---
lecture: 2
date: 2025-01-09
---
# Quotients
## Constructing Quotients
Have equivalence relation on $\mathbb{Z} \times (\mathbb{Z} \setminus \{ 0 \})$ given $(a,b) \sim (c,d)$ when $ad = bc$.

Then $\frac{\mathbb{Z} \times \mathbb{Z} \setminus \{ 0 \}}{\sim}$ is the set of rational numbers $\mathbb{Q}$ with addition and multiplication defined in an **obvious** way.
$\frac{a}{b} = \frac{c}{d}$
# Real Numbers
## Constructing Real Numbers
Problem with $\mathbb{Q}$:
For example Pythagoras' Theorem with sides 1 and 1 gives $\sqrt{ 2 }$
$$
\sqrt{ 2 } \notin \mathbb{Q}
$$
$$
\sqrt{ 2 } = \frac{a}{b}, \; a,b \in \mathbb{Z}, \; b \neq 0.
$$
Which is a contradiction

Yet we can find a sequence $\{ X_{n} \}$ of rational numbers such that $X_{n}^2 \to 2$ as $n \to \infty$.
$$x_{1} = 1, \; x_{2} = 1.4, \; x_{3} = 1.41$$
$$\sqrt{ 2 } = 1.4142\ldots$$
By a **sequence** $\{ x_{n} \}$ in a set $X$ we mean a function $f : \mathbb{N} \to X$ with $x_{n} \equiv f(n)$, and that a **function** or **map** $X \to Y$ between two sets ascribes one member of $Y$ to each member of $X$.

A sequence $\{ x_{n} \}$ in $\mathbb{Q}$ **converges** to $x \in \mathbb{Q}$, written $\lim_{ x_{n} \to \infty} = x$, if $\forall k \in \mathbb{N}$ $\exists N_{k} \in \mathbb{N}$ such that $| x - x_{n} | \lt \frac{1}{k}, \; \forall n \lt N_{k}$.

So what is $\sqrt{ 2 }$?

Real numbers are certain equivalence classes of **[[Rational Cauchy Sequences]]**

Two such sequences $\{ x_{n} \}$, $\{ y_{n} \}$ are equivalent if the "distance" $\lim_{ |x_{n} - y_{n}| \to \infty } = 0$ between them vanishes.
$$\{ x_{n} \} \in X \sim \{ y_{n} \} \in X$$
Their set of equivalence classes is the set of real numbers $\mathbb{R}$.
$$X \setminus \sim = \mathbb{R} \ni [\{ x_{n} \}]$$
$$[\{ x_{n} \}] + [\{ y_{n} \}] = [\{ x_{n} + y_{n} \}]$$
Get natural algebraic operations on $\mathbb{R}$ from $\mathbb{Q}$; check well-definedness. Then $\mathbb{Q} \subset \mathbb{R}$ as the classes containing constant sequences. An order $\gt$ on $\mathbb{R}$ is defined by declaring as positive those classes having sequences with only positive rational numbers.
$$x \gt y \implies x-y \gt 0$$
## Convergence of Real Numbers
A sequence of real numbers is said to converge to a real number if the "distance" between their representatives tend to zero.