Geometrical Arithmetic on a Two-Number Line.

A Geometry of Arithmetic

We'll work toward a geometrical description of arithmetic. In the last Post a number-segment on a Straight Line Segment(SLS) represented a number. A coordinate system is located, oriented, and scaled the segment. Representation of arithmetic on a SLS requires the representation of the second number in the arithmetic expression.  How does the number segment for the first segment locate, orient, and scale the number segment for the second number? That requires a second coordinate system, and it is built with the first number's number segment.

This may get a bit tedious, but I have to lay the basis out once and for all to make sure it works. At the end we will have a foundation to build subtraction on addition and division on multiplication. We also will have a strong footing for the physical model we introduced in a previous Post; we want a way to "see" arithmetic as an active process. 

Here is the outline of the geometrical process:

• Represent one number using a coordinate system picked for its convenience.
• For addition, take the outer end point of this number segment for the origin of a second coordinate system. The scale and orientation of the two coordinate systems are the same. 
• For multiplication, take the outer end point of this number segment for the outer end point of a second coordinate system's unit number-segment . Here the two coordinate systems have the same origin. 
• Represent the second number as a number-segment with the second coordinate system.
• Read out the coordinate of the outer end of this number-segment. It is the value of the sum or product.

What is traditionally executed as a numerical calculation is now a geometrical process in a one-dimensional space. It produces a numerical result from a geometrical process. Although somewhat disguised, this approach takes the number line apart to get at the tool used to construct it: a coordinate system. The geometric process then attaches a second coordinate system to the SLS. The result is then retrieved from this Two-Number-Line. 

For addition our approach superficially looks like the usual one on a number line, but ours is a unified treatment of addition and multiplication that follows from the two possible ways to represent the combination of two numbers in a superposition of two number systems on one line. I would argue it is the inevitable extension of the ideas underlying the traditional number line to incorporate both addition and multiplication. It is the representation of 

A more intuitive interpretation sees arithmetic as taking the representation of one number to make another representation of the other number. It is arithmetic on a Two-Number-Line.

There are many representations of addition and multiplication on the Number-Line. In all of them that I know about, multiplication relies on a representation of addition. This suggests addition  is somehow intrinsic to multiplication, confusing what multiplication does how computational algorithms work. It is a confusion of mechanism with the meaning and purpose.

On the other hand, on a Two-Number-Line, addition and multiplication are independent variants of one process; and together they exhaust the possibilities of that process; they neither overlap each other, nor do they leave room for any other independent variant in the process: it is not an accident that we learn two arithmetic operations, and that those operations are addition and multiplication. Arguably, once we want the one number that represents the possible results of the commutative interaction of two, it is inevitable. And finally it is easy to draw a picture of the process, and even to make a physical model of it. 

Notation

From the Post Semantics of Elementary Arithmetic Expressions

Addition: 

(to-this-Operand + add-this-Operand = Sum =

(to-this-N1 + add-this-N1)  x common-unit =
N3 x common-unit.

Multiplication: 

(by-this-Operand) x (multiply-this-Operand) = Product =  

(by-this-N1 x multiply-this-N2) x product-unit  = N3 x product-unit.

N1 and N2 are number operands.

• For addition N3 is the number result of evaluating the addition expression (to-this-N1 + add-this-N2).
• For multiplication N3 is the result of evaluating the number expressions:  (by-this-N1 x multiply-this-N2) .

In short we are now addressing the number side of the problem.
We are going to represent a number by a segment. For this we have to set a relative scale, the ratio of the length imposed on the Straight-Line-Space SLS) by coordinate system to the common-unit for addition or the product scale for multiplication. We shall assume it is 1; e.g. 1 inch on the SLS corresponds to 1 square foot, and etc.

Cartesian Coordinate System

For Co a Cartesian Coordinate System(CCS), with N the coordinate of P on a Straight Line Space Sp: 

• Co(P) = N, 
• invCo(N) = P .

That's it for what a coordinate system does.

For a CCS the points on Sp whose coordinates differ by a fixed amount are evenly spaced. 

Co is fixed on Sp by two points: P0 and P1 with coordinates 0 and 1 respectively.

• P0 is C's origin,
• One of: (a) P1 is to the right of P0, the points to the right of the origin have positive coordinates, those to left, negative.  (b) P1 is to the left of P0, it is oppositely oriented.

The distance from P0 to P1 is the scale of the coordinate system: e.g if P1 is one centimeter from the origin then any two points separated by one centimeter have coordinates that differ by 1.

A Number-Segment

A representation of a number, N1, on Sp, a SLS, is a geometrical figure that relates properties of the Sp to the number. With Co a CCS on Sp, the number can be represented as the geometrical figure: the line segment on Sp with its inner end at the origin P0 of Co, and its outer end at the point PN1 on Sp whose coordinate in Co is N1. Refer to it as a "number-segment" N1:Co : the representation of the N1 on Sp built with Co. It relates a number to the oriented distance of a point from the origin of a coordinate system.

The Geometrical Process

Three steps are needed in the process to evaluate (N1 op N2) for op one of  "+" an "x". The first and last steps are common, the second is the isolated step particular to op.

  

1) For one of the two numerical operands, say N1, with a preexisting coordinate system C0 and the coordinate N1 = N1, construct the number-segment N1:Co on a SLS, Sp. Its outer end is the point PN1 in Sp.

2) One variant for this step is for addition; the other, for multiplication. In both cases it produces a point PN2 at the outer end of a number-segment  N2:COPo, This number-segment is constructed with a coordinate system COPo particular to the operator op.

3) The coordinate of N2 of the point PN2 in Co is the Sum or Product N3 = N2.

Step 2 for addition.

• construct the coordinate system COPo for N2 = N2 as the translation of Co; ie, the origin 0:COPo = PN1 with the scale and orientation unchanged.
• PN2 is the outer end of the number-segment N2:COPo.

Step 2 for Multiplication.

• the origin of COPo and Co are the same point P0.
• the scale and orientation are fixed by N1 as 1:COPo = N1:Co; ie, the scale for COPo is N1 multiplying the scale that N1 was represented by in step 1. The relative orientation is determined by the sgn of N2.
• PN2 is the outer end of the number-segment N2:COPo.

Wrap Up 

It all worked out pretty much as advertised. Now that we have the notation, we can show another way addition and multiplication diametrically differ from one another.

For addition, the relation of the coordinate systems is a difference:

• Co(PN1) - COPo(PN1) = N1 = N1 , PN1 is the origin of COPo
• Co(P) - COPo(P) =  N1 , for all P  , <= Cartesian CS  
• Co(PN2) - COPo(PN2) =  Co(PN2) - N2 = N1 ,
• Co(PN2) = N1 + N2 .

For multiplication, the relation is as a ratio:

• Co(PN1) / COPo(PN1) = N1 = N1 , COPo(PN1) = 1 ,
• Co(P) / COPo(P) =  N1 , for all P  , <= Cartesian CS ,
• Co(PN2) / COPo(PN2) =  Co(PN2) / N2 = N1 ,
• Co(PN2)  = N1 x N2 .

Difference and ratio are the two ways numbers are compared in elementary education. One provides the additive comparison and the other is the multiplicative comparison. These are the two ways coordinates are compared in the Two-Number Line. Further,they reflect the two ways two Cartesian Coordinate Systems relate the points that define them.

For addition:

• PN1, the outer end of the number segment for N1 in the preexisting coordinate system Co, is the origin of coordinate system COPo used to construct N2's number-segment.

For Multiplication:

• The origin points of Co and COPo coincide.
• Now PN1, the outer end of the number segment for N1 in the preexisting coordinate system Co, is the outer end of the unit number-segment 1 in COPo. This fixes both the scale and the orientation of COPo. 

A coordinate system is determined by its origin, its orientation, and its scale. These three possibilities are exhausted by addition and multiplication. 

We conclude:

• two-operand operations can be represented on the Two-Number Line
• the two two-operand, operations, addition and multiplication, exhaust the possibilities;
• addition and multiplication are independent of one another; ie. either one can be defined independently of the other.
• subtraction and division can respectively be defined in terms of addition and multiplication. 

It is not an accident that addition and multiplication, along with their variants subtraction and division, command so much attention. Understand them.