Thanks for asking this profound and interesting question:
Do there exist numbers other than the reals and imaginaries?
Well, the notion of "what is a number" has expanded over the
centuries, and probably there is no one clear definition. So I will
leave the word "number" undefined for now and address your question
intuitively.
I would say the question was in once sense definitively answered by by
William Rowan Hamilton, whose letter in 1865 here
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html
resolved his research into seeking an extension of the complex
numbers.
(Note: An *imaginary* number is normally thought of as a number with
no real part, like 4i; a *complex* number can be anything of the form
x + y i
where x and y are real numbers and i=sqrt(-1).)
The quaternions are described here
http://www.pcisys.net/~bestwork.1/HamiltonQ/hamilton.htm .
As described in John Baez excellent article on Octonions:
http://math.ucr.edu/home/baez/Octonions/octonions.html ,
----begin excerpt-----
Most mathematicians have heard the story of how Hamilton invented the
quaternions. In 1835, at the age of 30, he had discovered how to treat
complex numbers as pairs of real numbers. Fascinated by the relation
between and 2-dimensional geometry, he tried for many years to invent
a bigger algebra that would play a similar role in 3-dimensional
geometry. In modern language, it seems he was looking for a
3-dimensional normed division algebra. His quest built to its climax
in October 1843. He later wrote to his son, ``Every morning in the
early part of the above-cited month, on my coming down to breakfast,
your (then) little brother William Edwin, and yourself, used to ask
me: `Well, Papa, can you multiply triplets?' Whereto I was always
obliged to reply, with a sad shake of the head: `No, I can only add
and subtract them'.'' The problem, of course, was that there exists no
3-dimensional normed division algebra. He really needed a
4-dimensional algebra.
Finally, on the 16th of October, 1843, while walking with his wife
along the Royal Canal to a meeting of the Royal Irish Academy in
Dublin, he made his momentous discovery. ``That is to say, I then and
there felt the galvanic circuit of thought close; and the sparks which
fell from it were the fundamental equations between ; exactly such as
I have used them ever since.'' And in a famous act of mathematical
vandalism, he carved these equations into the stone of the Brougham
Bridge...
----end excerpt------
Just as the complex numbers are formed from the real numbers by
adjoining and element i such that i*i = -1,
the quaternions are formed from the real numbers by adjoining elements
i,j,k
with the rules
ij=k
ki=j
jk=i
ii=jj=kk=-1
A general quaternion is of the form
x + y*i + z*j + w*k
Just as complex numbers are important to understand rotations in the
plane, so are quaternions important to understand rotations in
3-dimensional space.
However, the real numbers, the complex numbers, the quaternions, and
the octonions are the only four examples of normed division algebras.
Octonions are described in the paper by Baez above, and are much less
well-known the other three.
The definition of normed division algebra can be found in the
Wikipedia here: http://www.wikipedia.org/wiki/Division_algebra .
There is, however, one important sense in which, once we have the real
and the complex numbers we have "all" the numbers. How is this?
One way to think of the way that the complex numbers are formed from
the real numbers is that we start with the real numbers and a
polynomial
x^2+1
We ask: is there a value for x such that the polynomial is 0?
Well, the answer is NO if x must be real. So, to get the complex
numbers from the real numbers, we ask: what if we IMAGINE that there
WERE a root of this polynomial, call the root "i", and see what
products and sums we can form from this new element, keeping the
normal rules of commutativity, associativity, and so on.
That is, suppose we have a domain of entities X containing the real
numbers R and suppose:
x+y=y+x
x+0=x
x*1=x
x*y=y*x
x*(y*z)=(x*y)*z
Now, if X contains an element i such that i*i+1 = 0, then X must be
the complex numbers.
What if we try adding some other element to the complex numbers; can
we continue this process?
In other words, once we have the complex numbers C, can we choose a
polynomial P such that when we add a root of P to the complex numbers
we get a new domain?
The answer to this is quite deep, and it is no. Every polynomial over
the complex numbers has all its roots in the complex plane. This is
the "fundamental theorem of algebra".
There are many proofs of this fact, first proven by Gauss, here is a
good discussion: http://www.cut-the-knot.org/do_you_know/fundamental2.shtml
.
The most interesting and beautiful generalization of the notion of
"number" that I know is that of the so-called Conway numbers, or
surreal numbers.
Some nice web sites on surreal numbers are
http://www.tondering.dk/claus/sur12.pdf (You will need a pdf viewer,
http://www.adobe.com/prodindex/acrobat/readstep.html to read this.
Surreal numbers are particularly useful in understanding the values of
certain kinds of games .
Nowadays, mathematicians constantly invent new mathematical structures
in which certain formulae hold. The elements of the structures are not
always called numbers, but typically one is allowed to perform one or
more operations on them. Originally such structures were thought to
closely mirror aspects of physical reality, and those that did not
seem to mirror such aspects were called derogatory names, hence
"irrational number" and "imaginary number". Experience has taught us,
though, that often the most abstract and strange mathematical
structures that one can imagine are in fact just what is needed to
understand the world.
The branch of mathematics that deals with such structures is called
abstract algebra. Instances of such structures are groups, rings,
fields, vector spaces. Elements of each of these are "numbers" in some
sense. The best book I have read on algebra is probably the book by
Joseph Rotman called Introduction to Group Theory.
You have asked about a rather deep problem and I would be happy to
clarify.
Search strategy
----------------
quaternions
octonions
"fundamental theorem of algebra"
"surreal numbers"
Links:
-------
Abstract algebra: http://www.math.niu.edu/~beachy/aaol/ |