*"Wait, what are you going to do? Divide me by zero?"*- ― Gumball, The Amazing World of Gumball

The power to **divide by zero i.e. make affected entities equivalent and collapse multiple entities into one or generate uncertainty**. Sub-power of Mathematics Manipulation and Logic Manipulation. Can be interpreted as a generalized, more complicated version of Transmutation .

## Also Called

- N/0
- Divided by Zero
- Zero Divider

## Capabilities

The user is able to utilize the effects of division by zero, which has an apparent property of collapsing the affected system in question. *In a nutshell. It's most widely used effect is to turn many things into just one, irreversibly unless a more powerful power reverses it afterwards.*

Basically the major effect of this power is it can equalize entities and/or collapse them into one, so by making A = B users can achieve feats impossible using A and B together as distinct entities.

One typical way to use it is say you want to apply it on A (e.g. a tree), you then use this power via B (e.g. water). After that A = B (Tree and water now become one and the same) with A having the properties of B (e.g. there is no such thing as tree, there's only water.)

What you use to execute this power is important. Using water to divide tree by zero is not the same as using a tree to divide water by zero, because what you use to divide the target by zero will determine what the target will become or take traits on.

The user can also project an all absorbing energy sphere as a projectile or an expanding energy shield by performing a division by zero of nothingness against nothingness. The sphere is capable of eliminate anything in its path (save for objects with at least Nigh Omnipotent natures) by simply deleting them from existence. The sphere usually dissipates after a few seconds or longer if the user master this power.

The power when used on only one target (instead of two or more) will scramble the physical identity of the target. The outcome varies depending on the target, but it is clear that it no longer just has its original identity. Using on computer information or human (or similar) memory, the information will become unreadable gibberish. Using on physical objects such as fire, the object become a physical version of a wildcard, allow it to interact with other things with many properties all at once (e.g. can dissolve salt, can burn trees, can be molded etc.). Perhaps most importantly, using it on the location of anything will result in irreversible Nigh Omnipresence of said thing, unless its existence is being divided again by zero (See second final item in Description of a few applications).

The power when used in a large open grassland (Parks do count) has the powerful property of imbuing a "null" property to its targets, such that when the target interact with something of its opposite aspects, both will be erased from existence. Therefore users who teamed up in battle with another user who can arbitrary set the targets to be opposites to each other can quickly eradicate a large army in the blink of an eye.

## Description of a few applications

Due to its collapsing property, it can be used to make two entities/concepts equivalent. For example, make fire and water equivalent so they inflict the same effects, or Space-Time Manipulation by making space and time equivalent so that you can freely travel in time and/or make others forced to be dragged along a certain direction in space.

For example, division by zero can create black holes since black holes warp space-time to such an extent that the law of physics as we know it collapses at the center of it.

A more concrete example is say you have a base and you defend it using a steel wall. Now a user of this power will divide the wall by zero using a wooden stick. What happens is that your steel wall will become a pile of sticks and any future steel walls you built will become sticks. Thus you lost your protection. In this case we make **steel wall = wooden** **stick** by dividing the steel wall by zero using a wooden stick.

Another application which also make use of this "collapse the system" property is say you are a evil and fire user trying to fight a good water user. Given in the universe where this battle takes place has a rule such that water wins over fire. In order to turn the tables, you divide the concept "opposites" by zero. The outcome of this is that the many different types of opposites (polar, elemental, moral etc.) all collapsed into one type decided by the user, say moral. Now the law of this universe is circumvented and you will be unharmed by the water attacks since the elemental opposite aspect of it is removed, leaving only the moral aspect which depends on the moral aliment of the water user, which is good. Thus you now effectively end up only with evil vs good, which is not specified by the rules of this universe thus the outcome that you win becomes possible.

Another even direct example is based on the principle of the previous example. Say you are fighting against a user who has the ability to be in many places at once, or make a lot of clones. By dividing by zero you force all copies to be gone regardless of the nature of the duplication (unless because of plot reasons or the user is at least Nigh Omnipotent). This is actually very powerful because you can effectively prevent all teleportation or dimensional travel ability of the opponent (e.g. time travel, dimensional shifting, the ability to initiate a chronolock or omnilock etc. NB If the user is already omnilocked, then it does not work because of the properties of omnilock), provided that the opponent is less powerful than you and the story would not be affected resulting in plot control.

The scrambling ability can be used to rapidly spread something over a vast region in the blink of an eye, such as a virus or a wildfire. (By location of object/0)

## Definition

The power basically utilize the metaphorical effects when one tries to introduce division by zero into number system and suggested by the memes of the same name in order to achieve its purpose.

**Click here to skip this section if you are not interested in maths, despite it has been laymennized to be easy to understand. This basically just provide the background on what this power is based on.**

Division is the inverse of multiplication, that is if e.g. A × B = C then A = C/B where C/B is a unique number (Let's call it D for simplicity) such that D × B = C. For the case of zero in most number systems (reals, complex etc.) because of the axioms, or rules that build them result in a property that 0 × n = 0 for any number n in the number system, there is no unique number D such that D × 0 = n (In the case of 0/0 theoretically all numbers will satisfy it's requirement since it basically means it is a number D such that D × 0 = 0 thus it is indeterminate and tells no useful information, hence it is undefined). Thus division by zero is undefined.

In a sense it is the most pathological example of a map, where a single element is mapped to all elements in a mathematical space.

In more exotic number system such as the complex projective plane (Think the complex plane as a 2D sheet of paper. You then roll it up into a sphere and glue it at a point known as infinity) division by zero is possible and this will equal to infinity (note this infinity has no signs because -infinity and infinity are equivalent in this number system). However 0/0 is still undefined.

0/0 can be defined using a weird mathematics structure known as a wheel, by abandoning the notion that / is a binary operation (i.e. requires two inputs and output one result) but treated as a unitary operation such that /a.b=a./b. The result is that in general 0x ≠ 0, x - x ≠ 0 and terms with 0 will linger behind and can be group together e.g. 0x + 0y = 0xy. 0/0 however will turn any expressions that contains it into itself.

Another approach in defining division by zero is another weird structure known as a meadow, which make use of a rule x*(x^-1*x) = x and 0^-1*(x^-1)^-1 = 0^-1*x+(1-x*x^-1) to force /0 into a certain number. However in that case we lost the usual meaning for multiplicative inverse of 0 since 0^-1*0=0≠1, although division is now defined for all numbers.

Division by zero in the real number system is only possible if you relax one of its axioms else you end up having all numbers equal to each other, thus collapsing the entire real number line into just a single entity (Usually quoted as the trivial ring which means it is a set [r] such that r × r = r and r + r = r, a very boring mathematical structure.

**Definition ends here.**

## Applications

- Black Hole Creation
- Infinite Supply (If used on anything located on any spherical surface)
- Space-Time Manipulation
- Transmutation (Main application, more general, not just limited to physical entities, and its scale is always the entire reality. This is limited to avoid conflicting with Omnipotent beings).
- Nothingness Manipulation (As a projectile or energy field that removes matter or energy upon contact)
- Nigh Omnipresence (Whenever the location of the target is divided by zero)
- Nothingness Infusion (Affected target gained a property that can allow them to be erased from existence once a certain condition is met)

## Associations

## Limitations

- Tend to collapse entities, thus effects is usually one-sided. Can be destructive as other powers are often required to reverse the operation.
- Unskilled user will often found a black hole is left in its wake.
- Requires an extensive understand of the concepts involved in the collapse to decided accurately what to collapse, as due to the ambiguous nature of concepts, many unintended side effects can occur when the collapse is brought out carelessly (e.g. via "damage suffered"/0 via "healing himself", a person will be impervious to all kinds of damage, this include beneficial things such as blood transfusion (unless it can be done without a needle)
- Although extremely capable, it cannot match the power of Omnipotence or Nigh Omnipotence
- Entities who form the only example of their respective categories are immune to collapse with regard to that category; collapsing attempt with regard to another category by exploiting that fact that entities can belong to multiple categories, these aforementioned entities can be resistant to that depending on the Verse requirements
- The scrambling ability of this power on physical target (instead of just one aspect of it such as its location) will make the target susceptible to Uncertainty Manipulation, since its original identity or nature has been lost and can be anything possible, including its original self, thus uncertain.

## Known Users

- Anne Poole (
*Fine Structure*) - Thoma Avenir (
*Magical Record Lyrical Nanoha Force*) - 01 - The Magician (
*Pharos of the End*)

## Known Objects

- Null Driver (
*Iji*)