FOM: Complex i.
Dana_Scott@POP.CS.CMU.EDU
Dana_Scott at POP.CS.CMU.EDU
Wed Dec 17 21:12:59 EST 1997
I would like to add comments to one of the threads in this discussion
group.
(1) COMPLEX NUMBERS. Is the imaginary unit i distinguishable from -i?
Since there is an automorphism of the complex field taking one to the
other, as was remarked, there is clearly no algebraic distinction
possible. It gets even worse when one wants of pick ONE of the roots
of a higher-degree equation. The complex field cannot be made into an
ordered field, and so picking one element in an algebraically
meaningful way from a finite set seems confusing.
The proper answer has to do with orientation in geometry. Euclidean
geometry (of dimension 2, say) can be axiomatized, and, in the usual
form, there is no prefered orientation to the Euclidean plane.
Indeed, after suitable definitions, one proves there are two possible
orientations. The group of rigid motions has two components: those
that preserve orientation, and those that reverse it.
Based on Euclidean geometry, vectors can be introduced as equivalence
classes of ordered segments, and it is clear that THERE IS NO
PREFERRED BASIS for the resulting vector space implied. (The i vs. -i
problem is also related to the question: Can you give me a vector
space without giving me a basis?)
Next, complex numbers can be introduced geometrically as RATIOS of
plane vectors, say, a/b, where b is nonnull. Equivalence of ratios
means that we have a/b = c/d iff there is a rotation and a similarity
taking the triangle 0ab to the triangle 0cd -- with the vertices in
the order indicated. We then have at least one preferred unit among
ratios because a/a = b/b for any two nonnull vectors. Call this ratio
1. The ratio 0/a is constant as well. Call this 0.
Now if a and b are PERENDICULAR and congruent, and if the same goes
for c and d, then either a/b = c/d or a/b = -c/d. These ratios are
going to be the i or -i elements. But what does perpendicularity have
to do with the imaginary unit?
Well, we have to introduce algebra. Vectors can only be added and
given SCALAR multiples (by reals). Ratios, however, can be
multiplied. Indeed, we can essentially determine what PRODUCT means
by the formula
a/b * b/c = a/c
At least this works when b is nonnull. We can put any "product" in
this form as long as the second factor is nonzero just by rotating and
scaling the representatives of the second factor. Of course we want
the equation z 0 = 0 to hold by definition for any ratio z.
As for ADDITION, it comes from vector addition, and it is determined
by the formula:
a/b + c/b = (a + c)/b,
since the second term can always be chosen to have a representative
with the same "denominator".
Now, admittedly, theorems have to be proved to show that these
definitions of the algebraic operations are independent of the choices
of the representatives, and that we get the usual laws of algebra.
But this is true, and these are INTERESTING GEOMETRIC FACTS. In
particular, the negatives of ratios go back to negatives of vectors,
and we have
-a/b = (-a)/b
as the defining equation. Then (-a)/a = a/(-a) = -1.
Suppose now that a and b are nonnull, perendicular and congruent.
Then, by elementary geometry, a/b = b/(-a). But, by the definition of
our algebra
a/b * a/b = a/b * b/(-a) = a/(-a) = -1.
In other words the ratios a/b and -a/b both solve the equation z^2 = -1.
Unless we are working in the ORIENTED PLANE, we cannot say whether the
rotation from a to b in the ratio above is in the positive or negative
direction. But if we choose one of the roots of negative unity, THEN
we have determined the orientation. So the choice is exactly the
choosing of an orientation. And you can choose if you want, but you
have to choose. There is a difference between geometry with a
preferred orientation and geometry without. (What difference? Well
the groups of allowed rigid motions will be different.)
(2) QUATERNIONS. There is some business about i,j, k, right? But
what are they? And how do we choose them?
For a geometrically conceptual definition of quaternion algebra use
THREE DIMENSIONAL VECTORS. (Note, it's 3 and not 4.) Quaternions are
ratios of space vectors. We just have to be careful in defining the
equivalence a/b = c/d. For DEPENDENT vectors, this becomes just a
REAL SCALAR. For INDEPENDENT vectors, there is a plane through the
origin determined, and the triangles 0ab and 0cd have to be IN THE
SAME PLANE. This is essential to the definition.
The key fact about Euclidean 3D is that two distinct planes through
the origin INTERSECT IN A LINE. This means that the definitions of
product and sum can be taken over from the complex case word for word,
since rotations of one pair of representing vectors can always be
chosen so that one nonnull vector lies in any given plane. (Imagine
two intersecting planes, and you will see why.) Note, however, that
the there are many more ratios here, and in the verification of the
laws of algebra, one gets an associative division algebra which is
NONCOMMUTATIVE. (In the complex case, the commutative law is an
expression of the one-dimensional character of the plane rotation
group.)
Suppose now that a and b are nonnull, perendicular and congruent.
Then again, a/b satisfies z^2 = -1. But a and b imply a PARTICULAR
PLANE. It is true that -a/b also satisfies this equation, but so does
a similar ratio in EVERY OTHER PLANE. And the ratios from different
planes are DIFFERENT. The equation z^2 = -1 has a CONTINUUM of
solutions in the quaternions! So now choose i, j, and k. Try. Go
ahead. Try!
Note that this insight also shows that there are infinitely many ways
of embedding the complex numbers into the quaternions. Each plane
through the origin gives two ways. This interesting fact is not at
all obvious (to me at least) in the usual ijk-definition of
quaternions by pure algebra!
Now the complexes form and two-dimensional vector space, and the
quaternions form a four-dimensional vector space. We get the
dimension by choosing a BASIS of congruent vectors a, b, and c for the
3D vectors and taking i = a/b, j = b/c, and k = a/c, in order to
respect the convention that i j = k. However, j i = -k, as can be
seen geometrically by thinking of the three perpendicular planes and
where the unit vectors sit in them. But we have to do some work to
see that every quaternion q can be written (uniquely) in the form
q = t 1 + u i + v j + w k
for real scalars t,u,v,w. But this can be done by working out the
norm formula for quaternions of this special form and the formula for
their algebraic inverses. (Hint: To work out the 1ijk-version of a
ratio
(x a + y b + z c)/(p a + r b + s c),
where a,b,c are the basis vectors of the coordinate planes, rewrite
this ratio as a sum of three quaternions, and then invert each of the
three terms. This shows how 1 gets mixed in with i,j, and k.)
We can also see that choosing these unit quaternions is chosing three
perpendicular 90-degree rotations, which is much more structure than
just choosing one 90-degree rotation to be the "positive" one in the
plane case of the complexes.
Algebra is a great way to do geometry, but algebra often conceals
making special choices, and we have to work hard sometimes to isolate
the true expressions of the GEOMETRIC INVARIANTS. The story of the
complexes and the quaternions is a fine example that shows that too
trusting a adoption of algebra is not always the best geometric guide.
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