23. Distributive Law for Real Numbers
Summary: The distributive law for real numbers
implies that changes of scale or direction in coordinate systems
determined the by selection of unit vectors in the addition of vectors
in the line or plane do not affect the result. This essay shows the
converse, namely if the addition of vectors in the line or plane
is not affected by a change of scale and/or direction in coordinates
systems then the distributive law holds of the addition and
multiplication of real numbers.
Option: Readers may restrict directed line
segments to being a rational multiple of a unit length and assume or
require or changes of scale involve rational multiples rather than real
numbers.
Theorem: If a, b and c are real numbers then (a+b)c = ac+ bc
and
c(a+b) = ca+ cb.
First Proof: If c = 0 then both sides are equal and there
is nothing to prove. So assume without loss of generality that c is
non-zero.
Let m = c k where k is a non-zero vector.
Then both m and k can be employed as unit vectors
for a real number line.
Now
Therefore (a+b)c = ac+ bc. by the unique measurement assumption
for unit vectors. The equality c(a+b) = ca+ cb now follows as
multiplication is commutative.
Alternate Proof - Changing the Coordinate Scale
Let unit vector k be a unit vector for a straight line - a real
number line.
Each point P on a straight line may be identified with its position
vector, with a unique multiple pk with tail at the origin and
head at P. Let Q be another point likewise identified with it
position vector qk. Then P+Q = (p+q)k can be identified with the
position vector of another point T.
That being said, the addition of the position vectors is independent of
the selection of unit vector k. Let k = c m
where m is another nonzero vector. Then
P = pk = p (c m) = (pc) m
Q = qk = q (c m) = (qc) m
P+Q = (p+q) k = (p+q) (c m) = ((p+q)c)
m
But P+Q = (pc) m + (pc) m = (pc+qc)
m as well.
Therefore unique measurement assumption implies the two expression the
coefficients of m in the representations of P+Q, that is, in
((p+q)c) m = (pc+qc) m
must be equal. Therefore
(p+q)c = pc +qc.
The latter provides a second proof of the distributive law.
Remark: In the above proof, p and q are the coordinates of P
and Q relative to the choice of k = cm as a unit vector for
a coordinate system. Likewise pc and qc the coordinates of P and Q
relative to the choice of m as a unit vector for a coordinate
system. The distributive law implies the coordinates of sum P +Q can be
calculated relative to k and then transformed (multiplied by c) or the
addends can be transformed first and then added. So the sum of
vectors can be calculated directly or in any unit -vector based
coordinate system. The distributive law is equivalent to the latter
invariance.
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