An Overview of Numeristics and Equipoint Analysis

$$\def\ds{\displaystyle} \def\s#1{\mskip-#1mu} \def\ol#1{\overline{#1}} \def\8{\infty} \def\eqdef{\mathrel{{\bf:}\mkern1mu{=}}} \def\rmopn#1{\mathop{\rm #1}\nolimits} \def\Im{\rmopn{Im}} \def\Re{\rmopn{Re}} \def\sgn{\rmopn{sgn}}$$

Overview

This document gives an overview of four documents written by the author on numeristics and equipoint analysis. Numeristics is a number-based foundational theory of mathematics. Equipoint analysis is a system of calculus based on numeristic principles.

Each of the four source documents is monograph length. The present document summarizes key findings. Definitions, proofs, and secondary findings are found in the source documents.

The four source documents are:

• Numeristics: A number-based foundational theory of mathematics ([CN, CM]). Basic principles and concepts of numeristics. See the numeristics section below.
• Equipoint analysis: A numeristic approach to calculus ([CE, CM]). A system of calculus and analysis using numeristic principles and an extended number system. See the equipoint analysis section below.
• Divergent series: A numeristic approach ([CD, CM]). An alternative approach to the theory of divergent series using numeristic principles. See the divergent series section below.
• Repeating decimals: A numerisitic approach ([CR, CM]). Theorems and proofs about repeating decimals, and an extension of repeating decimals using numeristic principles. See the repeating decimals section below.
Numeristics

The first of the source documents on numeristics is Numeristics: A number-based foundational theory of mathematics [CN]. This document contains basic principles and concepts of numeristics. Key principles and results include the following.

Numeristics bases mathmatics purely on number. Numeristics starts with three numeric ultraprimitives or alternate ways of expressing the ultimate value of mathematics:

 Infinity Number, as with everything else, ultimately starts from the infinite. The infinite is inexhaustible and therefore only partly conceptualizable. Unity Unity, the number $1$, is the first mathematical manifestation. It expresses the unified nature of infinity. Zero The number $0$ represents the unmanifest quality of pure consciousness. Zero is called the absolute number, because it is the unmanifest state from which all manifestation begins.

In its multiple step phase, numeristics starts with the following primitives, which function somewhat as axioms. These primitives are generally those which existed in “classical mathematics,” by which we mean mathematics as it historically existed before the development of set theory, and which is currently taught at the primary, secondary, and upper division undergraduate levels.

• Natural, integral, rational, real, and complex numbers
• Addition, multiplication, exponentiation of these numbers and their inverses
• Usual commutative, associative, and distributive properties of these numbers
• Elementary equality and order relations
• Euclidan geometry
• Ordinary classical logic with quantifiers (first order logic)

Numeristics does not use sets. Instead, a numeristic class is a potentially multivalued number or number-like construction. Numeristic classes have a flat structure: Every number is a single valued class; a class containing a single number is identical to the number: $a = \{a\}$. A class containing multiple numbers distributes operations on it and statements about it over each constituent number: $\{2, 3\} + 1 = \{3, 4\}$.

The empty class or null class, denoted $\emptyset$ or $\{\}$, is a class with no values. For example, $1 \cap 2 = \{\}$.

One important consequence of the flat structure of numeristic classes is that the inverse of a function is always itself a function. For instance, the inverse of $f(x) = x^2$ is $f^{-1}(x) = \pm\sqrt x$. The inverse of a conventionally indeterminate form such as $\ds \frac00$ is the full class, the class of all numeric values. An inverse that has no values for a given point is well defined at that point as the empty class.

 O P S L C ϕ r r 1 Fig. 1: Projectively extended real numbers $r = \tan\frac\phi2$ O C S₁ S₂ E₁ E₂ L ϕ r 1 Fig. 2: Geometric relation of $0$ and $\pm \8$ in projectively extended real numbers

Numeristics uses several methods of adding infinite elements to the real numbers, the primary being the projectively extended real numbers (which is known to conventional mathematics). The projectively extended real numbers add one infinite element, $\8$, to the real numbers. Thus $\ds \frac10 = \frac20 = \8 = -\8$.

Figures 1 and 2 show how real number $r$ at position $L$ uniquely maps to some point $S$ on the circle, and the infinite element $\8$ maps to the point $P$ at the top of the circle.

All numeric operations are thus well defined. This includes infinite sums, which are not defined as limits but as sums with an actual infinite number of terms.

Equipoint analysis

The second of the source documents on numeristics is Equipoint analysis: A numeristic approach to calculus [CE]. Equipoint analysis is a system of calculus and analysis using numeristic principles and an extended number system. Key concepts and results include the following.

We define multiple levels of sensitivity. The real numbers are at the most coarse level of sensitivity and are called the folded numbers. Then we unfold each real number to a finer sensitivity level. Each unfolded number may be unfolded again to an even finer sensitivity level.

When we unfold a number, what is a single value at the coarser level becomes a multivalued class at the finer level. The unfolded members of 0 are a class of infinitesimals. The reciprocals of infinitesimals, which are the unfolded members of $\8$, are a class of infinite numbers.

Figure 3 is a diagram of the infinitesimals, through an infinitely magnifying microscope view.

We pick one infinitesimal which is not at the origin of the unfolded class and denote it $0'$, and we similarly pick one unfolded infinite and denote it $\8'$.

The extension of unfolded numbers allows direct arithmetic with infinitesimals and infinite numbers. Figure 4 shows how we use this to define the derivative as an actual ratio of infinitesimals: $$\frac{df(x)}{dx} \eqdef \frac{f(x + 0') - f(x)}{0'}.$$

Using this definition, the derivative of $x^2$ is calculated as follows: \eqalign{ f(x) &= x^2 \cr \frac{df(x)}{dx} &\equiv \frac{(x + 0')^2 - x^2}{0'} \cr &\equiv \frac{x^2 + 2\cdot0'x + 0'^2 - x^2}{0'} \cr &\equiv \frac{2\cdot0'x}{0'} \cr &= 2x. }

Similarly, as shown in Figure 5, an integral is defined as an actual sum of an infinite number of infinitesimals: $$\int_a^b f(x) dx \eqdef \sum_{k=1}^{\8'} f \left( a + \frac{k(b-a)}{\8'} \right) \frac{b-a}{\8'}.$$

The integral of $x^2$ is calculated as follows: \eqalign{ \int_0^u 2x dx &\equiv \sum_{k=1}^{\8'} 2 \frac{ku}{\8'} \frac{u}{\8'} \cr &\equiv \frac{2u^2}{\8'^2} \sum_{k=1}^{\8'} k \cr &\equiv \frac{2u^2}{\8'^2} \8' \frac{\8' + 1}2 \cr &\equiv u^2 \left( 1 + \frac1{\8'} \right) \cr &\equiv u^2 (1 + 0') \cr &= u^2. }

The direct arithmetic of unfolded numbers allows us to formulate simpler proofs. For example, the proof of the first fundamental theorem of calculus does not require the mean value theorem: \eqalign{ \int_a^b \frac{d f(x)}{dx} dx &\equiv \int_a^b \frac{f \left( x - \frac{b-a}{\8'} \right) - f(x)}{\frac{b-a}{\8'}} dx \cr &\equiv \frac{b-a}{\8'} \frac\8'{b-a} \sum_{k=1}^{\8'} f \left( a + \frac{k(b-a)}{\8'} + \frac{b-a}{\8'} \right) - f \left( a + \frac{k(b-a)}{\8'} \right) \cr &\equiv \sum_{k=1}^{\8'} f \left( a + (k+1) \frac{b-a}{\8'} \right) - f \left( a + k \frac{b-a}{\8'} \right) \cr &\equiv f \left( a + (\8'+1) \frac{b-a}{\8'} \right) - f \left( a + \frac{b-a}{\8'} \right) \cr &\equiv f \left( a + (b - a) + 0' \right) - f \left( a + 0' \right) \cr &= f(b) - f(a) \cr }

The equipoint approach allows us to easily discover that the natural logarithm can be expressed as a polynomial of order $0'$: $$\ln y = \frac{y^{0'}-1}{0'},$$ which shows that $\ds \int_1^t t^{-1} dt = \ln t$ is an instance of the general law $\ds \int_0^t t^n dt = \frac{t^{n+1}}{n+1}$ and not an exception.

The Dirac delta function $\delta(x)$ can be defined as the derivative of the Heaviside step function. It is a true function in equipoint analysis, but it is defined only in unfolded arithmetic, i.e. it is not the unfolding of any folded function.

Figure 6 shows the infinite value at $\delta(0)$. The microscope in this figure expands infinitely in the $x$ direction and contracts infinitely in the $y$ direction.

Unrestricted direct arithmetic in the derivative allows us to derive direct formulas for higher derivatives:

$$\ds f^{(n)}(x) = \frac{\sum_{k=0}^n {{n}\choose{k}} (-1)^{n-k} f(x+0'k)}{0'^n}.$$

The indefinite integral can be derived from this formula as a negative order derivative: $$\int f(x) dx = f^{(-1)}(x) = \sum_{k=1}^{\8'} f(x - 0'k) \; 0'.$$

Unrestricted direct arithmetic means that the derivative of any function is defined, including functions that have no conventional derivative. One example is the Dirac delta function, which as mentioned above can be defined as the derivative of the Heaviside step function. Another example is the Weierstraß function, which is a fractal, continuous everywhere but conventionally differentiable nowhere. The Weierstraß-like function $$W(x) \eqdef \sum_{n=0}^\8 \frac{\sin (2^n x)}{2^n} = \sin x + \frac{\sin 2x}2 + \frac{\sin 4x}4 + \dots$$ has the equipoint derivative $$\frac{dW(x)}{dx} = \sum_{n=0}^\8 \cos (2^n x).$$

Complex functions which violate the Cauchy-Riemann equations are also equipoint differentiable. For example, for the function $$M(x) \eqdef 3 \Re z + 2 \Im z$$ the derivative at each point is a class of complex numbers which depends on the unit of differentiation $0'$: $$\frac{\ds{}_{0'}\s2 d}{\ds{}_{0'}\s2 dz} M(x) = \frac{3 \cos \arg 0' + 2i \sin \arg 0'}{\sgn 0'}.$$

Equipoint analysis may improve the understanding of quantum renormalization. A very simplified example is to evaluate $$\int_{-\8}^{\8} x^2\;dx,$$ which is conventionally evaluated through limits. In the equipoint approach, we we avoid limits and directly write $$\frac1{\8'^3} \int_{-\8'}^{\8'} x^2\;dx = \frac23.$$

Divergent series

The third of the source documents on numeristics is Divergent series: A numeristic approach [CD]. This document describes an alternative approach to the theory of divergent series using numeristic principles. Key concepts and results include the following.

The conventional theory of divergent series was pioneered by Euler and given substantial expression by Hardy, whose posthumous work on the subject is considered a classic. Euler and Hardy each start out with the following result for $|a| > 1$: $$\sum_{k=m}^\8 a^k = a^m + a^{m+1} + a^{m+2} + a^{m+3} + \dots = \frac{a^m}{1-a}.$$

The conventional theory as expressed by Hardy then uses methods of summation, an approach which has several weaknesses:

1. An unworkable conception of weak equality.
2. The failure of all methods, except one expressed by Euler, to account for the above result.
3. A faulty understanding of Euler's method, including a circular definition in Hardy, and failure to realize this method as an extension of arithmetic.

Euler's method is the principle that we do not reject an algebraic process simply because it deals with divergent series. If we find an algebraic process that is valid for a convergent case, then we assume that that process is also valid for the corresponding divergent case.

The numeristic theory of divergent series firmly justifies Euler's method and not other methods of summation. The numeristic approach extends Euler's method to yield the following two appraoches:

1. Recursion, which is the Euler's method coupled with the understanding that most infinite series have at least two values, one finite and one infinite. For example, using recursion, we establish that $1 + 2 + 4 + 8 + \dots = \{-1, \8\}$.
2. Equipoint summation, which uses equipoint analysis. Equipoint analysis uses multiple levels of sensitivity to unfold real and complex numbers to multilevel numbers, functions, and relations. Equipoint summation uses these multiple levels to unfold the terms of an infinite series, which shows that the sum of a divergent series may depend on the mode of unfolding. For instance, $1 - 1 + 1 - 1 + \dots$ may have the value $\8$, $\frac12$, an arbitrary finite number, 1, or 0, depending on the mode of unfolding.

Other results include: \eqalign{ \sum_{k=-\8}^\8 a^k &= a^m + a^{m+1} + a^{m+2} + a^{m+3} + \dots = \left\{ 0, \8 \right\} \cr \sum_{k=0}^\8 e^{kix} &= 1 + e^{ix} + e^{2ix} + e^{3ix} + \dots = \left\{ \frac{1 + i\cot \frac{x}2}2, \8 \right\} \cr \int_0^\8 a^x dx &= \left\{ \frac{-1}{\ln a}, \8 \right\} \cr }

Repeating decimals

The fourth of the source documents on numeristics is Repeating decimals: A numerisitic approach [CR]. This document contains various theorems and proofs about repeating decimals, most of which are conventional, but also with some numeristic extensions. Key concepts and results include the following.

A repeating decimal is a convergent geometric series. Based on the results of [CD], a repeating decimal in the projectively extended real numbers is multivalued: \eqalign{ 0.\ol{d_1\dots d_P} &= \sum_{K=1}^\8 d_1\dots d_P \cdot 10^{-PK} \cr &= \left\{ \frac R{10^P - 1}, \8 \right\}, \cr } where $R = d_1\dots d_P$ is the repetend and $P$ is the period.

For example, \eqalign{ 0.333 \dots = 0.\ol3 = \sum_{K=1}^\8 3 \cdot 10^{-K} &= \left\{ \frac13, \8 \right\} \cr 0.999 \dots = 0.\ol9 &= \{ 1, \8 \} \cr }

A left repeating decimal has an infinite number of repeating digits to the left of the decimal point. This type of decimal is a divergent geometric series, which is also multivalued: \eqalign{ \ol{d_1\dots d_P} &= \sum_{K=0}^\8 d_1\dots d_P \cdot 10^{PK} \cr &= \left\{ -\frac R{10^P - 1}, \8 \right\}, \cr } and \eqalign{ \dots 333 = \ol3 = \sum_{K=0}^\8 3 \cdot 10^K &= \left\{ -\frac13, \8 \right\} \cr \dots 999 = \ol9 &= \{ -1, \8 \} \cr }

Right and left repeating decimals with the same repetend are negatives: \eqalign{ \ol{d_1\dots d_P}.\ol{d_1\dots d_P} &= \sum_{K=-\8}^\8 d_1\dots d_P \cdot 10^{-PK} \cr &= \{ 0, \8 \} \cr \ol3.\ol3 = \ol9.\ol9 &= \{ 0, \8 \} \cr }

When we allow infinite left decimals, every real number has an infinite number of representations.

Acknowledgment

The author wishes to thank the organizers, participants, and supporters of the Invincible America Assembly at Maharishi International University in Fairfield, Iowa for creating one of the best research environments in the world.

References
 CD K. Carmody, Divergent series: A numeristic approach, 500K. CE K. Carmody, Equipoint analysis: A numeristic approach to calculus, 600K. CI K. Carmody, Is consciousness a number? How Maharishi Vedic Mathematics resolves problems in the foundations and philosophy of mathematics, 100K. CM K. Carmody, Numeristic Mathematics, a printed collection of [CN], [CE], [CD], and [CR], Amazon USA, Canada, UK, Germany, France, Spain, Italy, Japan. CN K. Carmody, Numeristics: A number-based foundational theory of mathematics, 400K. CR K. Carmody, Repeating decimals: A numerisitic approach, 300K. CS K. Carmody, Summaries of published sources by Maharishi Mahesh Yogi on mathematics, including modern mathematics and Vedic Mathematics, 200K.