**What is an imaginary number?**

*by
Eckhard Hitzer*

*Department
of Physical Engineering, University
of Fukui, Japan*

December
2004

The
previous Japanese
emperor is said to have asked this question. Today many students and
scientists
still ask it, but the traditional canon of mathematics at school and
university
needs to be widened for the answer. We find it in the works of
Hamilton,
Grassmann and Clifford. Hamilton introduced quaternions i,j,k, with

i^{2}= j^{2}= k^{2}= ijk = - 1

and

ij = - ji = k, jk
= - kj = i, ki = - ik = j

for
3D rotations. Grassmann invented the
outer product of oriented line segments (vectors) **a**,**b** to give the
directed oriented area of the enclosed parallelogram:

**a**Θ**b **= - **b**Θ**a**.

Clifford
unified their work with the
geometric product

**ab = a**E**b **+ **a**Θ**b**,

leading
to geometric algebras.

In
two dimensions we
have orthogonal, unit vectors **e**_{1},**e**_{2} as
vector space basis with

**e**_{1}^{2}= **e**_{2}^{2}=
1, **e**_{1}E**e**_{2}_{} = 0.

The
associative geometric multiplication
of the oriented directed unit square

i =** e**_{1}**e**_{2}

gives:

ii = **e**_{1}**e**_{2}**e**_{1}**e**_{2}
= **e**_{1}(**e**_{2}**e**_{1})**e**_{2}
= **e**_{1}(**e**_{2}E**e**_{1}_{}+**e**_{2}Θ**e**_{1}_{})**e**_{2} =

= **e**_{1}(0- **e**_{1}Θ**e**_{2}_{})**e**_{2} = - **e**_{1}**e**_{1}**e**_{2}**e**_{2}
= - 1.

NB:
We only used** **

**e**_{2}_{}E**e**_{1}_{} =** e**_{1}E**e**_{2}_{} = 0 and **e**_{2}**e**_{1} = **e**_{2}Θ**e**_{1}_{} = - **e**_{1}Θ**e**_{2}_{} = - **e**_{1}**e**_{2}.

So
the square of the oriented unit area i
is - 1. Enough to satisfy the emperorfs curiosity!

But
todayfs
politicians ask for an application. As an answer we calculate:

i** e**_{1} =** e**_{1}**e**_{2}**e**_{1}
= - **e**_{1}**e**_{1}**e**_{2} = - **e**_{2}

and

i** e**_{2}
= **e**_{1}**e**_{2}**e**_{2} = **e**_{1},

which
is a clockwise 90rotation.
We can also
calculate (NB: the order!)

**e**_{1}i = **e**_{1}**e**_{1}**e**_{2}
= **e**_{1}

and

**e**_{2}_{}i = **e**_{2}**e**_{1}**e**_{2}
= - **e**_{1}**e**_{2}**e**_{2}
= - **e**_{1},

which
is an anticlockwise (mathematically
positiv) 90rotation.
For a general rotation in two dimensions, we simply add trigonometric
coefficients:

**a **( cos(alpha) + i sin(alpha) )

rotates
the real vector **a **by
alpha degrees.
Now even a politician can rotate vectors without using (or even
knowing)
matrices.

Of
what use may the
geometric product be for some new advanced technology venture business?
As an
application to *laser
beam
optics*
let us imagine a laser beam with
direction vector **a** hitting a
mirror surface element approximated with
unit normal vector **n** (**n**^{2}=1). We can write **a** in
components parallel and perpendicular to **n**:

**a **= **a _{|}**

Now

**a _{||}**

because
parallel vectors span no
parallelogram, and

**a**_{Ϋ}E**n **= 0,

because
of perpendicularity. So we must
have

**a _{||}**

and

**a**_{Ϋ}**n **=** **0 **+ a**_{Ϋ}Θ**n **= 0 - **n**Θ**a**_{Ϋ}
= - **na**_{Ϋ}.

Reflection
only changes the sign of **a**_{||}.
Therefore

**a**f = - **a _{||}**

= - **n**(**a _{||}**

is
the reflected vector. In a cavity we
may want to trace many reflections at a sequence of surface elements
with
normal vectors **n**_{1},** n**_{2},** **... **n**_{s}
which simply results in

**a**f=(- 1)^{s}** n**_{s }... **n**_{2}**n**_{1}**an**_{1}**n**_{2
}... **n**_{s}.

*Nanoscience*
is a modern buzz
word. On this scale mechanics meets quantum mechanics. Geometric
algebra
provides complete tools for both. From elementary geometry we know that
two
reflections at planes with normal vectors **n**,**m **enclosing the angle theta/2 result
in a rotation
by angle theta:

**a**f= **mn a nm**.

The
general *rot*ation
operat*or*
(*rotor*)
is

R = **nm **= **n**E**m **+ **n**Θ**m **= cos(theta/2) + i sin(theta/2) = exp(i
theta/2)

with
unit area element i in
the **n**,**m
**rotation**
**plane.
Two rotations are given by the geometric product of two rotors RRf. A second thetaf= 360=
2pi rotation
poduces

RRf = R exp(2pi i
/2) = R exp(pi i) = R (cos(pi)+i sin(pi))= R(- 1+i0)= - R.

The
rotor R itself behaves therefore like
the first known *quantum
particle*,
i.e. the electron
described by a Pauli spinor

y=
rho^{1/2} R.

Geometric
algebra
answers fundamental questions, which the traditional canon of
mathematics
taught at schools and universities canft. It further provides great
methodological simplifications and geometric insight in applications to
physics, molecular geometry, image processing, computer graphics,
robotics,
quantum computing, etc. Geometric algebra is an excellent candidate to
restructure mathematical syllabi on all (from school to post graduate)
levels. *I
propose therefore to establish a research
institute, dedicated to further develop geometric calculus (with
geometric
algebra as mathematical grammar) as a general tool for teaching,
research and
application.*

Geometric
Calculus - Research &
Development

http://modelingnts.la.asu.edu/GC_R&D.html

Cambridge
University Geometric Algebra
Research Group

http://www.mrao.cam.ac.uk/~clifford/

Geometric
Calculus in Fukui (Japan)

http://sinai.mech.fukui-u.ac.jp/gcj/gcjportal.html

Geometric
Calculus International

http://sinai.mech.fukui-u.ac.jp/gcj/gc_int.html

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