Numbers and Number Systems

Like it or not, theories will eventually come down to how you “add up” the Universe, so understanding some basic concepts about numbers and numerical systems is an essential building block. The descriptions following are “common sense” interpretations, based on historical and mythological use of numbers, not the mathematician's entries in encyclopedias.

There are three types of numbers, which are not apparent from their value but only from the context they are used in:

• Cardinal, a number that represents a quantity or magnitude. “I have 5 friends.”

• Ordinal, a number that represents a rank or position within a set (ordering). “Your seat is in row 5.”

• Nominal, a number that is just a name or identity for something, like a phone number.

In the context of the Reciprocal System, motion (a ratio) is composed solely of cardinal numbers, which is why Larson refers to that ratio as a scalar motion. “Scalar” means, “magnitude only.” Scalar motion does not contain ordinal nor nominal numbers.

Larson's coordinate space (or time) is composed of ordinal number he calls absolute locations, since their value is determined by an ordered set starting at zero.

One of the problems identified with both Larson's Reciprocal System and conventional theories is that they lack the proper context for numbers, particularly in mathematical equations. This has led to a great deal of confusion and misinterpretation in both systems. By pointing this out now, it is hoped that confusion can be minimized.

Natural Numbers (Cardinal numbers)

It all starts with simple counting. I have one, he has two, she has three. Larson's Reciprocal System and this reevaluation both start with the ancient Greek concept of Natural Numbers that begin with one and increase by single, whole amounts to a finite¹ maximum. Zero² and infinity³ are not included. Counting is the cardinal, scalar concept of magnitude only, not inferring any direction, geometry or coordinates.

Because the minimum quantity of the natural number system is unity, “one” becomes the reference point for natural ratios, the multiplicative inverse.

Whole Numbers (Ordinal numbers)

Once you can count, you can also “order”—what comes first, second, third… to define a sequence. Sequences imply direction, fewer to greater or vice versa. Once a sequence is established, it can point in two directions… you can always have more, but what about not having any? The ancient Greek argument of “how can nothing be something?” comes into play here, nothing is undefined—either you've got it, or you don't. It isn't until we impose order, a sequence, that the concept of “not having” comes into play and zero is born. The addition of zero to the set of natural numbers gives the concept of whole numbers, making a more complete system of representation, beginning with “I don't have any” through “I have this quantity.” But this is misleading!

Whole numbers are not cardinal quantities; they are ordinal distances from an origin of zero. Using whole numbers, you do not actually have “one” of something, but “one more than nothing.” It may sound like a trivial difference, but it is an important one⁴.

Integer Numbers

Now it's pay-back time! With the rise of trade and mercantilism, simple counting gave way to accounting, the tallying of quantities and the concept of debt (or owing). “You owe me two apples,” and when those apples were returned, the count of 2 went against the debt of 2, resulting in no debt or “nothing”—back to zero. This gave rise to negative quantities, those “owed,” and the integer number system, which included none (zero), the natural, counting numbers, and their opposite—the negative counting numbers, which are again, ordinal in structure.

Rational Numbers (Cardinal ratios)

Trade also gave rise to the concept of fractional parts, as individual items could be subdivided, such as one apple pie into 8 pieces. This resulted in the proportions expressed as the ratio—how many times one thing is contained in another—and the rational number system to represent it. Rational numbers are a ratio of natural numbers, meaning that the minimum quantity of both aspects⁵ is unity and the minimum ratio is 1/1.

Larson's concept of scalar motion is defined by a rational number. By assigning a unit of space to the numerator and of time to the denominator, he creates the concept of scalar speed⁶.

Real Numbers (Ordinal measure)

The use of counting numbers and ratios lead to the general system of measure, where numbers no longer represented quantities but relative amounts. If something is “4 miles distant,” that does not mean you have “4 miles” in a box marked “distant.” Relative measure required a starting point, or a “datum” as used in the Reciprocal System, from which measurement of quantity is made. Note the distinction here: quantity is absolute (cardinal); measurement is relative (ordinal). Given that quantity can have proportion (ratio of one quantity to another), measurement can also have proportion… these proportional measurements became the set of real numbers, which can express any measurement, when given a place to start.

Irrational Numbers (Ordinal ratio)

To this point, the number systems have been primarily linear in use. The ordering systems can be placed in a straight line, the ratios of rational numbers are analogous to slopes of a line—but still a straight line. But that is only one aspect… the other is the curve, when ordering becomes dependent upon two dimensions and is no longer commutative. When a curve interacts with a straight line, such as the ratio of a circumference to its diameter, another type of real number comes into existence, that of the irrational number—a number that cannot be expressed as a ratio of two integers, as it is not a ratio of quantities, but a ratio of measurements. Common examples of irrational numbers are π and the natural log, e. These numbers represent a different type of ordering system based on angles, with different forms of representation—polar sequencing.

Complex Numbers

The basic problem with mixing linear and polar numbering systems together was that they did not play by the same rules. Trigonometry was the first to correlate the two; later on, the concept of the imaginary number came into play and when joined with the linear became the complex number system, allowing both the linear and polar values to be represented on a linear graph known as the complex plane or Argand diagram. This simple concept allowed both linear and polar coordinates to be represented with a single “number,⁷” creating a global system of all the number systems preceding it.

Extended Complex Numbers

The final step is to close the system of complex numbers to include the counterpoint to nothing—that of totality (everything), or infinity⁸. Zero and infinity represent the numeric extremes of nothing to everything and are treated as boundaries—not limitations—in the system of extended complex numbers.

Summary

As explained, number systems have developed from simple, cardinal counting, through ordinal sequencing, measurement, proportions and mixing lines with curves. Though there are many other number systems for specialized branches of science and mathematics, RS2 only requires comprehension up to the extended complex number system to represent its universe of motion.