The Larson Research Center

In the previous two posts, I’ve sketched how, in the upcoming seminar, I will approach explaining the three properties of scalar numbers that correspond to the three properties of physical magnitudes - quantity, “direction,” and dimension. Essentially, we’ve seen that it is order in reciprocal progressions that defines numerical quantity with the dual “directions” observed in nature, and that these reciprocal quantities can be arithmetically, or algebraically, combined, by resorting to two, operational, interpretations of number, symbolized by a slash and a pipe symbol, and restricting the operations of multiplication (division) and addition (subtraction) to these interpretations, respectively.

Now we come to the question of dimension, the third property of physical magnitudes that we want to define as a corresponding property of numbers. At first this might seem impossible, because scalar quantities, even those with dual “directions,” can’t be rotated as physical quantities can. Yet, while this is certainly true, we need to remember that dimensions of physical magnitudes, i.e. length, width, and depth, are simply independent variables, and that it is through the operation of rotation that the independence of these fundamental physical dimensions is established. However, nothing precludes us from establishing independence of variable quantities through some means other than rotation.

For example, three points, separated in space, are independent points, in terms of their different locations, whether those locations are confined to a one-dimensional line, a two-dimensional plane, or a three-dimensional volume. As explained in the previous posts below, combining a positive and negative quantity, in the form of two reciprocal numbers, defines a “distance,” or an interval, between them; that is,

1|2 + 2|1 = 3|3 = 0, so it follows algebraically that

1|2 = -(2|1), and

2|1 = -(1|2).

Plotting these two quantities on a number line, we get

1|2———-0———-2|1 or -1———-0———-1

So, the difference, or interval, between them is two units

(1|2) - (2|1) = -1 - (1) = -2, or

(2|1) - (1|2) = 1 - (-1) = 2.

Hence, the algebra is non-commutative, because ordinary arithmetic is non-commutative (i.e. the order of operations matters in subtraction) . But the non-commutativity of the subtraction operation is tantamount to a conservation of the “direction” property of the reciprocal number, which we have defined without recourse to an imaginary number, in the form of the square root of -1. 

However, the question then arises, what is the square root of the negative quantity, 1|2? Is there a reciprocal number that when raised to the power of 2 equals 1|2? The answer is yes, there is, but to understand it requires us to delve into the meaning of raising numbers to powers and then extracting roots from them. What does it mean to raise the number 1 to the power of 2, and then extracting that power from it, as its root? We are taught in middle school that

1⁰ = 1
1¹ = 1
1² = 1x1 = 1
1³ = 1x1x1 = 1,

and we learn to think of the base number as a factor and the exponent as the number of factors in the product equal to the exponentiation. However, we are also taught to relate these numbers to the dimensions of a coordinate system (which Hestenes likens to catching a debilitating virus). This may be a little confusing to adults (children seldom question it), because, while 1¹ can readily be understood as a linear unit, 1² as an area unit, and 1³ as a cubic unit, in a 3D coordinate system, how is it that 1⁰ = 1? Logically it would follow that it should be analogous to a point at the origin of the coordinate system, but the origin has to be zero, not 1.

The way this is normally explained in terms of a binary operation is that, by a law of exponents, we understand that 1¹/1¹ = 1^1-1 = 1⁰ = 1, where 1 must be understood as a dimensionless number, a unit with no dimensions (so I guess zero is defined as 0¹/0¹?!!). Yet, to a jaded adult that seems  a little suspect, because 0 and 1 are quite different. Besides, if we go that route, it means that 1¹ = 1²/1¹, 1² = 1³/1¹, and 1³ = 1⁴/1¹, which also means that 1¹ * 1⁰ = 1¹, or a unit line times a unit point is a unit line, and 1¹ * 1¹ = 1², or a unit line times a unit line is a unit area, and 1² * 1¹ = 1³, or a unit area times a unit line is a unit volume, and 1³ * 1¹ = 1⁴, or a unit volume times a unit line is a what? A hypervolume? What’s that?

The only thing that we have accomplished, with this law of exponents, is a trade-off. We had no explanation, at one end of the tetraktys, and we traded it for no explanation at the other end of it! Besides that, in what sense is a point times a line equal to a line? Nevertheless we learn to glibly state that any number raised to the zero power is equal to one, without noting that this also requires us to believe that, in order to raise any number to the third power, we must define something as a unit that is clearly indefinable as a unit (i.e 1⁴).  Of course, we do it anyway, because, for most uses, it doesn’t affect us, and a point magnitude, somehow becoming a scalar multiplier of a line magnitude, makes sense in practice, if not in theory.

Fortunately, however, we don’t encounter the same theoretical problem with the dimensions of reciprocal numbers, because we can define dimensions, or powers of a number, as sets of dual “directions” inherent in the numbers. On this basis, we can describe four units using four numbers with increasing sets of “directions”:

1⁰:1⁰ = units with no dual “directions” (corresponding to geometric points)
1¹:1¹ = units with one set of dual “directions” (corresponding to geometric lines)
1²:1² = units with two sets of dual “directions” (corresponding to geometric areas)
1³:1³ = units with three sets of dual “directions” (corresponding to geometric volumes)

where the colon is used as a generic symbol of operation, representing either the slash, or the pipe, symbol of our two operational interpretations of number.

This clarification of the definition of numerical dimensions, as simply the difference in the number of sets of dual “directions,” in a given number, makes it possible to identify a numerical, or scalar, “geometry” with the customary vector geometry of Euclidean three space, when these scalar dimensions are independent variables, which is tantamount to the definition of orthogonality in spatial dimensions.

As Larson first pointed out, with what is now called Larson’s cube, there are a total of eight “directions” possible in a 3D magnitude. These “directions” are analogous to the eight vector directions in the cube, delineated by connecting the eight corners of the cube with four diagonal lines, intersecting at the origin of the cube, when it is formed from a stack of 2x2x2 cubes, as shown in figure 1 below:

Figure 1. The Eight Directions of Larson’s Cube

In the next post, we will analyze the cube in terms of eight scalar “directions,” which, as we will see, are eight 3D scalar magnitudes, or, what is tantamount to the same thing, eight 3D numbers, completing the generalization of number as magnitude.