Division by Zero

**Division by Zero** (ゼロ除算 *Zero josan*) is a Quirk used by Shirorei Asahi.

### Description

Asahi is able to utilize the effects of division by zero, which has the property of collapsing the affected entity in question. *In a nutshell. Its most widely used effect is to blend things, irreversibly unless a more powerful power reverses it afterwards.*

This power generally needs two entities. Let's take fire and water for example:

* **Blending**: The order matters (A/0)B =/= A(/0B)

Divide fire by zero with water will result in outcomes like fire that is cold to the touch, you can swim in it, or you can put out fires by tossing something burning into the fire.

Divide water by zero with fire will result in outcomes like you can light up a campfire by splashing water into a pile of wood.

* **Sphere of Annihilation**: When the two entities are both nothingness or nonexistence 0/0 = Erase

Projects an all absorbing energy sphere as a projectile or an expanding energy field. The sphere is capable of eliminate anything in its path by simply deleting them from existence. The sphere usually dissipates after a few seconds or longer if the user masters this power.

* **Scrambling**: When there is only one entity A/0 = Scramble

The power will scramble the physical and metaphysical identity of the entity. 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. This 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.

* **Eliminate Duplicates**: When there are duplicates and evasions (AAAAAA...)/0 = A

The power erases all notion of duplicates and erases all evasion countermeasures, making it very effective at fighting clones and mirroring. Teleportation, dimensional shifting, time travel. are examples of things that can be affected.

* **Null Infusion** (A/0)(B/0)(C/0)... = 0

The power also has the powerful property of imbuing a "null" property to its targets, such that when two or more null infused targets interact with each other, they will be erased from existence. Therefore if the user sets the targets to be opposites to each other they can quickly eradicate a large army in the blink of an eye.

* **Interaction With Nothingness and Nonexistence Based Powers** A0+A=A0, 0A+0B=0AB, 0/0+A=0/0

When erasure based powers collide with division by zero, the erasure powers usually loses its erasing ability, instead, they do weird things such as entities being impacted by erasure powers (the "erased" ones) will fuse together or being absorbed by another, while still retains the property of being "erased". Two similar entities, one being "erased" and one does not, the "erased" one will consume the unerased one.

**Dividing Itself By Zero** /0/0=/0/0

Dividing division by zero, or entities being infused with division by zero, does not do anything

## 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 an evil 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.

**Skip this section if you are not interested in mathematics, despite it has been laymennized/simplified to be easy to understand. This basically just provides 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 its 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.