Lecture 1
Introduction
Reading is not recommended before class, it;s hard.
Chapter 1: The real number and complex number systems
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Natural numbers: note by some conventions, is also a natural number
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IntegersL
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Rational numbers:
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Real numbers: the topic of chapter
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Complex numbers:
Theorem ( is irrational)
is false.
Proof
Suppose for contradiction, such that .
Let , where are not both even. (reduced form)
and , so , so is even, is even.
So is divisible by 4, is divisible by 4.
So is even. but they are not both even.
EOP
Theorem (No closest rational for a irrational number)
Let , Then does not have a largest element.
i.e. such that is false.
Remark: The book give a very slick proof trying to lean from these kinds of proofs takes some effort. (It is perfectly fine to write that solution this way...)
Thought process
Let , .
We want a such that and .
From , we know (this is a crude bound, ).
So one choice can be
Proof
, we can find a which is greater than zero () and construct a new number such that .
Here we construct a formula for approximate
Interesting...
We can also further optimize the formula by changing the bound of to , since
def sqrt_2(acc):
if acc==0: return 1
c=sqrt_2(n-1)
return c+((2-c**2)/(2*c+2))
Definition and notations for sets
Some set notation
use in this class.
- ,
- , and
- means and