Download Math 3345-Real Analysis — Lecture 01 8/31/05 1. What`s Real

Math 3345-Real Analysis — Lecture 01
8/31/05
1. What’s Real Analysis?
For the most part, I would say that real analysis is the study of the concepts needed to talk about differentiation and integration. Certainly, these concepts would include limits, continuity and distance, but more
fundamentally, we need to understand the real numbers. The properties of the real numbers, in fact, are
fundamental to understanding most areas of mathematics.
I think of the properties of the real numbers in two categories. There are the algebraic properties, and then
there are the geometric ones. In algebraic terms, the set of real numbers is an ordered field. Being a field
means that we have an addition and a multiplication that have the properties we generally take for granted
(e.g. the associative property, the commutative property, the distributive property, etc.). We also have a
concept of order. This means that we can take any two distinct real numbers, and say that one is greater
than the other. Furthermore, the ordering and the operations (addition and multiplication) go together. For
example, if x − y is a positive number (i.e., is greater than zero), then we know that x is greater than y. If
x is positive and y is negative, then we know that xy is negative (i.e., is less than zero).
The geometric properties are based mainly on the concept of distance. For example, we know that the
integers are equally spaced. The distance between 3 and 5, for example, is 2. With a notion of distance,
we can talk about numbers being small or large, and we can talk about numbers being close together or far
apart. The geometry of the real line is nicely related to the algebra. For example, we know that 2 · 3 = 6.
That’s algebra. The number 3.01 is close to 3. That’s geometry. Together, 2 · 3.01 = 6.02, and we see that
a small geometric change in one of the factors results in a small geometry change in the product. If we
couldn’t depend on this, how messed-up would that be?
Can you think of any area of mathematics that is totally unrelated to the things I’ve just mentioned? That’s
what real analysis is all about. Most of the other major areas of pure mathematics (algebra, geometry,
topology, logic, set theory, combinatorics, etc.) study isolated aspects of mathematical structures. Real
analysis brings everything together.
I’m not going to worry a lot about defining what the real numbers are. You’ve been introduced to most
of it. The focus for this class will be on how things fit together. With this in mind, we’re going to take a
somewhat non-politically correct definition for the set of real numbers.
Definition 1. The real numbers, denoted by R, is the set of all those numbers that have an infinite
decimal representation.
Today, we’ll talk about decimal representations in more detail. Note also that we’ll assume that we’re
working in base ten (unless it’s specifically made clear that we’re not). On the other hand, any other base
would work out the same way.
2. Decimal Form
We’re starting with the fact that all the real numbers can be written in decimal form. For example,
(1)
5
= 1.25
4
Virtually all the real numbers (whatever that means) cannot be written as a finite decimal. For example,
(2)
1
= 0.33333 . . .,
3
1
2
an infinite decimal number with repeating 3’s. It’s common to put a line over the digits as an indication
that they repeat, so we could write one-third as
(3)
1
= 0.3 = 0.333 = · · ·
3
The set of real numbers, R, has some important subsets. Two simple ones are the integers and the natural
numbers.
Definition 2. The natural numbers (sometimes called the counting numbers) will be denoted by N. These
are the whole numbers starting with 1. (Some people start with 0, but we’ll start with 1.)
(4)
N = { 1, 2, 3, 4, 5, . . .}
If we throw in 0 and the negatives of the natural numbers, we get the integers, denoted by Z.
(5)
Z = { . . . , −3, −2, −1, 0, 1, 2, 3, . . .}
Note that any integer can be written as an infinite decimal. For example
(6)
3 = 3.00000 . . . = 3.0
Another extremely important subset for us is the set of rationals.
Definition 3. The rational numbers, denoted by Q, is the set of real numbers that can be written as an
integer fraction. In set notation, we would write
n
o
a
(7)
Q = q ∈ R q = with a, b ∈ Z and b 6= 0 .
b
We can show that any repeating decimal can be written as a fraction, and therefore, all repeating decimals
must be rational numbers. For example, suppose x has the infinite decimal form
(8)
x = 0.814681468146 . . . = 0.8146
The sequence of digits 8146 repeat indefinitely.
1.
What is the decimal expansion for 10,000x?
2.
Certainly, 10,000x − x = 9,999x, but what do you get when you subtract the decimal forms?
What we know, therefore, is that 9,999x = 8146.000 . . ., and so
8146
.
9000
We can reduce this, of course, but the main thing is that x can be written as a fraction. If you think about
it, you’ll see that this trick will work for any repeating decimal. If you do a few long divisions, it will also
become clear that any fraction will have a repeating decimal.
(9)
x=
Basic Principle 1. The rational numbers are those real numbers that can be expressed as a fraction. The
set Q, therefore, is precisely the set of all repeating decimals (and terminating decimals, which repeat 0’s).
The last of our principle subsets consists of all the rest of the numbers. All the real numbers that are not
rational are said to be irrational.
Definition 4. The irrational numbers, denoted by I, is the set of real numbers that are not rational.
Basic Principle 2. The irrational numbers are those real numbers that cannot be expressed as a fraction.
The set I, therefore, must be the set of all non-repeating decimals.
3
An example of an irrational number is
√
(10)
2 = 1.414213562 . . .
√
It is a fact that 2 cannot be written as a fraction (see the end of this document for a proof), and so
this decimal cannot repeat. On the other hand, if we were to truncate this decimal, for example after four
decimal
smaller
√ places, we would have 1.4142, a rational number (since it terminates). This must be strictly
√
than 2. If we were to add 0.0001 to get 1.4143, we would get a rational number larger than 2. In other
words,
√
(11)
1.4142 < 2 < 1.4143
3.
√
1
Find a pair of rational numbers (terminating decimals) that are 1,000,000
apart and have 2 between
them.
√
Note that we can always find two rational numbers that have 2 between them and differ only at any digit
we want.
Another √way of saying this is that we have an increasing sequence of rational numbers that get infinitely
close to 2.
√
(12)
1 < 1.4 < 1.41 < 1.414 < 1.4142 < 1.41421 < 1.414213 < · · · < 2
√
Almost as easily, we can find a decreasing sequence of rational numbers that get infinitely close to 2.
√
(13)
2 > 1.5 > 1.42 > 1.415 > 1.4143 > 1.41422 > 1.414214 > · · · > 2
For problems 4 and 5, note that π is irrational, and π = 3.141592653 . . .
4.
As with equation (12), list the first seven terms of an increasing sequence of rational numbers that get
infinitely close to π.
5.
As with equation (2), list the first seven terms of a decreasing sequence of rational numbers that get
infinitely close to π.
Basic Principle 3. At any level of approximation (e.g., rounding to ten decimal places), the rational
numbers are sufficient.
2.1. Between any two irrationals, there is at least one rational.
√
√We can continue our exploration of
irrationals and rationals. Consider the two irrational numbers 2 and 3 3.
√
2 = 1.414213562 . . .
(14)
√
3
3 = 1.442249570 . . .
These numbers are fairly close together, and it’s easy to find a rational between them. Since they both start
with 1.4 . . ., I’ll look at the second decimal places, which are a 1 and a 4. Certainly, 2 is between them, so
1.42 is a rational number in between. So are 1.43, and 1.415. There are, in fact, infinitely many rationals
between these two irrationals.
For any two irrationals, no matter how close, they must differ at some digit in their decimal expansions. We
can then find a rational between the two numbers.
4
2.2. Between any two rationals, there is at least one irrational. We can play a similar game to find
irrationals between rationals. For example, take the two rationals 2.516 and 2.517. They’re pretty close, but
it’s easy to find an irrational between them. One way is to just
√ tag on a sequence of non-repeating decimals.
If we want something specific, we could use the digits from 2. We know those don’t repeat.
(15)
2.516 < 2.516141421356 . . . < 2.517
√
Since we don’t really know all the digits of 2 (although we can prove that they don’t repeat) we could
write this number more explicitly
√
√
2
2
(16)
2.516141421356 . . . = 2.516 + 0.000141421356 . . . = 2.516 + 4 = 2.516 +
.
10
10,000
I forgot if I assigned any homework, or not. Let’s say that I didn’t.