Alternative to imaginary numbers?

In this video, starting at 3:45 the professor says

There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some structure that we can explain by other means.

What is the "other means" that he is referring to?

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I don't know what this could be referring to. In my experience the complex numbers help uncover structure in a problem, not conceal it. – Qiaochu Yuan Jun 19 '11 at 23:45
It's an engineer talking, so i wouldn't read too much into it. – Joe Jun 19 '11 at 23:46
I think that the professor who said that is Seamus Garvey. I recommend e-mailing him to ask. nottingham.ac.uk/engineering/people/seamus.garvey – Jonas Meyer Jun 20 '11 at 0:12
Thanks Jonas, I sent him an email. Hopefully I will get an interesting reply. I will post his reply here. – picakhu Jun 20 '11 at 0:25
@picakhu Why not simply refer him to this question so that he can reply here? – Bill Dubuque Jun 20 '11 at 0:44

I don't know what "other means" the fellow has in mind, but there are a couple of ways to do what complex numbers do without ever introducing imaginary units:

1. If you know about rings, ideals, and quotient rings, then you can work in ${\bf R}[x]/(x^2+1)$ which has an element, $x+(x^2+1)$, which does whatever you want your imaginary unit to do.

2. If you know about matrices, the set of all matrices of the form $$\pmatrix{a&b\cr -b&a\cr}$$ with $a,b$ real does everything you need, with $\pmatrix{0&1\cr-1&0\cr}$ playing the role of the imaginary unit.

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I guess it can't hurt to also mention the complex structure on arbitrary $2n$-dimensional real vector space where the role of the imaginary unit is played by an operator $J: V \to V$ such that $J^2 = {\rm Id}_V$. Of course, for $n=1$ this is nothing else than what $i \cdot$ does but still... – Marek Jun 20 '11 at 0:50
Make that $J^2 = -\text{Id}_V$. – Scott Carnahan Jun 20 '11 at 17:36

The following is his response.

Hi,

I was alluding to Clifford Algebra (some people call it geometric algebra). See a paper by Chris Doran, Stephen Gull and Anthony Lasenby with a title something like "Imaginary Numbers are not Real ..."

The complex numbers are a sub-algebra of the simplest of all Clifford Algebras, Cl_2. Moreover the "vector" nature of complex numbers is captured by the complementary sub-algebra.

SDG

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Far out, man! Complex numbers are discounted in favor of Clifford Algebra. – GEdgar Jun 20 '11 at 16:45
It really is true that Clifford algebra reveals the geometric structure better, even moreso at higher dimensions--for instance, the even sub-algebra in Cl_3 gives the quaternion representation for rotations in three dimensions in a very intuitive way. On the other hand, representing N-dimensional geometry requires a 2^N-dimensional Clifford algebra, so projective 3D coordinates, common in computer graphics, would require 16 coefficients, rather than the standard 4. – camccann Jun 21 '11 at 19:20
How cool: SDG responding! Thanks, @picakhu for pursuing this...Perhaps SDG would like to "drop in" and visit math.SE periodically... – amWhy Jun 26 '11 at 23:06

Seamus Garvey is making a lot of sense here and alluding to something pretty deep. Geometric algebra is a unifying concept that can seem like magic for someone who has never seen them before. It scoops up complex numbers, quarternions, exterior algebra, spinors, and host of other tools that previously seemed unrelated.

To see their use in physics check out:

and to see a mathematical approach that leads you down the spinor path check out:

or something a bit older and explicit is:

I can't stress enough how the Clifford Algebra concept brings it all together. One works with all these separate tools and you get a feeling that it's all related but it's not often that they're presented as such. Now don't get me wrong -- Clifford algebras are not a magic bullet and complex analysis will always be in your tool box. That's a fact. But to see how it all links up go the CA route.

You won't be sorry.

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Complex numbers are often great explainers and illuminators. Here is a canonical example. We have $${1\over 1 + x^2} = \sum_{k=0}^\infty (-1)^n x^{2n}.$$ A bright calc student will be prompted to ask, "What is the deal here? Why does the series suddenly stop converging at $\pm 1$? The function on the left-hand side is differentiable to any order on the entire line."

The complex plane reveals the answer. The function $f(z) = 1/(1 + z^2)$ has poles at $\pm i$. So, the distance from the center of the Taylor series to the place where it first has an analytical nasty (a pole here) is 1. All of a sudden, this mysterious "stoppage of convergence out of the blue" becomes an entirely natural phenomenon.

I fail to see merit in this guy's idea that complex numbers are somehow unnatural.

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Professor Garvey is not arguing against the value of complex numbers. I think it is premature to judge the merit of the papers being referred to without knowing what they are. This nice example was also given in a thread where it is more relevant: math.stackexchange.com/questions/4961/… – Jonas Meyer Jun 20 '11 at 0:34
Thanks, @Jonas...my point, as well, in comments preceding the OP. I credit Professor Garvey for being willing to admit that there are credible (even "superb"arguments) which challenge the necessity of imaginary numbers. – amWhy Jun 20 '11 at 0:47

Maybe he meant the following: A complex number $z$ is in the first place an element of the field ${\mathbb C}$ of complex numbers, and not an $a+bi$. There are indeed structure elements which remain hidden when thinking in terms of real and imaginary parts only, e.g., the multiplicative structure of the set of roots of unity.

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