View Question
Q: other numbers besides reals and imaginaries? ( Answered ,   1 Comment )
 Question
 Subject: other numbers besides reals and imaginaries? Category: Science > Math Asked by: placain-ga List Price: \$5.00 Posted: 23 Oct 2002 21:51 PDT Expires: 22 Nov 2002 20:51 PST Question ID: 89089
 ```Do there exist any other numbers besides the reals and the imaginaries? (I don't count such values as epsilon, aleph-0, or aleph-1 as true "numbers".) If not, can this be proven?```
 ```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/```
 ```I thought this was a really nice answer! Done in a timely way and with surprisingly comprehensive detail. regards, mathtalk-ga```