0

0 (zero) is a number which represents "nothing", or having no "thing". For example having "0 apples" is the same as saying one has "no apples", and having 0 of a quantity, such as a substance, is the same as saying one has does not have any of the substance.

0 is the least Ordinal and Cardinal Number in Cantor's Theory of the Transfinite Numbers, and 0 is the "simplest number" in Conway's Theory of the Surreal Numbers. In both cases, 0 is used as the generating case of all other numbers.

## In Mathematics

In mathematics, 0 possesses several unique properties. It is the unique signless number, being neither positive nor negative. It is the position from which the absolute magnitude of all numbers is measured. It's the only number equal to it's own negative, and it's the only number a distance of 0 from 0.

In addition, Zero possesses the **Zero Property of Multiplication** which states 0 of anything is the same as 0:

0 \* x = x \* 0 = 0

Zero also possesses the Zero Property of Addition which states adding or removing 0 from anything is the same as taking no action at all:

x + 0 = 0 + x = x

x - 0 = x

## In Zeroid Theory

In Zeroid Theory, Zero is the "honorary Zeroid", also called the "Zeroth Zeroid" or "Zeroid-Zero", as it was both the first discovered Zeroid, as well as the model on which all other Zeroids are based. It is also the unique Zeroid that is tethered to the nominal notion of nothingness or lack of anything. Other zeroids may be *informally* be thought as representing different extremes of *zeroness* but in fact none of them are physically realizable and have no corporal representation. As such, from a philosophical point of view, Zero is the only Zeroid which can independently be proven to exist.

Zero is the only Zeroid which can exist in physical reality, which makes it unique even amongst the Zeroids. As such a distinction needs to be made between Zero and the Non-Zero Zeroids. In ordinary mathematics, There are no Non-Zero Zeroids, and either Zero is the only Zeroid, or all Zeroids would be regarded as Equal to Zero.

In Zeroid Theory, Zero is recognized as the smallest possible physical quantity, but not the smallest abstract size possible, and not the only distinct Zeroid Possible. Once we define what is meant by size, it is possible to hypothesize sizes which are smaller than zero. Sizes that are smaller than Zero are said to be of *subfinite size* (in contrast with the notion of the *transfinite*), and numbers of such size may be called *introverted,* while numbers larger than Zero may be called *extroverted*.

Zero is the unique division point in Zeroid Theory, for classifying two main types of Zeroids. Zeroids which are smaller than Zero are called *micro-zeroids*, while Zeroids which are larger than zero are called *macro-zeroids.* The *micro-zeroids* were discovered first, and originally called *hyper-zeroes,* among other names.

Zero is the only Zeroid that *actually exists* on the numberline. All other zeroids, and especially the *micro-zeroids,* can not exist on the numberline, because they are no distance away from 0, and they are not at 0 itself either. The situation is not believed to change no matter how much one zooms into 0. Even infinitesimal distances will not reveal the micro-zeroids. If one choses a point on a zoomed in numberline to the right of 0 to represent 0 distance from 0, then every point on the numberline to the left and right, and in the center, becomes 0. The Micro-Zeroids do not exist on this numberline either.

Zero is still regarded as the point at which the subfinite is measured, as the micro-zeroids are said to be closer to Zero than Zero is. Their distance may be described as the difference of Zero and the micro-zeroid in question, and the sum is paradoxically held to be smaller than 0, yet larger than the micro-zeroid being "removed". This distance may also be written as a sum of 0 and the micro-zeroid in so called "Normal Form".

In Zeroid Theory, the properties that always hold in ordinary mathematics are explicitly violated. Let mz stand for an arbitrary micro-zeroid. The Zero Property of multiplication is violated. The product of a subfinite number and 0 is always that subfinite number and never zero:

mz \* 0 = 0 \* mz = mz

The Zero Property of Addition is violated. A subfinite Number plus 0 is not equal to that *subfinite* number:

0 + mz ≠ mz

Subfinite numbers, unlike the finite, infinitesimal, and infinite numbers, are not equal to the distance they are from 0. Instead the distance to Zero is always larger than the size of the subfinite number itself, but still smaller than Zero. With micro-zeroids the action of adding or subtracting Zero is not synonymous with taking no action at all. Subfinite numbers are effected by the addition or subtraction of 0. Furthermore, there exists critical numbers, called *simplexes* which are subfinite numbers that one can not reach from 0 by adding or removing any amount, including any subfinite number.

In Zeroid Theory, Zero still retains all it's properties as long as it never comes in contact with any non-zero Zeroids. Thus none of these properties violate any of the common sense notions of 0, such as 1+0 = 1 or 0-0 = 0 , 0+0 = 0 , 0\*0 = 0 etc.

In Zeroid Theory, division by 0 is not necessarily undefined. For example mz/0 = mz in all known cases. This is a consequence of the fact that 0 \* mz = mz. None of the usual rules for division are violated if zero never interacts with non-zero zeroids. 0/0 for example is still indeterminate, and 1/0 is still undefined.

Zero acts as the titular example of Zeroid Properties such that the non-zero Zeroids inherit many of these properties. Some examples follow:

x \* 0 = 0 : for x larger than 0 --> x \* mz = mz : for x larger than mz

0 \* 0 = 0 --> mz \* mz = mz

0 + 0 = 0 --> mz + mz = mz

0 - 0 = 0 --> mz - mz = mz

0/0 = Indet --> mz/mz = Indet

1/0 = Undef --> 1/mz = Undef