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*Your Journey Begins Here!*

(From here you will eventually be able to navigate to any other number on the Wiki)

## Introduction

**0** (**zero**) is the smallest possible ordinal and can be understood as the Empty Set, the simplest possible set, the smallest nature number. It is also the smallest possible cardinal number. It is also the smallest whole number.

It is significant as the smallest number to appear in 0to Videos (as the name implies), and is unambiguously chosen as a starting point, suggesting either an Ordinal or Cardinal paradigm. 0 is not however considered the "smallest number" on the Wiki however. Smaller numbers, dubbed Hyper-Zeroes, Micro-Zeroids, or simply Zeroids, have been defined and some of their properties have been investigated. For an example and to begin to understand the concept you can check out the entry on \_.

Although 0 has often been maligned throughout history, and it's existence denied, it is not a "fictional number" (one could only imagine what mathematicians who rejected 0 would think of Hyper-Zeroes!).

0 is also notable for showing up in a lot of important Ordinal and Cardinal notations as it's used to denote the first of some property. Notable examples include Epsilon-0, Gamma-0, and Aleph-0 from professional mathematics. Some NO! numbers that use 0 in their name include Absolute Null (not to be confused with Absolute Infinity) and Unbeatable Naught.

0 is the smallest Nullum-Class Number by definition. Numbers which are greater than 0 are positive numbers. Numbers which are less than 0 are negative numbers. Thus 0 acts as the dividing point between the negatives and positives. The Nullum-Class is composed of 0 and the Positive Integers, and can be said to represent the Class of Finite Multiplicities or Cardinalities. To continue up through the Nullum-Class hierarchy you can Click the link in the Infobox for Larger Numbers. Although under development, this will eventually allow you to gradually ascend up the chain of Finite Cardinalities and learn some things along the way from googology. 0 is the smallest Nullology number, marking the end of Nullology. Nullology has the special name for 0, which is: Zeero.

## Notable Properties

0 is a unique Number in mathematics in a lot of ways. It is often however, misunderstood. 0 essentially is a mathematical construct which means having "no things" of a thing. From this definition it follows that having "no apples" is essentially the same as having "no oranges", because in both cases one can more simply say one has "nothing", that is "0". Despite this, in day-to-day life, the qualitative aspect matters, and 0 miles per hour is not the same as having 0 apples, or 0 days of vacation. In each case these could not be free interchanged in natural language and would lead to such nonsensical notions of having a speed of "0 apples" being equivalent to "0 miles per hour". It is however non-arbitrary say one has a "0 speed" since the units of speed here do not matter. 0 mph is the same as 0 kph , or meters per second, or inches per minute, etc.

In mathematics 0 may be multiplied by any "number" to obtain 0. This is under the understanding that having "0" of any number is still just "0", whether it be "0 ones" "0 twos" "0 threes" etc. This means that 0 is definitionally a common multiple of every integer.

When multiplying 0 by 0, we obtain 0. This can seem confusing. What does it mean to have "nothing" of "nothing"? To understand this imagine boxes that contain 0 cookies within them. If we have say 100 boxes of 0 cookies, we still have 0 cookies, but if we have 0 boxes of 0 cookies ... we still have 0 cookies.

Since 0 represents nothing, adding or subtracting it from any other number has "no" effect on it.

We can also attempt to describe fractions with 0 as a numerator as in 0/2 , 0/3 , 0/4, etc. As is the case with almost everything with 0, it doesn't actually matter what we have "0 of", whether that be 0 halves, 0 thirds, or 0 quarters, the end result is that we still have nothing.

Another way to look at it is 0/2 is attempting to divide 0 "in half". However, if we think about it in terms of trying to find something when multiplied by 2 that gives us 0, 0 gives us a ready made answer: two zeros is zero, so "half of zero" as opposed to "zero halves", is still just 0.

Lastly we can attempt to divide numbers by 0.

Expressions like 1/0 tend to be confusing because it is not clear what a denominator of 0 is suppose to mean in terms of "units", like wholes, halves, thirds and so on. However we can think of 1/0 as trying to figure out how many times 0 can go into 1. The problem is any number of 0s is still 0, so there is no answer. We have an undefined expression with no possible answer.

0/0 presents yet another difficulty in that any number of 0s can "fit within" 0. And so instead of there being no answer, there is everything as the answer. Any number is correct. 0/0 = 0 , 0/0 = 1 , 0/0 = 2 , 0/0 = 3, etc.

0^0 presents difficulties that are similar to 0/0. In some contexts it is treated as being equal to everything, while in others it is sometimes treated as equal to 1, on account of the fact that the limit of x^x as x approaches 0 from the right is 1.