Hi Marty, Kim, and all those who need some math revision (for the  
seventh step of the Universal Dovetailer Argument).



I think, from the preceding post, that it is a good time to recall  
some basic definitions. And to introduce new one.

The set of natural numbers, or positive integers is N = {0, 1, 2,  
3, ...}

New and important definition: a set is said to be closed for an  
operation, when the operation, applied to elements of the set, does  
not lead outside the set.

The set of natural numbers N is said to be close for the operation of  
addition and multiplication.

The set of integers, Z = {... -3, -2, -1, 0, 1, 2, 3, ...}

Z is closed for the operation of addition, multiplication, and  
subtraction. But not closed for division.


The set of rational numbers Q = {n/m such that n and m belong to Z}.
Here we mean by "n/m" the value of the division of n by m. I mean that  
1000/500, and 4/2, and 66/33, all represent the same rational number 2.

The set of rational numbers is closed for addition, multiplication,  
subtraction, and division.

We have seen that the rational numbers can always been written under  
the form of periodic decimal. For example

2 is 1.999999....   (exercise prove this!)

I will not allow you to write 2 = 2,00000.... (why? because it is ugly  
and it will make our life harder later). Some mathematicians allow  
this, though.

1/3 = 0.3333333...

Alas the square root of two, which is also the length of the diagonal  
of square with side 1, cannot be written as n/m (I will prove this  
soon), so we have to extend the notion of number if we want that all  
length are measurable by numbers. We will just add all all infinite  
decimal numbers to Q. We get

R the set of (so-called) real numbers. R is the set of all numbers  
measuring possible distance (length) between any points of the usual  
geometrical line.
I will perhaps provide cleaner definition later.

I hope that you see that the set N is included in Z, which is included  
in Q which is included in R.

R is of course closed for addition, multiplication, subtraction and  
division, like Q, but is closed for the operation of limit. This will  
be explained later.

We will not need to go beyond the real numbers, for the seventh step  
of UDA. So, what follow is purely for education and entertainment.


Entertainment:

To do quantum mechanics, or just electricity theory, we would need the  
complex numbers C. They extend into the plane. The 2-dimensional  
geometry. They make it possible to define the Mandelbrot set very  
quickly too. I may talk on them later.

But if C extends R, it means obviously that R is included in C. Note  
that N, Z, Q, R, can be seen as subsets of the so called "real line",  
or one dimensional space. But with C, the numbers extend into the  
plane (the two dimensional space)

Do we need still some other numbers?

Hamilton sought to find numbers  extending into the three dimensional  
space, for a long time, and realize there are none! But he discovered,  
in 1843, the numbers which are extending in the four dimensional  
space, called the quaternions. They are used to compute rotations in  
three dimensional spaces, and most algorithms used to rotate real  
objects (like a satellite in space, or a spaceship in a video game)  
are using quaternions (or rational approximations 'course).

Graves, in 1843,  and Cayley, published in 1845, independently  
discovered the "next sort" of numbers: the octonions, which lives in  
eight-dimensional spaces. Some pure mathematician were proud of having  
at last find the beautiful Number that no applied mathematicians would  
ever find a use for! But they did recently found applications in the  
attempt to marry General relativity with Quantum mechanics.

Are we finished yet?

A case could be made for the sedenions, which lives in the 16- 
dimensional space. Sedenions have still an entry in Wikipedia, for  
example. I don't know if they have found some concrete application,  
except for a vague souvenir of a reference I read sometimes ago.

We have N included-in Z included-in Q included-in R included-in C  
included-in H included-in O included-in S, where H is the set of  
quaternions, O the set of octonions, and S the set of sedenions.

Yet, you see again the power of 2. The dimensional geometry of the  
type of numbers seems to follow the sequence

1, 2, 4, 8, 16, ....   (that is power of 2: 2^n for n = 0, 1, 2, 3, ...)

Have you all seen the many apparitions of that that sequence in the  
Mandelbrot set zooms?

Of course the sequences 1, 2, 4, 8, 16, ... appears in all bifurcation  
diagrams, like those from the theory of chaos. See a picture here:
http://sprott.physics.wisc.edu/chaos/abschaos.htm

Bruno

http://iridia.ulb.ac.be/~marchal/




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