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 02 = 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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Received on Fri Jul 24 2009 - 18:41:59 PDT