II.   Divine Systems

The PRIMARY RULES of the ELEMENTAL CONCEPT of NUMERICS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. NUMERICS is a FOUNDATION CLASS of ELEMENTAL CONCEPTS associated with the study of the principles and rules of definition by which NUMBERS are defined, classified and used in valid statements defined by the EIKOS Language System.

2. NUMERICS is constructed of eighteen [18] ELEMENTAL CONCEPTS including: NUMERICS, NUMBER, UNISET, PERFECT NUMBER, IMPERFECT NUMBER, PERFECT UNIQUE NUMBER, PERFECT SIMILAR NUMBER, PERFECT IDEA NUMBER, IMPERFECT UNIQUE NUMBER, IMPERFECT SIMILAR NUMBER, IMPERFECT IDEA NUMBER, PERFECT RATIO NUMBER, IMPERFECT RATIO NUMBER, GEOMETRIC RATIO NUMBER, SEQUENCE RATIO NUMBER, CONSTANT RATIO NUMBER, COMPLEX RATIO NUMBER and IMPERFECT-PERFECT RATIO.

The PRIMARY RULES of the ELEMENTAL CONCEPT of NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. All numbers are both unique concepts and objects in themselves.

2. All numbers may be represented symbolically.

3. All numbers are real by virtue of their existence.

4. The degree of reality of a number is dependent upon the degree to which the number represents real world objects and/or measurements and relationships of real world objects.

5. The set of all numbers may be defined as the UNISET.

The PRIMARY RULES of the ELEMENTAL CONCEPT of UNISET of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. All NUMBERS are a SET of 1 (one).

2. All numbers of the UNISET may be defined as existing between the prime numbers of 0 (zero) and 1 (one) by some multiplying degree (ratio).

3. 0 (zero) is a MEMBER of 1 (one) and itself.

4. 1 (one) is a SET and a MEMBER of itself.

5. The SUM of these PROPERTIES is known as the UNISET.

6. If a single individual MEMBER of a SUBSET ceased to exist, the total set being UNISET would cease to exist.

7. The existence of UNISET is therefore dependent on the EXISTENCE of each individual MEMBER of the SET for itself to exist.

The PRIMARY RULES of the ELEMENTAL CONCEPT of PERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. A PERFECT NUMBER is any positive number including zero that can be expressed as a ratio of itself or other positive numbers.

2. Infinity, Six, Two, One and Zero are examples of perfect prime numbers related to themselves, 1 and other numbers.

3. All PERFECT NUMBERS may be defined as either UNIQUE, SIMILAR or IDEA (theoretical).

4. UNIQUE PERFECT NUMBERS– can represent uniquely real objects (e.g.1, 3,5, 7, 9, 11 etc)and are found most commonly in the unique measurement of real objects.

5. SIMILAR PERFECT NUMBERS– can only represent collective real objects (e.g. 2, 4, 6) and are found most commonly as sets of real world objects.

6. IDEA PERFECT NUMBERS– cannot represent real objects (e.g. 2.5)

7. PERFECT NUMBERS are equivalent in part to integers in mathematics as the term incorporates the set of positive numbers. However, as integers may also contain negative numbers, the term Integer is never permitted within Eikos Language System.

The PRIMARY RULES of the ELEMENTAL CONCEPT of IMPERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. An IMPERFECT NUMBER is any positive or negative number excluding zero that cannot be expressed as a ratio of itself.

2. Pi is an imperfect unique number expressing the level of perfection of geometric configuration of perfect numbers in a circle.

3. All imperfect numbers may be defined as either UNIQUE, SIMILAR or IDEA (theoretical).

4. UNIQUE IMPERFECT NUMBERS– can represent uniquely real ratios (e.g. pi, e)

5. SIMILAR IMPERFECT NUMBERS– can only represent collective real ratios (e.g. 180º)

6. IDEA IMPERFECT NUMBERS– cannot represent real ratios or real objects (e.g. -1.2)

7. IMPERFECT NUMBERS are equivalent in part to integers in mathematics as the term incorporates the set of numbers. However, as integers may also contain natural numbers, the term Integer is never permitted within Eikos Language System.

The PRIMARY RULES of the ELEMENTAL CONCEPT of UNIQUE PERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. A UNIQUE PERFECT NUMBER is a number that can be expressed as a ratio of itself uniquely.

2. A UNIQUE PERFECT NUMBER is divisible only by itself and 1.

3. The simplest UNIQUE PERFECT NUMBERS are 0 and 1.

4. The largest UNIQUE PERFECT NUMBER is infinity.

5. All UNIQUE PERFECT NUMBERS are a less than the UNISET (total sum of all numbers) ultimately derived from a UNIQUE PERFECT NUMBER.

6. Existence tends towards a UNIQUE PERFECT NUMBER.

7. UNIQUE PERFECT NUMBERS exist within the UNISET as a perfect pattern of randomness called the PRIME (itself an imperfect unique number).

8. The PRIME may be defined as unique pattern of existence of UNIQUE PERFECT NUMBERS and SIMILAR PERFECT NUMBERS within the UNISET related to a multiple ratio of the simplest synergetic number 6.

The PRIMARY RULES of the ELEMENTAL CONCEPT of SIMILAR PERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. A SIMILAR PERFECT NUMBER is a number that can only be expressed as a ratio of itself as a collective of perfect unique numbers.

2. A SIMILAR PERFECT NUMBER is divisible by itself, UNIQUE PERFECT NUMBERS, SIMILAR PERFECT NUMBERS and 1.

3. The simplest SIMILAR PERFECT NUMBER is 2.

4. The largest perfect similar number is one less than infinity.

5. All SIMILAR PERFECT NUMBER greater than two (2) may be defined as the SUM of two (2) PERFECT UNIQUE NUMBERS.

6. Existence is the interaction of UNIQUE PERFECT NUMBERS and SIMILAR PERFECT NUMBERS a tending towards a UNIQUE PERFECT NUMBER.

7. SIMILAR PERFECT NUMBER may be defined as SYNERGETIC. SYNERGETIC are numbers equal to the sum of all their possible divisors except itself. The smallest SYNERGETIC numbers are 6, 28, 496 and 8128.

8. SIMILAR PERFECT NUMBER may be defined as SYMPATHETIC. SYMPATHETIC are two numbers where each is the sum of all the possible divisors of the other. The smallest SYMPATHETIC pairs are (220, 284), (1184,1210) and (17,296 18,416).

9. A SIMILAR PERFECT NUMBER is divisible by two (2) if the last digit is even. A SIMILAR PERFECT NUMBER is divisible by three (3) if the SUM of its digits is divisible by 3.

10. A SIMILAR PERFECT NUMBER is divisible by four (4) if the last two digits are divisible by 4. A SIMILAR PERFECT NUMBER is divisible by five (5) if it ends in 5 or 0.

11. A SIMILAR PERFECT NUMBER is divisible by six (6) if the number is divisible by 2 as well as 3. A SIMILAR PERFECT NUMBER is divisible by eight (8) if the SUM of its last three digits is divisible by 8.

12. A SIMILAR PERFECT NUMBER is divisible by nine (9) if the SUM of its last three digits is divisible by 9. A SIMILAR PERFECT NUMBER is divisible by ten (10) if it ends in zero.

13. A SIMILAR PERFECT NUMBER is divisible by twelve (12) the number is divisible by 3 as well as 4.

The PRIMARY RULES of the ELEMENTAL CONCEPT of IDEA PERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. An IDEA PERFECT NUMBER is a number that cannot express a real OBJECT but can be defined as a ratio of itself and other numbers.

2. An IDEA PERFECT NUMBER is always SIMILAR in its expression of ratio of itself to other numbers as collections of SETS of more than one.

3. An IDEA PERFECT NUMBER cannot be a uniquely expressed ratio NUMBER, as this is the proper classification of a UNIQUE IMPERFECT NUMBER.

The PRIMARY RULES of the ELEMENTAL CONCEPT of UNIQUE IMPERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. An UNIQUE IMPERFECT NUMBER is a number that represents real ratios but can only be expressed as a ratio of itself uniquely.

2. An UNIQUE IMPERFECT NUMBER is divisible only by itself and 1.

3. All UNIQUE IMPERFECT NUMBER may be defined as a DECIMAL. All UNIQUE IMPERFECT NUMBER DECIMAL will always have an INFINITE number of DECIMAL points.

4. The most perfect UNIQUE IMPERFECT NUMBER is Pi.

5. Existence is an UNIQUE IMPERFECT NUMBER.

The PRIMARY RULES of the ELEMENTAL CONCEPT of SIMILAR IMPERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. A SIMILAR IMPERFECT NUMBER is a number that can only be expressed as a collection of real ratios.

2. A SIMILAR IMPERFECT NUMBER is divisible by itself, UNIQUE PERFECT NUMBERS, SIMILAR PERFECT NUMBERS and 1.

3. The simplest SIMILAR IMPERFECT NUMBER is 1 Degree.

4. All SIMILAR IMPERFECT NUMBER may be defined as a FRACTION or a DECIMAL. All FRACTIONS may be converted to a DECIMAL or vice versa.

5. All SIMILAR IMPERFECT NUMBER DECIMAL will always have a FINITE number of DECIMAL points.

The PRIMARY RULES of the ELEMENTAL CONCEPT of IDEA IMPERFECT NUMBERS of the NUMERICS FOUNDATION CLASS of the EIKOS Language System are:

1. An IDEA IMPERFECT NUMBER is a number that cannot express a real OBJECT nor defined as a ratio of itself in theory as as a collection of perfect unique numbers.

2. The simplest IDEA IMPERFECT NUMBER is -1