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--- math104-s22:notes:lecture-1 [2022/01/17 18:59]
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+++ math104-s22:notes:lecture-1 [2022/01/19 09:28] (current)
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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?
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 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
+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: \\
+. 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
+. About rational roots. Why does  $x^2=2$  have no rational roots? Try Ross p12 Ex 2.2. How about 2.7?
+. 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).

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