Z*

Z\* (Z Prime), is the current Primary Model of Zeroid Theory. It is informally codified in a number of non-standard conceptions/interpretations of distance, size, and arithmetic operations and relations. Z\* is not yet fully understood, but a partial list of it's definitions along with it's underlying assumptions, called it's Zeroid Postulates (borrowing from the term in geometry), is provided and will be improved upon as the Model becomes better understood. To understand the current state of Zeroid Theory, and how Zeroids behave with each other and with other types of numbers under the four basic arithmetic operations, it will be helpful to have a grasp of these postulates and how they interact with each other. It is not known if the postulates are consistent with each other. It is however generally suspected that the concepts of distance and size break down at the Zeroid level, and some adjustments to one, the other, or both, is required to create a consistent system.

The Postulates are written with the assumption that the reader is familiar with the basics of arithmetic on real and complex numbers, and with principles of Euclidean Geometry.

We begin with a set of basic definitions, then we will go over the Postulates, which are split into The Size Postulates and The Distance Postulates.

In definitions, words or phrases which are in bold are terms with specialized meaning within Z\* that require definition.

## Zeroid Definitions

**Zeroid Definition I. Number**

*A **Number** is a mathematical entity composed of two parts, a **sign (II)** and a **size (III)**.*

**Zeroid Definition II. Sign**

*The **Sign** of a number can be thought of as either the direction one travels in a straight line to go from 0 to the numbers "position", or as a "color" that distinguishes it from other numbers of the same **size (III)**.*

**Zeroid Definition III. Size**

*The **Size** of a number is half the **distance (VI)** from itself to its **negation (IV)**.*

**Zeroid Definition IV. Negation**

*The **Negation** of a number, a, is either (-1)\*a or a\*(-1).*

**Zeroid Definition V. Magnitude**

*The **Magnitude** of a Number is it's **distance (VI)** from 0 to itself.*

**Zeroid Definition VI. Distance**

*The **Distance** between two numbers, a and b, is the **normalization (XIII)** of either of their differences: a-b or b-a.*

**Zeroid Definition VII. Self-Distance**

*The **Self-Distance** of a number is the difference of the number from itself: a - a.*

**Zeroid Definition VIII. Route**

*A **Route** is a path formed of straight lines where each line joins adjacent numbers in a sequence of numbers.*

**Zeroid Definition IX. Total Distance**

*The **Total Distance** of a Route is the sum of the distances of it's straight line segments.*

**Zeroid Definition X. Normoid**

*A **Normoid** is number, N, such that N\*a and a\*N are the same size as a but may or may not have the same sign as a.*

**Zeroid Definition XI. Rotoroid**

*A **Rotoroid** is a normoid, R, such that, for any number a, R\*a and a\*R are the same size as a but they both differ in the sign of a.*

**Zeroid Definition XII. Identity Normoid**

The ***Identity Normoid**, or The **Identoid**, sometimes simply called **The Norm**, is the (unique) Normoid, usually denoted by 1, such that 1\*a and a\*1 are the same size and the same sign as a.*

**Zeroid Definition XIII. Normalization**

*The **Normalization** of a number, a, is the multiplication of it by the Rotoroid, R' , chosen such that after a has been **standardized (XXIII)**, st(a)\*R' is positive.*

**Zeroid Definition XIV. Signless**

*A number, a, is said to be **Signless** if it is either (1) **equal (XV)** to it's own negation, that is, -a=a, or (2) for any normoid N, a\*N=a and N\*a=a.*

**Zeroid Definition XV. Equality**

*If two numbers, a and b, have the same sign and size, then they are equal and we may write: a=b or b=a. If they are the same size but of different sign, they are equal if and only if they are **signless (XIV)**, in which case we may treat them as if they are the same number, and may omit the sign.*

**Zeroid Definition XVI. Large Number**

*A **Large Number** is any number whose size is larger than The Norm.*

**Zeroid Definition XVII. Normal Number**

*A **Normal Number** is any number whose size is the same as The Norm. A Normal Number is said to be neither Large nor Small.*

**Zeroid Definition XVIII. Small Number**

*A **Small Number** is any number whose size is smaller than The Norm.*

**Zeroid Definition XIX. Exofinite Number**

*An **Exofinite** number is a number whose size is larger 0.*

**Zeroid Definition XX. Subfinite Number**

*A **Subfinite** number is a number whose size is smaller than 0.*

**Zeroid Definition XXI. Mixfinite Number**

*A **Mixfinite** number is a number which is the sum or difference of an exofinite number and a subfinite number.*

**Zeroid Definition XXII. Simplex**

*A **Simplex** is a number, s, which can not be expressed as the sum of two numbers, both smaller than itself, nor as the sum of two or more simplexes all of a different sizes than s.*

**Zeroid Definition XXIII. Complex**

*A **Complex** is a number which can be expressed as the sum of two smaller parts.*

**Zeroid Definition XXIII. Standardization**

The ***Standardization** of a sum is when it as been simplified as far as it can be simplified such that it is a sum of distinct simplexes and the least number of additional complexes.*

**Zeroid Definition XXIV. Positive Number**

*A **Positive Number** is any number greater than 0 or alternatively to the right of 0 on any conceivable number line.*

**Zeroid Definition XXV. Negative Number**

A ***Negative Number*** *is any number less than 0 or alternatively to the left of 0 on any conceivable number line.*

**Zeroid Definition XXVI. Critical Number**

*A **Critical Number** is either a small number which can not be made any smaller by multiplying it by anything the same size or larger than itself, or a large number which can not be made any larger by multiplying it by anything the same size or smaller than itself.*

**Zeroid Definition XXVII. Zeroid**

*A **Zeroid** is a critical simplex small number.*

**Zeroid Definition XXVIII. Macro-Zeroid**

*A **Macro-Zeroid** is a exofinite zeroid.*

**Zeroid Definition XXIX. Micro-Zeroid**

*A **Micro-Zeroid** is a subfinite zeroid.*

**Zeroid Definition XXX. Zeroid Normal Form**

***Zeroid Normal Form**, which may be abbreviated as ZNF, is a sum of zeroids such that all zeroids are distinct, and the later a zeroid occurs in the sum the smaller it is.*

## Zeroid Postulates

### Size Postulates

**Zeroid Postulate I. Multiplication is Commutative**

a\*b = b\*a

**Zeroid Postulate II. Multiplication is Associative**

(a\*b)\*c = a\*(b\*c)

**Zeroid Postulate III. Preservation of The Multiplicative Identity**

*1 is the unique universal element which when multiplied by any other number, returns that number. Alternatively we may state this postulate as: (1) There Exists an Identity Normoid, (2) It is unique, and (3) It is the number +1. Symbolically we may write:*

∀*x* : 1\**x* = *x*\*1 = *x*

**Zeroid Postulate IV. Preservation of Signed Multiplication**

*For any two numbers, x and y, if there exists a pair of normoids, a and b, such that:* sign(x)=sign(a) *and* sign(y)=sign(b), *then:* sign(x\*y)=sign(a\*b), *independent of the sizes of x and y.*

**Zeroid Postulate V. Signless Multiplication**

*Anything times a signless number, also a signless number times anything, is also a signless number. Let a number be written as [θ](a), where the expression inside the brackets is the sign and the number written next to it is the size. If a number is signless we may leave the brackets empty. We can then denote our postulate as follows:*

[θ](a)\*[ ](z) = [ ](a\*z)

[ ](z) \* [θ](a) = [ ](z\*a)

**Zeroid Postulate VI. The Large Number Postulate**

*Any Number, times a Large Number, will be larger than or the same size as The Original Number. Let a be any number, M be a large number, and let a<<=>b mean, a is smaller than or the same size as b. In this case we may write:*

a <<=> a\*M

**Zeroid Postulate VII. The Small Number Postulate**

*Any Number, times a Small Number, will be smaller than or the same size as The Original Number. Let a be any number, m be a small number, and let a<<=>b mean, a is smaller than or the same size as b. In this case we may write:*

a\*m <<=> a

**Zeroid Postulate VIII. Invertibility of Multiplication**

*If and only if a\*b = c, we may say c/a = b. If b does not exist, we say c/a is **undefined**. If c/a has one unique solution, b, then it is equal to b. If there are multiple solutions, then c/a is **Indeterminate** and is not a number. In this case it may treated as being equal and the same size to any of it's solutions, but because it is not a number, The Transitivity of Equality nor The Transitivity of Size are applicable and can not be used to argue distinct numbers are equal or the same size.*

**Zeroid Postulate IX. Trichotomy of Size Relations**

*For two numbers, x and y, one of the three must hold: Either (1) x is smaller than y, (2) y is smaller than x, or (3) x and y are the same size. If we use a<<>b to mean "a is smaller than b", and a<=>b to mean "a is the same size as b", then we may denote this trichotomy as follows:*

x<<>y *or* y<<>x *or* x<=>y

### Distance Postulates

**Zeroid Postulate X. Addition is Commutative**

a+b = b+a

**Zeroid Postulate XI. Addition is Associative**

(a+b)+c = a+(b+c)

**Zeroid Postulate XII. Multiplication Distributes over Addition**

a\*(b+c) = a\*b + a\*c

**Zeroid Postulate XIII. Limitation of Additive Identity**

*If a number, ζ , is a **non-zero zeroid** or a **sum containing a non-zero zeroid** then we may **NOT** **assume** 0 is the additive Identity. That is, we may **NOT assume** ζ+0=ζ or ζ-0=ζ . Otherwise, we assume the Additive Identity remains unchanged.*

**Zeroid Postulate XIV. Preservation of Geometric Distance**

*The closest **point in space** to a point in space is the **point** itself, which is always a distance of 0 from itself in any direction. Any other point in space is a distance larger than 0 from the point, exclusively in one unique direction.*

**Zeroid Postulate XV. Assertion of Subfinite Distances**

*For any chosen point in space there exists at least one **entity** closer to it than any point in space.*

**Zeroid Postulate XVI. Absolute Centrality**

*The closer something is to 0, the smaller it is.*

## See Also

0

1

Zeroid Theory

Arcanum

Sanctuary