Lecture 1: Sets, Real Numbers, Fields, Ordered Fields

October 30, 2012

The real number system is a set \{a,b,c, ...\} on which the operations of addition and multiplication are defined so that every pair of real numbers (say a and b) has a unique sum and product, both real numbers, with the following properties:

Given any (and all) a,b,c \in \mathbb{R}, we have

(A) a+b=b+a and a*b=b*a (Commutative laws)

(B) (a+b)+c=a+(b+c) and (a*b)*c=a*(b*c) (Associative laws)

(C) a*(b+c)=a*b+a*c (Distribute law)

(D) There exist distinct real numbers 0 and 1 such that a+0=a and a*1=a, for all a.

(E) For each a there is a real number -a such that a+(-a)=0 and if a \neq 0, there is a real number \frac{1}{a} such that a*\frac{1}{a}=1

A set on which two operations are defined so as to have properties (A) – (E) is a field.

Motivation: Think about these properties for a second before moving on. (A) establishes what the operations do, (B) then asks what happens when we include more elements, a “double operation” if you like. (C) then mixes the two operands into one. (D) and (E) are closely related. Suppose we want to add an element to our given real number that… gives us our real number? So we are adding nothing, or 0. Suppose we have the same idea for our other operand, multiplication, we have the real number 1 to do this for us.

Then we ask, can we find real numbers that give us what we just discovered, namely 0 and 1 from addition and multiplication? For that we have -a and \frac{1}{a} with a \neq 0 for the second condition.

Exercise: Can you think of another example of a field?
Exercise: Do the set of integers \mathbb{Z} form a field? Verify through (A) – (E).

The real number system is ordered by the relation <;, which has the following properties:

(F) For each pair of real numbers a and b, exactly one of the following is true: a <; b, a=b, b<;a.

(G) If a<;b and b<;c then a<;c. (Transitive)

(H) If a<;b, then a+c<;b+c for any c and if b<;c then a*c<;b*c.

A field with an order relation satisfying (F) – (H) is an ordered field. The real numbers (and rationals) form an ordered field.

Suppose we get bored of the real numbers and define a new field satisfying everything \mathbb{R} does but is defined as such. For any a,b \in \mathbb{R} we define a new number z as z = a+b*\imath, where \imath^2 = -1. We call this set \mathbb{C}, the set of complex numbers.

Exercise: Verify that \mathbb{C} is a field.
Exercise: Is \mathbb{C} an ordered field? (Use (F) – (H) to prove order or give a counterexample)
(Hard) Exercise: Why have we defined \imath^2 = -1 and not \imath=\sqrt{-1}?

Questions to think about:

Are all fields ordered fields?
Are finite fields ordered?
What do these two above questions (if and when answered) say about ordered fields? (Hint: Infinite number of…)

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The Motivation Behind Real Analyis

October 21, 2012

When I ask anyone who is not a mathematician (and even some who are) what a “number” is, they never define what a “number” is, but say something along the lines of “well a thing you add to get another thing, like 2+2=4”. But here we had defined “+”, which I do not know of, have not defined and do not have much care for; I asked a different question. We see numbers as “left and right” on some big “interval”, this is also wrong. We will see this by the end of the first lecture.

The motivation behind this course is to build some proper rigour into understanding what, how and why analysis is/operates/works.

(When probability is taught wrong, it leads to bad knowledge of quantum physics, statistics(bizarrely), stochastic models, differential equations. Anything with randomness.)

When real analysis is taught wrong, it leads to bad knowledge of everything taught later on in maths: probability, calculus (vector calculus too), geometry, complex analysis, Fourier analysis, group theory, just a few to name, really cannot be taught without some specific and excellent knowledge of real analysis.

It makes sense (which through out this blog I will be saying… a lot) to begin with my notes for the course “An Introduction To Real Analysis”.

But… What if you have no mathematical knowledge? No worries. The prerequisites are nothing. Especially not calculus (integration or differentiation). But you should have some mathematical ability, what do we assume? That you know that a “real number” is. That is all. You do not even need to know what addition or multiplication are. To be honest; I hope you do not.

If you do have some mathematical knowledge, the notes will not be so interesting but the excercises should keep (some are very difficult, some very easy) you awake. And if you have a lot of mathematical knowledge, you may wish to simply download these notes for your own need; to teach others as a lecturer, as a prerequisite to your course, or just to have some real analysis notes, which is always handy.

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