Lecture Notes

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====== Lecture 1 ======

Exercises: 
  - Ross 1.10, 1.12
  - Read Ross 'Rational Zero Theorems', then do 2.1, 2.2, 2.7
  - Try proving Ross Theorem 3.1, 3.2, by yourself, without read his proof. It is a good exercise for logical deduction. Yes, the result may sounds obvious for $\Q$, but you need to prove them for **any** ordered field, which you have no idea what it looks like.



===== Script =====

Welcome to Math 104, your first analysis class. You have learned about calculus, knows all about integration, perhaps also the Stokes formula, Green's formula, namely, all the useful things. What do you want to gain from this course? 

In some sense, this course is a training of critical thinking. Namely, why do you believe what other people told you? Why cannot it be otherwise? Is 1+1 = 2 happens to be true in our world, or does it have to be true? (does this sentence even make sense?) Why do we want to challenge the received wisdom? (otherwise, there is no innovation). What's the benefit to challenge the received wisdom? (The invention of GPS, requires general relativity, and requires the notion of Riemannian manifold to describe curved spacetime, requires one to go beyond the Euclidean geometry, and requires one to give up the axioms that 'two parallel lines in space never intersects'). 

On the other-hand, it is also good to know certain dead-ends. Like, anyone knows the conservation law of energy will not attempt to invent the 'perpetual motion machine'. So, it is useful to know the rule, and where do those rules come from. And how to discover new rules. If you wish, this is like the 'classics' in literature. You are unlikely to discover something new, but the logical training you get will save you some time in the future. 

Long story short, what's this class is about? As you have seen in the syllabus, there are three parts: limit, metric space topology and calculus (integration, differentiation). We will roughly spend one month each. The topic about topology might be new, and it takes some getting used to.

Today, I want to discuss about 'numbers'. By number, we have the following 3 systems
$$ \N = \{0,1,2 \cdots \} $$
$$ \Z = \{ \cdots, -2,-1,0,1,2,\cdots \} $$
$$ \Q = \{ m / n \mid m, n \in \Z, n \neq 0 \} $$

$\N$ is a semi-group, with addition, but no inverse. $\Z$ is a group (abelian group), more over, it is a commutative ring, with a multiplication.  $\Q$ is a 'field', with division as well. 

Why do we need more? When you try to solve equation, say $x^2 = 1$ or $x^2=2$, we sometimes get two solutions, and sometimes get zero solutions in $\Q$. (the problem is that, $\Q$ is not algebraicallly closed). There is also a problem, given a set $E \In \Q$ with an upper bound, it is possible that, there is most economical 'upper bound', or sharpest upper bounded living in $\Q$. Related to this problem is that, $\Q$ is not 'complete' (namely Cauchy sequence in $\Q$ may not converge to a number in $\Q$). 

Introducing the real number $\R$ solves the second problem. And there is no other option, namely, $\R$ is the unique **ordered** field containing $\Q$ that is **complete**. 

Remark: there are other ways to enlarge $\Q$. What other option do you know? 

Discussion questions: \\
1. About mathematical induction: let $P(n)$ denote a statement depending on a natural number $n$, if we can prove two things that: (a) $P(0)$ is true, and (b)  $P(n)$ implies $P(n+1)$, then, we know $P(n)$ is true for all $n \in \N$.

Try Ross p6, 1.10, 1.12

2. About rational roots. Why does $x^2=2$ have no rational roots? Try Ross p12 Ex 2.2. How about 2.7? 

3. There are two ways to introduce real numbers, one is through completion of $\Q$ with respect to a metric (as Tao-I) did, the other is through 'Dedekind cut' (as Ross section 6 did).
Lecture Notes

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