2.3 KiB
lecture, date
| lecture | date |
|---|---|
| 2 | 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.