Editing Division by zero (section)

==Higher mathematics==
The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as a framework to support the extension of the realm of numbers to which they apply. For instance, to make it possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set of [[integer]]s in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to the [[rational number]]s. During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (is ''undefined'') in the whole number setting, this remains true as the setting expands to the [[real number|real]] or even [[complex number]]s.<ref>{{harvnb|Klein|1925|page=63}}</ref>

As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers.<ref>{{harvnb|Klein|1925|page=26}}</ref> Similarly, when the realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't a rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers.

In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on [[set theory]]. First, the natural numbers (including zero) are established on an axiomatic basis such as [[Peano axioms|Peano's axiom system]] and then this is expanded to the [[ring of integers]]. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of [[ordered pair]]s of integers, {{math|{(''a'', ''b'')}<nowiki/>}} with {{math|''b'' ≠ 0}}, define a [[binary relation]] on this set by {{math|(''a'', ''b'') ≃ (''c'', ''d'')}} if and only if {{math|1=''ad'' = ''bc''}}. This relation is shown to be an [[equivalence relation]] and its [[equivalence class]]es are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying [[Transitive relation|transitivity]]).<ref>{{harvnb|Schumacher|1996|page=149}}</ref><ref>{{harvnb|Hamilton|1982|page=19}}</ref><ref>{{harvnb|Henkin|Smith|Varineau|Walsh|2012|page=292}}</ref>

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.

===Non-standard analysis===
In the [[hyperreal number]]s, division by zero is still impossible, but division by non-zero [[infinitesimal]]s is possible.<ref>{{citation|last=Keisler |first=H. Jerome |title=Elementary Calculus: An Infinitesimal Approach |url=https://people.math.wisc.edu/~hkeisler/calc.html |orig-year=1986 |year=2023 |publisher=Prindle, Weber & Schmidt |pages=29–30}}</ref> The same holds true in the [[surreal number]]s.<ref>{{citation |url=https://books.google.com/books?id=tXiVo8qA5PQC |title=On Numbers and Games |edition=2nd |last=Conway |first=John H. |date=2000 |orig-year=1976 |publisher=CRC Press |isbn=9781568811277 |page=20}}</ref>

===Distribution theory===
In [[Distribution (mathematics)|distribution theory]] one can extend the function <math display="inline">\frac{1}{x}</math> to a distribution on the whole space of real numbers (in effect by using [[Cauchy principal value]]s). It does not, however, make sense to ask for a "value" of this distribution at ''x''&nbsp;=&nbsp;0; a sophisticated answer refers to the [[singular support]] of the distribution.

===Linear algebra===
In [[matrix (mathematics)|matrix]] algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can be [[matrix addition|added]] and [[matrix multiplication|multiplied]], and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by its [[Invertible matrix|inverse]]. Not all matrices have inverses.<ref>{{citation|last=Gbur |first=Greg |author-link=Greg Gbur |title=Mathematical Methods for Optical Physics and Engineering |pages=88–93 |year=2011 |isbn=978-0-521-51610-5 |publisher=Cambridge University Press|bibcode=2011mmop.book.....G }}</ref> For example, a [[zero matrix|matrix containing only zeros]] is not invertible.

One can define a pseudo-division, by setting ''a''/''b''&nbsp;=&nbsp;''ab''<sup>+</sup>, in which ''b''<sup>+</sup> represents the [[Moore–Penrose inverse|pseudoinverse]] of ''b''. It can be proven that if ''b''<sup>−1</sup> exists, then ''b''<sup>+</sup> = ''b''<sup>−1</sup>. If ''b'' equals 0, then b<sup>+</sup> = 0.

===Abstract algebra===
In [[abstract algebra]], the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a [[commutative ring]], which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called [[Localization (commutative algebra)|localization]]. However, the localization of every commutative ring at zero is the [[trivial ring]], where <math>0 = 1</math>, so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.

Nevertheless, any number system that forms a [[commutative ring]] can be extended to a structure called a [[Wheel theory|wheel]] in which division by zero is always possible.<ref>{{citation |year=2004 |last=Carlström |first=Jesper |title=Wheels: On Division by Zero |journal=Mathematical Structures in Computer Science |doi=10.1017/S0960129503004110 |volume=14 |issue=1 |pages=143–184 |doi-broken-date=1 November 2024 |url=http://www2.math.su.se/reports/2001/11/ }}</ref> However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element <math>1</math>, and if the original system was an [[integral domain]], the multiplication in the wheel no longer results in a [[cancellative semigroup]].

The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as [[Ring (mathematics)|rings]] and [[Field (mathematics)|fields]]. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a [[skew field]] (which for this reason is called a [[division ring]]). However, in other rings, division by nonzero elements may also pose problems. For example, the ring '''Z'''/6'''Z''' of integers mod 6. The meaning of the expression <math display="inline">\frac{2}{2}</math> should be the solution ''x'' of the equation <math>2x = 2</math>. But in the ring '''Z'''/6'''Z''', 2 is a [[zero divisor]]. This equation has two distinct solutions, {{math|1=''x'' = 1}} and {{math|1=''x'' = 4}}, so the expression <math display="inline">\frac{2}{2}</math> is [[Defined and undefined|undefined]].

In field theory, the expression <math display="inline">\frac{a}{b}</math> is only shorthand for the formal expression ''ab''<sup>−1</sup>, where ''b''<sup>−1</sup> is the multiplicative inverse of ''b''. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when ''b'' is zero. Modern texts, that define fields as a special type of ring, include the axiom {{math|0 ≠ 1}} for fields (or its equivalent) so that the [[zero ring]] is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.