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Complex Numbers .. the end of the line?

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Owen Densmore

Complex Numbers .. the end of the line?

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Integers, Rationals, Reals .. these scalars seemed to be enough for quite a while. Addition, subtraction, multiplication, division all seemed to do well in that domain.

But then came the embarrassing questions that involved the square root of negative quantities and the brilliant "invention" of complex numbers (a + bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e. that a polynomial of degree n has n roots .. but the roots must be allowed to be complex.

The obvious question is "what next"? I.e. if we look at complex numbers at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that are needed for operations beyond polynomial equations?

This led me to think of linear algebra .. after all, there we are comfortable with n-tuples and we can apply any algebra we'd like to them (likely limiting them to be fields).

Wikipedia shows this:

http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers

which illustrates an interesting job of integrating complex numbers into matrix form, not surprising 2x2, although the matrices are the primitives in this algebra, not 2-tuples.

3D transforms do get us into quaternions which wikipedia

http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions

considers a generalization of complex numbers.

So the question is: are there higher order numbers beyond complex needed for algebraic operations? Naturally n-tuples show up in linear algebra, over the fields N,I,Q,Z and C. But are there "primitive" numbers beyond C that linear algebra, for example, might include?

What's next? And what does it resolve?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Arlo Barnes

Re: Complex Numbers .. the end of the line?

Sadly, I am not going to answer your question, because I am still focussing in my current education on vanilla complex number geometries anyway.

Instead, I am going to comment on "are there higher order numbers beyond complex needed for algebraic operations" by emphasizing 'needed' - I always considered math as methods that could be applied to various hypothetical structures/ideas to provide an interesting train of thought. If this is a useful perception of mathematics(and if it is not, please feel free to say so), then would there be a necessary but as-yet undiscovered need for any particular concept? Would it not be better to say, "are there number(data?)-structures that provide for interesting algebras not yet considered?"

-Arlo James Barnes

Carl Tollander

Re: Complex Numbers .. the end of the line?

In reply to this post by Owen Densmore

Consider Baez on Octonions - talks about what the issues are. Beyond me for now. Suspect you are about to pop out of algebra and end up someplace else as interesting.

Carl

On 1/23/12 5:38 PM, Owen Densmore wrote:

Integers, Rationals, Reals .. these scalars seemed to be enough for quite a while. Addition, subtraction, multiplication, division all seemed to do well in that domain.

But then came the embarrassing questions that involved the square root of negative quantities and the brilliant "invention" of complex numbers (a + bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e. that a polynomial of degree n has n roots .. but the roots must be allowed to be complex.

The obvious question is "what next"? I.e. if we look at complex numbers at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that are needed for operations beyond polynomial equations?

This led me to think of linear algebra .. after all, there we are comfortable with n-tuples and we can apply any algebra we'd like to them (likely limiting them to be fields).

Wikipedia shows this:

http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers

which illustrates an interesting job of integrating complex numbers into matrix form, not surprising 2x2, although the matrices are the primitives in this algebra, not 2-tuples.

3D transforms do get us into quaternions which wikipedia

http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions

considers a generalization of complex numbers.

So the question is: are there higher order numbers beyond complex needed for algebraic operations? Naturally n-tuples show up in linear algebra, over the fields N,I,Q,Z and C. But are there "primitive" numbers beyond C that linear algebra, for example, might include?

What's next? And what does it resolve?

-- Owen
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Roger Critchlow-2

Re: Complex Numbers .. the end of the line?

In reply to this post by Owen Densmore

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

-- rec --

On Mon, Jan 23, 2012 at 5:38 PM, Owen Densmore <[hidden email]> wrote:

Integers, Rationals, Reals .. these scalars seemed to be enough for quite a while. Addition, subtraction, multiplication, division all seemed to do well in that domain.

But then came the embarrassing questions that involved the square root of negative quantities and the brilliant "invention" of complex numbers (a + bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e. that a polynomial of degree n has n roots .. but the roots must be allowed to be complex.

The obvious question is "what next"? I.e. if we look at complex numbers at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that are needed for operations beyond polynomial equations?

This led me to think of linear algebra .. after all, there we are comfortable with n-tuples and we can apply any algebra we'd like to them (likely limiting them to be fields).

Wikipedia shows this:
http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers

which illustrates an interesting job of integrating complex numbers into matrix form, not surprising 2x2, although the matrices are the primitives in this algebra, not 2-tuples.

3D transforms do get us into quaternions which wikipedia
http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions

considers a generalization of complex numbers.

So the question is: are there higher order numbers beyond complex needed for algebraic operations? Naturally n-tuples show up in linear algebra, over the fields N,I,Q,Z and C. But are there "primitive" numbers beyond C that linear algebra, for example, might include?

What's next? And what does it resolve?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Arlo Barnes

Re: Complex Numbers .. the end of the line?

Actually, I can think of one application for which quaternions and such are not enough: 3D fractals. I will have to find the thread on fractalforums.com, but it describes the creation of the MandelBox and MandelBulb in accessible language but technical detail, as the story of an artist being unsatisfied with quaternions, which take Mandelbrot-type fractals and make them 'look like whipped cream', as they only preserve their ragged nature in one dimension, and the others are something like rotations. So the artist in question figured out something that looked more Mandelbrottish, I forget how, but it might be interesting.

-Arlo

Roger Frye-3

Re: Complex Numbers .. the end of the line?

In reply to this post by Owen Densmore

On Jan 23, 2012, at 5:38 PM, Owen Densmore wrote:
> The obvious question is "what next"? I.e. if we look at complex numbers at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that are needed for operations beyond polynomial equations?

The Fundamental Theorem of Algebra states that complex numbers suffice. But that only means if you all you need is to do is express the solutions of polynomial equations. Abel showed that they do not suffice to solve quintics. Trigonometric functions allow easy solution of cubic equations with real roots, and Ramanujan used theta functions extensively.

Hamilton felt the need for quaternions, which are convenient for 3-D transformations. There are generalizations in many directions: hypergeometric functions, Hestenes geometric algebra, Carl pointed to Baez and octonions, which go on to Clifford Algebras. Penrose has long advocated spinors as fundamental. But conventional mathematical physics chose to generalize in the direction of linear operators and functional calculus.

Carl said it nicely as
> Suspect you are about to pop out of algebra and end up someplace else as interesting.

-Roger

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Owen Densmore

Re: Complex Numbers .. the end of the line?

Administrator

In reply to this post by Owen Densmore

Arlo:

...Would it not be better to say, "are there number(data?)-structures that provide for interesting algebras not yet considered?"

Yes indeed. I was fumbling for a way to say that but ran out of steam!

Roger Critchlow:

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Now that is interesting, and nice to know this is a broader conversation than I had known. GA's .. gotta look into them and their unification of complex numbers and vectors.

Roger/Carl:

Suspect you are about to pop out of algebra and end up someplace else as interesting.

As you say, I think this is the more fruitful approach.

All: The Cayley Dickson generalizations discussed in wikipedia: R C H O did present an "answer" in that there are successful numeric extensions, that complex numbers "are not alone". As much as I wish computer graphics had used them for their transformations rather than 4-tuples (homogeneous coordinates) and 4-matrices, I'm not sure just how quaternions differ in theory from linear algebra, which simply started in on generalized n-tuples.

In other words, simple n-tuple algebras might have put all these generalizations from R into a single framework. Why *aren't* complex numbers simply our first use of 2-tuples, unified with the rest of linear algebra. Possibly the answer is that, yes linear algebras uses n-tuples, but they focus on very different matters such as linear independence, spanning sets, projections, subspaces, null spaces and so on.

Fun! So now I hope I can find some interesting problems that ONLY can be handled with some of these non linear algebraic higher number systems. Interestingly enough, I believe all of the extensions mentioned, as well as all of linear algebra, have the same cardinality .. the continuum, right?

Thanks for the insights,

-- Owen

Bruce Sherwood

Re: Complex Numbers .. the end of the line?

In reply to this post by Roger Critchlow-2

This link to an Oersted Medal talk is indeed of great interest. The
author, the theoretical physicist David Hestenes, built on the
foundation laid by mathematicians in the 19th century and in an
important sense completed their work on what is called "Geometric
Algebra", a framework which unifies much of the math done by
physicists, by providing geometric representations in all areas that
complement the algebra.

An analogy: The introduction of vectors by Gibbs made many things
easier to do and to say. Not only were many things easier to do, more
importantly the vector concept provided powerful new ways of thinking.
GA is like that. Some things that are very effortful with vectors
become much easier with GA, but more importantly it opens up new ways
of thinking and, as mentioned above, unifies many maths (plural) that
are usually seen as completely separate. Incidentally, Hestenes feels
that it's unfortunate that Gibbs took a piece out of GA and missed the
full point, but it's only the Gibbs vectors that most physicists know
about. For example, in the Gibbs view there are two kinds of vectors,
the regular kind and "axial" vectors. In GA there's only one kind of
vector; what has been called an "axial" vector is actually a 2D
"bivector" representing a planar element whose magnitude is its area.
An example is the cross product of two vectors.

For me, a striking example of the unifying power of GA is this: The
Pauli spin matrices were taught to me as special 2x2 matrices, special
to quantum mechanics, for describing the spin state of an electron. In
the GA framework, these matrices pop out as just a natural part of
living in a 3D world! Nothing particularly to do with quantum
mechanics! Stunning.

There are additional GA links on my home page, http://www4.ncsu.edu/~basherwo/.

I hasten to say that I am alas not an expert on GA, just a fan
observing from a distance. Also, I've been told that something called
"differential forms" has much of the same flavor and power, and I know
absolutely nothing about that.

Bruce

On Mon, Jan 23, 2012 at 11:43 PM, Roger Critchlow <[hidden email]> wrote:
> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
>
> -- rec --
>

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Frank Wimberly

Re: Complex Numbers .. the end of the line?

Differential forms are (covariant) tensors. That is they are multi-linear
functionals defined on n-tuples of vectors. I wonder if tensor analysis
provides a framework for many of the mathematical concepts discussed in this
thread.

Frank
---
Frank C. Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505

[hidden email] [hidden email]
505 995-8715 (home) 505 670-9918 (cell)

-----Original Message-----
From: [hidden email] [mailto:[hidden email]] On Behalf
Of Bruce Sherwood
Sent: Tuesday, January 24, 2012 10:12 AM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Complex Numbers .. the end of the line?

This link to an Oersted Medal talk is indeed of great interest. The
author, the theoretical physicist David Hestenes, built on the
foundation laid by mathematicians in the 19th century and in an
important sense completed their work on what is called "Geometric
Algebra", a framework which unifies much of the math done by
physicists, by providing geometric representations in all areas that
complement the algebra.

An analogy: The introduction of vectors by Gibbs made many things
easier to do and to say. Not only were many things easier to do, more
importantly the vector concept provided powerful new ways of thinking.
GA is like that. Some things that are very effortful with vectors
become much easier with GA, but more importantly it opens up new ways
of thinking and, as mentioned above, unifies many maths (plural) that
are usually seen as completely separate. Incidentally, Hestenes feels
that it's unfortunate that Gibbs took a piece out of GA and missed the
full point, but it's only the Gibbs vectors that most physicists know
about. For example, in the Gibbs view there are two kinds of vectors,
the regular kind and "axial" vectors. In GA there's only one kind of
vector; what has been called an "axial" vector is actually a 2D
"bivector" representing a planar element whose magnitude is its area.
An example is the cross product of two vectors.

For me, a striking example of the unifying power of GA is this: The
Pauli spin matrices were taught to me as special 2x2 matrices, special
to quantum mechanics, for describing the spin state of an electron. In
the GA framework, these matrices pop out as just a natural part of
living in a 3D world! Nothing particularly to do with quantum
mechanics! Stunning.

There are additional GA links on my home page,
http://www4.ncsu.edu/~basherwo/.

I hasten to say that I am alas not an expert on GA, just a fan
observing from a distance. Also, I've been told that something called
"differential forms" has much of the same flavor and power, and I know
absolutely nothing about that.

Bruce

On Mon, Jan 23, 2012 at 11:43 PM, Roger Critchlow <[hidden email]> wrote:
> http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
>
> -- rec --
>

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Joshua Thorp

Re: Complex Numbers .. the end of the line?

In reply to this post by Roger Critchlow-2

Thanks Roger, interesting paper.

I have always been fascinated at the relationship between the language of a mathematics and corresponding science that can be described with it.

--joshua

On Jan 23, 2012, at 11:43 PM, Roger Critchlow wrote:

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

-- rec --

On Mon, Jan 23, 2012 at 5:38 PM, Owen Densmore <[hidden email]> wrote:

Integers, Rationals, Reals .. these scalars seemed to be enough for quite a while. Addition, subtraction, multiplication, division all seemed to do well in that domain.

But then came the embarrassing questions that involved the square root of negative quantities and the brilliant "invention" of complex numbers (a + bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e. that a polynomial of degree n has n roots .. but the roots must be allowed to be complex.

The obvious question is "what next"? I.e. if we look at complex numbers at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that are needed for operations beyond polynomial equations?

This led me to think of linear algebra .. after all, there we are comfortable with n-tuples and we can apply any algebra we'd like to them (likely limiting them to be fields).

Wikipedia shows this:
http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers

which illustrates an interesting job of integrating complex numbers into matrix form, not surprising 2x2, although the matrices are the primitives in this algebra, not 2-tuples.

3D transforms do get us into quaternions which wikipedia
http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions

considers a generalization of complex numbers.

So the question is: are there higher order numbers beyond complex needed for algebraic operations? Naturally n-tuples show up in linear algebra, over the fields N,I,Q,Z and C. But are there "primitive" numbers beyond C that linear algebra, for example, might include?

What's next? And what does it resolve?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Russ Abbott

Re: Complex Numbers .. the end of the line?

This is way outside my area of competence -- to the extent that I still have one -- but I remember reading about Conway's Surreal numbers, which may be of interest.

-- Russ

On Tue, Jan 24, 2012 at 10:21 AM, Joshua Thorp <[hidden email]> wrote:

Thanks Roger, interesting paper.

I have always been fascinated at the relationship between the language of a mathematics and corresponding science that can be described with it.

--joshua

On Jan 23, 2012, at 11:43 PM, Roger Critchlow wrote:

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

-- rec --

On Mon, Jan 23, 2012 at 5:38 PM, Owen Densmore <[hidden email]> wrote:

Integers, Rationals, Reals .. these scalars seemed to be enough for quite a while. Addition, subtraction, multiplication, division all seemed to do well in that domain.

But then came the embarrassing questions that involved the square root of negative quantities and the brilliant "invention" of complex numbers (a + bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e. that a polynomial of degree n has n roots .. but the roots must be allowed to be complex.

The obvious question is "what next"? I.e. if we look at complex numbers at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that are needed for operations beyond polynomial equations?

This led me to think of linear algebra .. after all, there we are comfortable with n-tuples and we can apply any algebra we'd like to them (likely limiting them to be fields).

Wikipedia shows this:
http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers

which illustrates an interesting job of integrating complex numbers into matrix form, not surprising 2x2, although the matrices are the primitives in this algebra, not 2-tuples.

3D transforms do get us into quaternions which wikipedia
http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions

considers a generalization of complex numbers.

So the question is: are there higher order numbers beyond complex needed for algebraic operations? Naturally n-tuples show up in linear algebra, over the fields N,I,Q,Z and C. But are there "primitive" numbers beyond C that linear algebra, for example, might include?

What's next? And what does it resolve?

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Frank Wimberly

Re: Complex Numbers .. the end of the line?

In reply to this post by Owen Densmore

This is a message from Dean Gerber. For some reason it didn’t reach the List when he sent it. I forward it at his request. I will certainly attend the lecture he offers.

Algebras

Owen--
I think what you are looking for is the theory of algebras, generally known as non-associative algebras. These structures are vector spaces V(F) defined over a field of scalars F satisfying the usual axioms of a vector space with respect to the operations of vector addition and scalar multiplication, and an additional binary operation of vector multiplication (called a product) that is distributive with respect the vector space operations. To be specific, let x,y,z be vectors in V(F), let a,b be scalars in F, and denote the product of x and y by x!y. Then V(F) and the product (!) define an algebra if and only if

i) x!y is in V. (x!y is a vector - the very meaning of "binary composition")
ii) a(x!y) = (ax)!y = x!(ay). ( Scalar multiplication distributes with vector multiplication)
iii) x!(y + z) = x!y + x!z and (y + z)!x = y!x + z!x. (Vector multiplication distributes with vector addition)

Since a vector space is always equivalent to a set of tuples, this provides the multiplication of tuples you are looking for. For an n-dimensional vector space the generic (general) product is completely defined by n-cubed parameters, known as the structure constants. Specific choices of these parameters from the field F define specific algebras and the properties of these algebras vary greatly over the possible choices. For example, for n = 2 there are 8 free parameters and the complex numbers represent a single point in this 8 dimensional space of structure constants. That particular choice implies that the algebra of complex numbers is itself field. Generally, algebras have no properties other than i) to iii) above, i.e. they are generally not commutative or associative.

The Caley-Dickson procedure is a process by which a "normed" algebra can be extended to a normed algebra of twice the dimension. The only real one dimensional normed algebra is the real number field itself. The Caley-Dickson extension is just the complex numbers as a 2 dimensional algebra, and it is also a field; in fact the only 2 dimensional field..

The Caley-Dickson extension of the complex number algebra is the 4 dimensional quaternion algebra. But, the quaternions are NOT a field: they are not commutative even though they are associative and a division algebra. They are often known as a "skew" field,

The Caley-Dickson extension of the quaternions is the 8 dimensional octonion algebra, and these are neither commutative or associative, but they are a division algebra.

The next step gives the nonions of dimension 16 at which point we lose the last semblance of a field because they are not commutative, not associative, and not a division algebra. Thus, if we want fields, the complex numbers are indeed the end point. All real division algebras are of dimensions 1,2,4, or 8! There are many division algebras in the dimensions 2,4,8, but only in n = 2 are all of them classified up to isomorphism.

I could go on, if you could gather up an audience of at least ten for a (free) three hour blackboard lecture with two breaks. For an audience of fewer than ten I would have to collect ten hours of Santa Fe minimum wages for prep and lecture time. Its a beautiful subject with a very colorful history, and includes the quaternions, octonians, Lie algebras, Jordan algebras, associative algebras, everything mentioned by the FRIAM commentariat.

Regards- Dean Gerber

From: [hidden email] [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, January 24, 2012 9:34 AM
To: Complexity Coffee Group
Subject: Re: [FRIAM] Complex Numbers .. the end of the line?

Arlo:

...Would it not be better to say, "are there number(data?)-structures that provide for interesting algebras not yet considered?"

Yes indeed. I was fumbling for a way to say that but ran out of steam!

Roger Critchlow:

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Now that is interesting, and nice to know this is a broader conversation than I had known. GA's .. gotta look into them and their unification of complex numbers and vectors.

Roger/Carl:

Suspect you are about to pop out of algebra and end up someplace else as interesting.

As you say, I think this is the more fruitful approach.

Thanks for the insights,

-- Owen

Rich Murray-2

Re: Complex Numbers .. the end of the line?

With only an intuitive, skating on soap bubble films, grasp, I still
enjoyed reading all these posts -- look forward to some kind of
computer interactive game learning process to convey the widest most
comprehensive framework to unify all these partial frameworks -- I
suspect it will have to intimately include a fractal reality in which
every tiny region is in one-to-one correspondence with the whole --
since I experience that each of us is all of single entire unified
creative hyperinfinity -- Rich Murray, Imperial Beach, CA 91932

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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Grant Holland

Re: Complex Numbers .. the end of the line?

In reply to this post by Frank Wimberly

Dean, Frank, Owen,

That would be 3 hours delightfully spent. Sign me up.

Thanks! -
Grant

On 1/24/12 8:21 PM, Frank Wimberly wrote:

This is a message from Dean Gerber. For some reason it didn’t reach the List when he sent it. I forward it at his request. I will certainly attend the lecture he offers.

Algebras

Owen--
I think what you are looking for is the theory of algebras, generally known as non-associative algebras. These structures are vector spaces V(F) defined over a field of scalars F satisfying the usual axioms of a vector space with respect to the operations of vector addition and scalar multiplication, and an additional binary operation of vector multiplication (called a product) that is distributive with respect the vector space operations. To be specific, let x,y,z be vectors in V(F), let a,b be scalars in F, and denote the product of x and y by x!y. Then V(F) and the product (!) define an algebra if and only if

i) x!y is in V. (x!y is a vector - the very meaning of "binary composition")
ii) a(x!y) = (ax)!y = x!(ay). ( Scalar multiplication distributes with vector multiplication)
iii) x!(y + z) = x!y + x!z and (y + z)!x = y!x + z!x. (Vector multiplication distributes with vector addition)

Since a vector space is always equivalent to a set of tuples, this provides the multiplication of tuples you are looking for. For an n-dimensional vector space the generic (general) product is completely defined by n-cubed parameters, known as the structure constants. Specific choices of these parameters from the field F define specific algebras and the properties of these algebras vary greatly over the possible choices. For example, for n = 2 there are 8 free parameters and the complex numbers represent a single point in this 8 dimensional space of structure constants. That particular choice implies that the algebra of complex numbers is itself field. Generally, algebras have no properties other than i) to iii) above, i.e. they are generally not commutative or associative.

The Caley-Dickson procedure is a process by which a "normed" algebra can be extended to a normed algebra of twice the dimension. The only real one dimensional normed algebra is the real number field itself. The Caley-Dickson extension is just the complex numbers as a 2 dimensional algebra, and it is also a field; in fact the only 2 dimensional field..

The Caley-Dickson extension of the complex number algebra is the 4 dimensional quaternion algebra. But, the quaternions are NOT a field: they are not commutative even though they are associative and a division algebra. They are often known as a "skew" field,

The Caley-Dickson extension of the quaternions is the 8 dimensional octonion algebra, and these are neither commutative or associative, but they are a division algebra.

The next step gives the nonions of dimension 16 at which point we lose the last semblance of a field because they are not commutative, not associative, and not a division algebra. Thus, if we want fields, the complex numbers are indeed the end point. All real division algebras are of dimensions 1,2,4, or 8! There are many division algebras in the dimensions 2,4,8, but only in n = 2 are all of them classified up to isomorphism.

I could go on, if you could gather up an audience of at least ten for a (free) three hour blackboard lecture with two breaks. For an audience of fewer than ten I would have to collect ten hours of Santa Fe minimum wages for prep and lecture time. Its a beautiful subject with a very colorful history, and includes the quaternions, octonians, Lie algebras, Jordan algebras, associative algebras, everything mentioned by the FRIAM commentariat.

Regards- Dean Gerber

From: [hidden email] [[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, January 24, 2012 9:34 AM
To: Complexity Coffee Group
Subject: Re: [FRIAM] Complex Numbers .. the end of the line?

Arlo:

...Would it not be better to say, "are there number(data?)-structures that provide for interesting algebras not yet considered?"

Yes indeed. I was fumbling for a way to say that but ran out of steam!

Roger Critchlow:

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Now that is interesting, and nice to know this is a broader conversation than I had known. GA's .. gotta look into them and their unification of complex numbers and vectors.

Roger/Carl:

Suspect you are about to pop out of algebra and end up someplace else as interesting.

As you say, I think this is the more fruitful approach.

All: The Cayley Dickson generalizations discussed in wikipedia: R C H O did present an "answer" in that there are successful numeric extensions, that complex numbers "are not alone". As much as I wish computer graphics had used them for their transformations rather than 4-tuples (homogeneous coordinates) and 4-matrices, I'm not sure just how quaternions differ in theory from linear algebra, which simply started in on generalized n-tuples.

In other words, simple n-tuple algebras might have put all these generalizations from R into a single framework. Why *aren't* complex numbers simply our first use of 2-tuples, unified with the rest of linear algebra. Possibly the answer is that, yes linear algebras uses n-tuples, but they focus on very different matters such as linear independence, spanning sets, projections, subspaces, null spaces and so on.

Fun! So now I hope I can find some interesting problems that ONLY can be handled with some of these non linear algebraic higher number systems. Interestingly enough, I believe all of the extensions mentioned, as well as all of linear algebra, have the same cardinality .. the continuum, right?

Thanks for the insights,

-- Owen
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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Owen Densmore

Re: Complex Numbers .. the end of the line?

Administrator

In reply to this post by Frank Wimberly

+1! I'd love to attend. Lets see if we can reach quorum.

Dean may not be on the list (he may be on wedtech, discuss, or sfx_graphics instead), or he may have multiple email addresses and sent from one not associated with his friam membership. That's the typical problem getting bounces from the list.

-- Owen

On Tue, Jan 24, 2012 at 8:21 PM, Frank Wimberly <[hidden email]> wrote:

This is a message from Dean Gerber. For some reason it didn’t reach the List when he sent it. I forward it at his request. I will certainly attend the lecture he offers.

Algebras

Owen--
I think what you are looking for is the theory of algebras, generally known as non-associative algebras. These structures are vector spaces V(F) defined over a field of scalars F satisfying the usual axioms of a vector space with respect to the operations of vector addition and scalar multiplication, and an additional binary operation of vector multiplication (called a product) that is distributive with respect the vector space operations. To be specific, let x,y,z be vectors in V(F), let a,b be scalars in F, and denote the product of x and y by x!y. Then V(F) and the product (!) define an algebra if and only if

i) x!y is in V. (x!y is a vector - the very meaning of "binary composition")
ii) a(x!y) = (ax)!y = x!(ay). ( Scalar multiplication distributes with vector multiplication)
iii) x!(y + z) = x!y + x!z and (y + z)!x = y!x + z!x. (Vector multiplication distributes with vector addition)

Since a vector space is always equivalent to a set of tuples, this provides the multiplication of tuples you are looking for. For an n-dimensional vector space the generic (general) product is completely defined by n-cubed parameters, known as the structure constants. Specific choices of these parameters from the field F define specific algebras and the properties of these algebras vary greatly over the possible choices. For example, for n = 2 there are 8 free parameters and the complex numbers represent a single point in this 8 dimensional space of structure constants. That particular choice implies that the algebra of complex numbers is itself field. Generally, algebras have no properties other than i) to iii) above, i.e. they are generally not commutative or associative.

The Caley-Dickson procedure is a process by which a "normed" algebra can be extended to a normed algebra of twice the dimension. The only real one dimensional normed algebra is the real number field itself. The Caley-Dickson extension is just the complex numbers as a 2 dimensional algebra, and it is also a field; in fact the only 2 dimensional field..

The Caley-Dickson extension of the complex number algebra is the 4 dimensional quaternion algebra. But, the quaternions are NOT a field: they are not commutative even though they are associative and a division algebra. They are often known as a "skew" field,

The Caley-Dickson extension of the quaternions is the 8 dimensional octonion algebra, and these are neither commutative or associative, but they are a division algebra.

The next step gives the nonions of dimension 16 at which point we lose the last semblance of a field because they are not commutative, not associative, and not a division algebra. Thus, if we want fields, the complex numbers are indeed the end point. All real division algebras are of dimensions 1,2,4, or 8! There are many division algebras in the dimensions 2,4,8, but only in n = 2 are all of them classified up to isomorphism.

I could go on, if you could gather up an audience of at least ten for a (free) three hour blackboard lecture with two breaks. For an audience of fewer than ten I would have to collect ten hours of Santa Fe minimum wages for prep and lecture time. Its a beautiful subject with a very colorful history, and includes the quaternions, octonians, Lie algebras, Jordan algebras, associative algebras, everything mentioned by the FRIAM commentariat.

Regards- Dean Gerber

From: [hidden email] [mailto:[hidden email]] On Behalf Of Owen Densmore
Sent: Tuesday, January 24, 2012 9:34 AM
To: Complexity Coffee Group

Subject: Re: [FRIAM] Complex Numbers .. the end of the line?

Arlo:
...Would it not be better to say, "are there number(data?)-structures that provide for interesting algebras not yet considered?"

Yes indeed. I was fumbling for a way to say that but ran out of steam!

Roger Critchlow:
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Now that is interesting, and nice to know this is a broader conversation than I had known. GA's .. gotta look into them and their unification of complex numbers and vectors.

Roger/Carl:

Suspect you are about to pop out of algebra and end up someplace else as interesting.
As you say, I think this is the more fruitful approach.

All: The Cayley Dickson generalizations discussed in wikipedia: R C H O did present an "answer" in that there are successful numeric extensions, that complex numbers "are not alone". As much as I wish computer graphics had used them for their transformations rather than 4-tuples (homogeneous coordinates) and 4-matrices, I'm not sure just how quaternions differ in theory from linear algebra, which simply started in on generalized n-tuples.

In other words, simple n-tuple algebras might have put all these generalizations from R into a single framework. Why *aren't* complex numbers simply our first use of 2-tuples, unified with the rest of linear algebra. Possibly the answer is that, yes linear algebras uses n-tuples, but they focus on very different matters such as linear independence, spanning sets, projections, subspaces, null spaces and so on.

Fun! So now I hope I can find some interesting problems that ONLY can be handled with some of these non linear algebraic higher number systems. Interestingly enough, I believe all of the extensions mentioned, as well as all of linear algebra, have the same cardinality .. the continuum, right?

Thanks for the insights,

-- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org