11. 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. 

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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  (ac+bc) k 

	=  (ac) k  + (bc)  k	as scalar multiplication distributives over vector addition:  (p+q)k = pk + qk for all real numbers p and q
	= a (c k) + b(c k)	as (pq)k = p(qk) for all real numbers p & q (that is, because multiplication of vectors by signed is associative)
	=  a  m + b m	as  m = c k
	= (a+b)m	as scalar multiplication distributivesover vector addition:  (p+q)k = pk + qk
	= (a+b)(c k)	as  m = c k
	= [(a+b)c ] k	as (pq)k = p(qk) for all real numbers p & q (that is, because multiplication of vectors by signed is associative)

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. 

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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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Number Theory

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Start of Number Theory

Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law  Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions

Number Theory
Continued

Decimal Place Value
Place Value Reinforcement
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting  Whole No.  Factors
Arithmetic Videos
Square Roots  & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II

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