« Approfondissements de lycée/Infini et processus infinis » : différence entre les versions

 

====Exemple====

 

:'''N'''   '''B'''

:1   -1

and so on.

 

 

 

... -3,-2,-1, 0, 1, 2, 3, ...

 

 

 

:'''Z'''   '''N'''

:  1   3

: -2   4

 

#Is the number of even numbers the same as the natural numbers?

#What about the number of square numbers?

#Using the idea of one to one correspondance prove that infinity + 1 = infinity, what about infinity + A where A is a ''finite'' set? What about infinity plus C where C is a countably infinite set?

 

In this section we will look to see if we can find a set that is '''bigger''' than the countable infinity we have looked at so far. To illustrate the idea we can imagine a story.

 

So how do we go about counting Q'? If we try counting the first row then the second and so on we will fail because the rows are infinite in length. Likewise if we try to count columns. But look at the diagonals. In one direction they are infinite ( e.g. 1/1, 2/2, 3/3, ...) but in the other direction they are finite. So this set is countable. We count them along the finite diagonals, 1/1, 1/2, 2/1, 1/3, 2/2, 3/1....

 

# Adapt the method of counting the set Q' to show that thet Q is also countable. How will you include 0 and the negative rationals? How will you solve the problem of multiple entries representing the same number ?

# Show that <math> \infty \times \infty = \infty </math> (provided that the infinites are both countable)

 

 

So far we have looked at N, Z, and Q and found them all to be the same size, even tough N is a subset of Z which is a subset of Q. You might be beginning to think "Is that it? Are all infinities the same size?" In this section we will look at an set that is ''bigger'' than N. A set that ''cannot'' be put into a one to one correspondence with N no matter how it is arranged.

Now imagine you measure a book and found it to be 10.101010101010cm. You'd be pretty surprised wouldn't you? But this is exactly the sort of result you would need to get if the book's length were rational. Rational numbers are dense (you find them all over the number line), infinite, yet much much rarer than real numbers.

 

It's good to get a feel for the size of infinities as in the previous section. But to be really sure we have to come up with some form of proof. In order to prove that R is bigger than Q we use a classic method. We assume that R is the same size as Q and come up with a contradiction. For the sake of clarity we will restrict our proof to the real numbers between 0 and 1.We will call this set R' Clearly if we can prove that R' is bigger than Q then R must be bigger than Q also.

 

We have done what we set out to do. We have constructed a number that is in R but is not on the "list of all members of R". This means that R is bigger than any list. It is not listable. It is not countable. It is a bigger infinity than Q.

 

 

There are but they are difficult to describe. The set of all the possible combinations of any number of real numbers is a bigger infinity than R. However trying to imagine such a set is mind boggling. Let's look instead at a set that looks like it should be bigger than R but turns out not to be.

This defines a one to one correspondence between the points in the plane and the points in the line. (Actually, for the sharp amongst you, not quite one to one. Can you spot the problem and how to cure it?)

 

#Prove the the number of points in a cube is the same as the number of points on one of its sides.

 

 

We shall end the section on infinite sets by looking at the Continuum hypothesis. This hypothesis states that there are no infinities between the natural numbers and the real numbers. Cantor came up with a number system for transfinite numbers. He called the smallest infinity <math>\aleph_0</math> with the next biggest one <math>\aleph_1</math> and so on. It is easy to prove that the cardinality of N is <math>\aleph_0</math> (Write any smaller infinity out as a list. Either the list terminates, in which case it's finite, or it goes on forever, in which case it's the same size as N) but is the cardinality of the reals = <math>\aleph_1</math>?

The hypothesis is interesting because it has been proved that "It is not possible to prove the hypothesis true or false, using the normal axioms of set theory"

 

If you want to learn more about set theory or infinite sets try one of the many interesting pages on our sister project [[en:wikipedia]].

*[http://en2.wikipedia.org/wiki/Ordinal_number ordinal numbers]

*[http://en2.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel Hilbert's Hotel]

 

The theory of infinite sets seems weird to us in the 21st century, but in Cantor's day it was downright unpalatable for most mathematicians. In those days the idea of infinity was too troublesome, they tried to avoid it wherever possible.

 

Note that we are not dividing 1 by infinity and getting the answer 0. We are letting the number ''n'' get bigger and bigger and so the reciprocal gets closer and closer to zero. Those 18th Century mathematicians loved this idea because it got rid of the pesky idea of ''dividing by infinity''. At all times ''n'' remains finite. Of course, no matter how huge ''n'' is, 1/''n'' will not be ''exactly'' equal to zero, there is always a small difference. This difference (or error) is usually denoted by &epsilon; (epsilon).

 

<blockquote style="padding: 1em; border: 2px dotted purple;">

When we talk about infinity, we think of it as something big. But there is also the infinitely small, denoted by &epsilon; (epsilon). This animal is closer to zero than any other number. Mathematicians also use the character &epsilon; to represent anything small. For example, the famous Hungarian mathematician Paul Erdos used to refer to small children as epsilons.

</blockquote>

 

Lets look at the function

:<math> \frac{x^2 + x}{x^2} </math>

:<math>\lim_{x \to \infty}\frac{1}{x}(\sin x) = 0 </math>

 

 

 

#<math>\lim_{x \to \infty}(2x^2 -x^4) </math>

 

Consider the infinite sum 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + ....

Do you think that this sum will equal infinity once all the terms have been added ? Let's sum the first few terms.

Here is another way of looking at it. Imagine a point on a number line moving along as the sum progresses. In the first term the point jumps to the position 1. This is half way from 0 to 2. In the second stage the point jumps to position 1.5 - half way from 1 to 2. At each stage in the process (shown in a different colour on the diagram) the distance to 2 is halved. The point can get as close to the point 2 as you like. You just need to do the appropriate number of jumps, but the point will never actually reach 2 in a finite number of steps. We say than in the limit as n approaches infinity, S_n approaches 2.

 

The ancient Greeks had a big problem with summing infinite series. A famous paradox from the philosopher Zeno is as follows: