Math Newsletter number 10;Wednesday, September 29, 2010.
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Vector Numbers

In Geometry, a vector is a quantity that has magnitude and
direction.

In algebra, the components of a finite dimensional vector is
a list of numbers for which addition, and one or more
multiplications have been defined. Usually, an algebraic
formula for the magnitude of the vector is also defined.

Example 1:
Define A vector components to be a list of two elements,
[a,b], where "a" and "b" are non-negative integers.

Define addition within a set of these vectors by

[a1,b1] + [a2,b2] = [a1+a2,b1+b2].

Define multiplication on this set by

[a1,b1] * [a2,b2] = [a1 a2 + b1 b2, a1 b2 + b1 a2 ]

What is the zero element of this vector arithmetic.

Clearly [0,0] is a zero element since,
[a,b] + [0,0] = [a+0,b+0] = [a,b]
and
[0,0] + [a,b] = [0+a,0+b] = [a,b].

We may write [0,0] = 0, where 0 is the additive identity.
The result of adding 0 to a quantity is equal to the
quantity.

What is the multiplicative identity of this vector
arithmetic?

[a,b] * [c,d] = [a,b]
[a c + b d, a d + b c] = [a,b]
a c + b d = a
a d + b c = b

Suggests that c = 1 and d = 0.

[1,0] is a multiplicative identity of this vector arithmetic.

We may write, [1,0] = 1, where 1 symbolizes the
multiplicative identity. The result of multiplying a
quantity by 1 is equal to the quantity.

Are there negative numbers in this vector arithmetic?

[0,1] + [1,0] = [1,1]

With the rules we have so far set up, there is not any way to
get
negative numbers in this vector arithmetic. We had specified
that the components never be negative.

However, we can make one simple rule that changes this.

We make the rule that for every non-negative integer, b,
that [b,b] = [0,0].

It has been proven in abstract algebra classes, for this
particular set of vectors, that this rule, [b,b] = [0,0] is
safe. It will not give rise to contradictions.

With this rule, [0,1] + [1,0] = [1,1] = [0,0].

Thus [0,1] = -1.

Example 2:
Define a vector components to be a list of two elements, [a,b], where
"a" and "b" are integers.

Define addition within a set of these vectors by:

[a1,b1] + [a2,b2] = [a1 b2 + b1 a2, b1 b2].

Define multiplication within a set of these vectors by:

[a1,b1] * [a2,b2] = [a1 a2, b1 b2]

[a,b] + [c,d] = [a,b]

[a d + b c,b d] = [a,b]

b d = b ==> d = 1.
a d + b c = b and d = 1
--> a + b c = a --> b c = 0

Since b represents an arbitrary number,
and b c = 0, it must be that c = 0.

We may identify [0,1] with 0

[0,1] = 0

What is the multiplicative identity?

Multiplying by the multiplicative identity
is the same as not multiplying.

[a,b] * [c,d] = [a,b]

a c = a
b d = b

c = 1
d = 1

The multiplicative identity is [1,1].

We may identify [1,1] with 1.

[1,1] = 1.

In abstract algebra, it is proved that we could, without

for each non zero integer b,

[b,b] = [1,1] =1

We identify 1 with [1,1].
Thus 2 is identified with
[1,1] + [1,1] = [1*1 + 1 * 1, 1 * 1] = [2,1]

We Defined multiplication within a set of these vectors by:

[a1,b1] * [a2,b2] = [a1 a2, b1 b2]

Note that

[2,1] * [1,2] = [2*1, 1*2] = [2,2] = [1,1].

2 * [1,2] = 1

Thus we may identify [1,2] with 1/2.

We have demonstrated a model for the rational numbers in
terms of integer vectors.

For integers b and c, [b,c] is identified with b/c.

Example 3:
Define a vector components to be a list of three elements,
[x,y,z], where x,y, and z are real numbers.

[x1,y1,z1] + [x2,y2,z2] = [x1+x2,y1+y2,z1+z2]

Define two multiplications.

[x1,y1,z1].dot.[x2,y2,z2] = x1 x2 + y1 y2 + z1 z2

[x1,y1,z1].cross.[x2,y2,z2]
=[y1 z2 - z1 y2, z1 x2 - x1 z2, x1 y2 - y1 x2]

This implements the classical vector analysis arithmetic,
much used in physics and engineering.

Example 4; Define a vector components to be a list of two
elements, [A,B], where A and B are numbers.

Let d be some constant number.

[A1,B1] + [A2,B2] = [A1+A2, B1+B2]

Define multiplication by

[A1,B1] * [A2,B2]
= [A1 A2 + d B1 B2, A1 B2 + B1 A2 ]

We define the magnitude of
[A,B] to be equal to

sqrt([A,B]*[A,-B])
= sqrt([A**2 - d B**2, A B - B A])

If A and B are integers, or real numbers, or some subset
of the complex numbers, then A B - B A = 0.

We may identify

[A**2 - d B**2,0] with (A**2 - d B**2).

Note that if d is positive, then

A**2 - d B**2 might be negative.

In this vector arithmetic, the magnitude might be the square
root of a negative number.

As usual, when it's possible to define consistently a
magnitude,
we automatically get the rule,

The magnitude of the product is the product of the
magnitudes.

Since square roots are not convenient to handle, we define,

Norm([A,B]) = square of magnitude([A,B])

Norm([A,B]) = A**2 - d B**2.

This vector arithmetic is unusual in that the Norm might be
negative.

We may apply our product rule, the Norm of the product is the
product of the Norms.

[A,B] * [A,B] = [A**2 + d B**2, 2 A B]
Norm([A,B]) = A**2 - d B**2

Norm of product
= Norm([A**2 + d B**2, 2 A B])
= (A**2 + d B**2)**2 - d (2 A B)**2

Product of Norms
= (A**2 - d B**2)**2

Yielding the identity, for all numbers d,A, and B,
(A**2 + d B**2)**2 - d (2 A B)**2 = (A**2 - d B**2)**2