The 5 C's of Real Analysis
1. Completeness (of real numbers) and Completeness of metric spaces
2.
Convergence (of sequences)
3. Continuity (of functions)
4. Compactness
(of sets)
5. Connectedness (of sets)
The 3 C's of the New Sciences
1. Chaos (in dynamical systems)
2. Complexity (algorithmic,
computational, combinatorial)
3. Computability (or lack of
computability)
If comoputable, there exists a decimal
expansion
example of a non-computable number: a conditional decimal
expansion.
Number Classifications
- Natural
- Integer
- Rational
- Irrational
algebraic-roots of polynomials with integer coefficients
(countable)
transcendental- irrational numbers that are non-algebraic
Computable- a number alpha such that for every n, the nth
digit of the decimal (or binary, ternary, etc.) expansion of alpha, can be
determined by an algorithm (in infinitely many steps):
-rational
-algebraic
-(some) transcendental
Computable
numbers are countable (can count by counting the algorithms that determine
them)
Computable property is only possessed by countably many numbers
The importance of countability was that it limited what was possible
in numerics.
In Class Excercise:
Can you come up with an algorithm to calculate the nth digit of the
expression of the number Pi? (i.e. can you find a pattern in the long division?)
Is the number Pi therefore countable?
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