and multiplication
- If \(a \in F\), then exclusively \(a \in F^+\), \(a = 0\), or \(-a \in F^+\).
Theorem. In any ordered field \(F\), the following hold
- \(1 \in F^+\)
- \(\mathbb{N} \subseteq F^+\)
- If \(a \in F^+\), then \(\frac{1}{a} \in F^+\)
Definition. The order relation \(a > b\) and \(b < a\) is defined by \(a - b \in F^+\).
Theorem. If \(a, b, c \in F\), then
- One of \(a > b\), \(a = b\), or \(a < b\) hold (trichotomy)
- If \(a > b\) and \(b > c\), then \(a > c\) (transitivity)
- If \(a > b\), then \(-a < -b\)
- If \(a > b\) and \(c > 0\), then \(ac > bc\)
- If \(a > b\) and \(c < 0\), then \(ac < bc\)
- If \(a > b > 0\), then \(\frac{1}{b} > \frac{1}{a} > 0\)
Definition. Let \(S\) be a nonempty subset of ordered field \(F\). Then, \(S\) is bounded above if there exists some \(u \in F\) such that \(s \leq u\) for all \(s \in S\). Then, said element \(u\) is an upper bound of \(S\).
Definition. Let \(S\) be a nonempty subset of ordered field \(F\). Then, \(S\) is bounded below if there exists some \(u \in F\) such that \(s \geq u\) for all \(s \in S\). Then, said element \(u\) is a lower bound of \(S\).
Definition. Let \(S\) be a nonempty subset of ordered field \(F\). Then, \(S\) is bounded if it is bounded both above and below.
Definition. Given field \(F\) and nonempty subset \(S \subseteq F\), an element \(u \in F\) is a supremum or least upper bound of \(S\) if \(u\) is an upper bound of \(S\), and given any other upper bound \(v\), then \(u < v\)
Definition. Given field \(F\) and nonempty subset \(S \subseteq F\), an element \(u \in F\) is an infimum or greatest lower bound of \(S\) if \(u\) is a lower bound of \(S\), and given any other lower bound \(v\), then \(u > v\)
Definition. Given an ordered field \(F\), the field has the supremum/infimum property if given any nonempty subset \(S\), if \(S\) is bounded above/below, \(S\) has a supremum/infimum.
Section 2.2 - Absolute Value and the Real Line
Definition. Absolute value is defined as normal (piecewise). Multiline function in LaTeX are hard.
Theorem. Given any \(a, b \in \mathbb{R}\), we know that
- \(|a| > 0\) for \(a \neq 0\)
- \(|ab| = |a||b|\)
- \(|a + b| \leq |a| + |b|\)
Corollary. Given \(a, b \in \mathbb{R}\), then \(||a| - |b|| \leq |a - b|\).
Remark. Every field has at least one absolute value function.
Theorem. In an ordered field \(F\), for any \(r > 0\), we know that
- \(|x = r\) if and only if \(x = r\) or \(x = -r\)
- \(|x < r\) if and only if \(-r < x < r\)
- \(|x > r\) if either \(x > r\) or \(x < -r\)
Definition. The standard distance function or metric on the real numbers \(\mathbb{R}\) given \(a, b\) is \(|a - b|\).
Theorem. For any real numbers \(a, b, c\),
- \(|a - b| > 0\) if and only if \(a \neq b\) and \(|a - b| = 0\) if and only if \(a = b\)
- \(|a - b| = |b - a|\)
- \(|a - c| \leq |a - b| + |b + c|\)
Definition. A set together with a function satisfying these three properties is known as a metric space.
Definition. The \(\varepsilon\)-neighborhood of \(a \in \mathbb{R}\), denoted \(V_\varepsilon(a)\) is the set of all real numbers \(x \in \mathbb{R}\) such that \(|x - a| < \varepsilon\). That is,
Decimals. Let \(x \in \mathbb{R}\) such that \(x > 0\). By the archimedean property, there exists some \(b_0 \in \mathbb{N} \cup {0}\) such that \(b_0 < x < b_0 + 1\). We can repeat this to see
Definition. The decimal expansion of \(x\) is denoted \(b_0.b_1 b_2 b_3 \ldots\).
Section 2.5 - Intervals
Definition. A subset \(I\) is an interval if and only if, given \(a, b \in I\), then \([a, b] \subseteq I\).
Definition. Intervals \(I_1, I_2, \ldots, I_n, \ldots\) are nested if and only if \(I_1 \subseteq I_2 \subseteq \ldots \subseteq I_n \subseteq \ldots\).
Theorem. Nested Intervals Property. If \(I_n = [a_n, b_n]\) is a set of nested intervals that are closed and bound, then there exists some number \(z \in \mathbb{R}\) such that \(z \in I_n\) for all \(n\).
Theorem. If \(a < b\), then the interval \([a, b]\) is an uncountable set.
Corollary. \(\mathbb{R}\) is uncountable.