Number

The "Numbers" are a set of 100 Xyz Monsters that are included in the show Yu-Gi-Oh! ZEXAL. Supposedly pieces of the mysterious entity Astral, the protagonist of the show, Yuma Tsukumo, must acquire all 100 and solve the mysteries behind these powerful creatures.

Description
Here is the information regarding the Numbers, in both the canon series, and in other stories in other universes.

ZEXAL
When Yuma Tsukumo unlocked the Door with his Emperor's Key, he opened a portal between his own dimension and another, allowing Astral to come into his world. Yuma found himself trapped within a sphere of blank cards, which shortly shattered away. By crossing into Yuma's world, Astral's memories got transformed into the hundred "Number" cards and scattered in unknown directions. In order to regain his lost memories, Astral teamed up with Yuma. These cards that take the form of what whoever finds them desires the most, design themselves to fit their original owner's Decks, and keep that form forever. Some Duelists may have more than one "Number" card that fit their Deck. Their card text is written in a strange alien language that only those who wield them can understand (in the dub, all "Numbers" appear normally like other cards seen with the exception of episode 2 which Number 39: Utopia is blurred). When they are Summoned, their respective numbers are printed on their users, appearing in such places as the forehead, neck, left and right hands (in Cameron Clix's case, it appeared under his right eye and in Number 56's Holder's case, it appeared in his left shoulder). When Summoned, they use a galaxy-like "Overlay Network" portal in contrast to regular Xyz Monsters, who use a red spiral portal. All Xyz Monsters are said to come from another dimension and are made of antimatter. This would include the "Numbers". They have been shown to be very dangerous, as their users are easily possessed by their power (with the exception of Yuma due to his partnership with Astral and Kaito who has powers similar to those of Astral's). It is said that these "Number" monsters cannot be destroyed with cards in this world. "Number" monsters can only be defeated by other "Number" monsters. However, when two or more "Numbers" confront themselves in a Duel they'll fight each other to consume one another. When a "Number" is defeated in a Duel by Yuma, Yuma obtains it, and its power is absorbed into Astral. Also when Yuma defeats a "Number" monster Astral gains another part of his memory. Astral mentions there are 99 Numbers, but according to Kaito Tenjo, there are 100 Numbers. As Astral already had 1 of the Numbers, "", it is possible Astral was referring to the other 99 Numbers.

Alternate Numbers
However, there are also alternate Numbers from other universes. In some realities, these Numbers have replaced other Numbers from the original Duel Monsters universe. For example, in one alternate reality, Kotori, Yuma's love interest, gains Number 13: Paradise to help out Number 39: Utopia. Plus, some duelists are able to control normal Numbers as well without having the Number take them over, such as Chris Muir, who has Number T: Chrono Star Jiffy. Since these Numbers are from different universes, some Numbers in one reality will be completely different Numbers in others, allowing for different cards to have the same Number.

Abilities
In the anime, "Numbers" have a few "special" abilities that are unique. "Numbers" can only be destroyed by other "Numbers". The "Numbers" amplify the darkness and desires of humans, whether these cause good or bad. When a "Number" is defeated, they are absorbed into Astral, and can be used by any duelist whom Astral is currently with. Each "Number" contains a piece of Astral's memory, which he recovers each time he absorbs one. They are also known to manipulate their user into summoning them as soon as possible, sometimes to the point of cheating as seen with Rikuo and Kaio. Numbers also manipulate their users into protecting them at any cost, as seen when "Number 83: Galaxy Queen" made Fuya Okudaira believe that she was his mother. Thus, he equipped her with "Cursed Seal of Dead Max" to protect her from being destroyed by battle, and changed the attack target to his "D.D. ESPer Star Robin" via "Star Robin is Forever!", causing him to lose the Duel. A person possessed by a "Number" easily becomes addicted to their power and desires to obtain more "Numbers". It is, however, possible for some people to resist this effect, at least temporarily, as Tetsuo Takeda managed to do when he used Number 39: Utopia to save Yuma from "Number 96: Black Mist". Interestingly, whenever a "Number" is Xyz Summoned, they initially appear as a strange object, what is possibly a sealed form, until they transform into the actual monster and gain their Number on their bodies.

Special Abilities
Some Numbers have abilities unique to them. For example, "" gave unbeatable luck to until his "" was destroyed, but it returned when the card was given to Mayumi, transferring its luck to her. All Numbers are sentient to some degree (evidenced by the voice that speaks to them), but only a handful are capable of broader thinking, such as Utopia, Black Mist, Paradise, Mirror's Dark Half, Rampart, and Matrix. In some fanfictions, the abilities of the Numbers can even give the equivalent of superpowers, such as Number 48: Godspeed Mirage granting super speed, or Number 37: Zero Passion Pandemic causing nearby others to become very sick.

Subtypes
Furthermore, there are several different subsets of the Numbers, which will be described below.

Chaos Numbers
The "Chaos Numbers" are a new set of Xyz Monsters. They seem to be upgrades of Numbers, but how and why is not yet certain. All that is known is that they can use the previous Number as an Xyz Material, along with the previous Number's materials, have special support monsters called "Zexal Weapons", and have rather powerful effects that can only be activated at certain times of the duel.

A breed of Chaos Numbers can also be created from certain "Rank-Up-Magic" cards, among them being "Chaos Over-Hundred Numbers" used by the Barians.

Ranked-Up Numbers
Outside of cards that are designed to Summon "Chaos Xyz" monsters and Chaos Numbers, Numbers that are affected by "Rank-Up-Magic" cards become very distant from their roots, tending to associate more with the origin of their Rank-Up Magic.

Cross Numbers
The "Cross Numbers" are an upgraded style of Ranked-Up Numbers introduced in a fic taking place in the future of ZEXAL. Yuko Tsukumo, daughter of Yuma and Kotori, has the ability to use the ZEXAL power to combine two Numbers into one being in dire situations. The Numbers of these monsters are the products of their Fusion Materials' Numbers, with the letter "X" before the number to indicate its status as a Cross Number. These monsters are effectively fusions of 2 (or sometimes 3) Xyz Monsters, and possess the combined Ranks of its components. One such Number is "Number X663: Utopia Rider," which combines the offensive abilities of "" and the defensive abilities of "".

Jiffy's Cross Numbers
Adding to the mix are the Cross Numbers created by adding in "Number T: Chrono Star Jiffy", a curious little Xyz Monster that contains a power similar to "". Jiffy can easily combine with other Numbers to increase her own power, and her Cross Numbers have slightly different effects. Like a regular Cross Number, each effect needs a specific Number to be utilized in the Xyz Summon to activate, but if a specific Xyz Material is detached to pay for the effect, an additional effect is usually triggered.

Arcadia's Cross Numbers
Not unlike Jiffy, there is another Number with power similar to "Neos": "Number 100: Arcadia", a 101st Emperor's Key Number created between the energies of Number 39 and 13, taking the form of what could be seen as their child. Arcadia is capable of Rank-Crossing with other Numbers without the need of a "Rank-Cross-Magic" card, making him the original "Rank-Cross" monster. His Cross forms enhance his power with their powers, the other Numbers becoming armor of sorts and influencing his weapons. Unlike other Cross Numbers, whose Numbers are read in normal Japanese fashion, Arcadia's Cross Numbers are read as "## Hundred" (## Hyaku).

Verz Numbers
The "Verz Numbers" are only used by Yamoto in Taylor Gorrell's fanfiction universe. Through the use of his "Rank-Up-Magic Infestation Evolution" card, Yamoto can corrupt the Numbers he has into "Neoverz" monsters. Although the TCG term for the Verz Archetype is, Taylor Gorrell prefers to use Verz, so in the TCG, Verz Numbers are denoted as "Number V##" on English cards. All Verz Numbers are "Neoverz" monsters.

Notably, all Verz Numbers retain the Numbers' invincibility by battle with non-Numbers (or some variation of it) in the TCG/OCG.

Flight Numbers
The "Flight Numbers" are only used by Commander Cincinnatus Foster, formed through his "Rank-Up-Magic Cross Into The Blue" card. "Flight Numbers" are used alongside the "FXyz" cards due to their shared "Rank-Up-Magic", and like the "FXyz"s, the "Flight Numbers" are all WIND. These cards are noted as "Number F##" on their card names. More information to come.

Imperial Numbers
The "Imperial Numbers" are formed through "Rank-Up-Magic Royal Fire Formation - Kogumaza", thus giving them allegiance to the ""s. All "Imperial Numbers" are FIRE. These cards are noted as "Number I##" on their card names. More information to come.

Half Numbers
The "Half Numbers" are a set of 100 Numbers that will appear in future fanfic of ZEXAL, identified by the ".5" that follows the number. In this yet unknown alternate reality of this fanfic, the original Numbers have split to form themselves and these Half Numbers, obtaining one shows bits and pieces of the memory it holds, while obtaining both the original and the half will show the whole memory. Some of them follow the same theme as their original half, i.e. and Number 33.5: Chronomaly Pyramid Golem, and others take on their own form, i.e.  and Number 50.5: Patriot's Firestorm.

United Numbers
The United Numbers are a form of evolution used by Simon Notroson, Yuma, and a few of their friends in Rocket Knight 777's Zexal universe. They are created when two Numbers are overlayed together, or "fused", creating a form that seems to embody elements of both monsters. These card have a "U" preceding the Number, like the Chaos Numbers have a C preceding the Number. The Number itself is the sum of the two combined Numbers. The Numbers of the cards used in the unification appear in the relative spot they appeared on their original forms, or are placed in a spot appropriate to the new monster. More info to come.

True Numbers
True Numbers, or TNumbers, are a variation of United Numbers, where a Number and its corresponding Half Number are overlayed together, or "fused", to unlock their full and true power. More information coming soon.

Numbers From Other Worlds
There are some Numbers that do not originate from the standard universe, and therefor do not follow the usual conventions of Numbers, the Half Numbers and some types of Ranked-Up Numbers mentioned above being one such example. There are a few Numbers that originate from the, and go past 100, like "" and ". It is known the planet of Hearth, which lies in a pocket dimension separate from Earth, has a single Number: "Number 169: Hearth Phoenix, Divine Solar Avatar". Other odd examples of Numbers that do not follow the usual order include "Number Q: Chrononaut" and Number α: Variable Fusionist".

Minus Numbers
The "Minus Numbers" (AKA Negative Numbers, "negative" effectively translates to "minus" in Japanese) are an alternate set of Numbers created by the Barians of the above-mentioned future of ZEXAL. They are denoted (both in English and Kanji) as having a negative number in both their names and artwork. Most Minus Numbers themselves are not actual cards, but they are formed from the card "Rank-Up-Magic Minus Operation" by converting existing Xyz Monsters (except Numbers). The focus of the Minus Numbers is to gather the original Numbers, as Minus Operation successfully pulls 45 of them from. Each Minus Number, pre-determined depending on the copy of Minus Operation held, is capable of locating the original with the same Number. Some Minus Numbers have no visual relation to the positive counterparts, while others have a much more visible link (Ex. and Number -39: Purgatory). There are no Minus Multiplication Numbers, but Minus Numbers can be substituted for their positive counterparts via “Rank-Cross-Magic Minus Fusion”.

The Over-Hundred Numbers in the actual canon are Minus Numbers in ZEXAL - The Future, such as Galaxy-Eyes Tachyon Dragon being "Number -7" instead of "Number 107".

Order Numbers
A variation of the Minus Numbers, Order Numbers are the Minus equivalent of Chaos Numbers (as Order is the opposite of Chaos), Summoned through "" (which has a different effect in the story where Order Numbers take place). More information to come.

Galaxy Numbers (Number G)
A variation of the normal Number C (Chaos Numbers). They are Summoned by "Rank-Up-Magic Galaxion Force". Most of these are Number monsters that originate from the Time World. They are tens to hundred times bigger than their normal size. Awakening these Numbers is similar to awakening Chaos Numbers, but a bit different. More information coming soon.

Lost Numbers
Finally, there are the "Lost Numbers", a very small group of Xynchro Monsters. All of them are "00", and they all have more specific requirements than "Multinetwork God Yotta Byte". However, they are very few and very far between, though their power is quite impressive. The only one known of, "Lost Number 00: Biobeast Zonoid", is virtually unstoppable with an unnegatable negation ability.