Division by zero as a ''real number'', though, results in some major problems. You see, division is defined as the ''inverse operation to multiplication'' -- that is, ''a'' / ''b'' is a number that, when multiplied by ''b'', produces ''a''. Multiplication is, in turn, defined[[note]]at least in the real number system[[/note]] as ''hyperoperation of addition'', that is, ''a'' * ''b'' = ''b'' copies of ''a'' (or ''a'' copies of ''b'', which gives the same result) added together. Addition has an ''identity element'' zero (0), such that for all ''a'', ''a'' + 0 = 0 + ''a'' = ''a''. So, ''a'' * 0 = 0 * ''a'' = ''a'' copies of zero added together. But [[DepartmentOfRedundancyDepartment zero plus zero is always equal to zero]], so no matter how many copies of zero are added, the result will always be what was initially there -- zero. This, naturally, leads to some problems when you want to find a number ''a'' / 0 such that (''a'' / 0) * 0 = ''a'' if your ''a'' is nonzero, as all such numbers[[note]]provided they are real[[/note]] will make that expression evaluate to zero, not ''a''. You might be tempted to think that ''a'' / 0 for some nonzero ''a'' is some infinite value, but nonzero ''a'' / (any infinite number) evaluates to an ''infinitesimal'', which is a number infinitely close to zero but not zero. One way to demonstrate that these aren't the same as zero is to think of both infinities -- 1 / positive infinity = 0+, 1 / negative infinity = 0-, and if both of these were equal then that would mean that their multiplicative inverses would be as well, i.e. positive and negative infinity would be equal, a clear contradiction.

Real numbers form a [[http://en.wikipedia.org/wiki/Field_(mathematics) field]], which is an algebraic structure consisting of a set that is closed[[note]]using these won't result in a number outside of it[[/note]] under two operators + and *, each having commutative[[note]]''a'' + ''b'' = ''b'' + ''a'' and similarly for *[[/note]] and associative[[note]]''a'' * (''b'' * ''c'') * (''a'' * ''b'') * ''c'' and similarly for +[[/note]] properties, respective identity elements 0 and 1[[note]]''a'' + 0 = ''a'', ''a'' * 1 = ''a''[[/note]], respective ''inverse'' operators - and /[[note]]''a'' + -''a'' = 0, ''a'' * /''a'' = 1[[/note]], and a distributive property[[note]]''a'' * (''b'' + ''c'') = (''a'' * ''b'') + (''a'' * ''c'')[[/note]] relating them. That isn't the whole definition, though -- the axioms rather conspicuously state that every ''nonzero'' element has a multiplicative inverse, leaving /0 undefined. The real numbers are not the ''only'' totally ordered field, however, and nowhere ''near'' the largest -- the largest (and most general) one is usually called the [[http://en.wikipedia.org/wiki/Surreal_number surreal number system]], which contains all real numbers, all infinitesimals (an infinite number, in fact, directly adjacent to every real number ''[[MindScrew and every infinitesimal]]''), and all ''transfinite'' numbers (numbers greater than any finite value). Every number that can even ''begin'' to be imagined in one dimension, including division by ''countless''[[note]]literally -- see [[http://en.wikipedia.org/wiki/Cardinality_of_the_continuum cardinality of the continuum]][[/note]] infinitesimals, and guess what? ''They still can't properly handle a value describable in two symbols!''

The [[http://en.wikipedia.org/wiki/Real_projective_line real projective line]] (real numbers with infinity thrown in) ''does'' allow division by zero, but "division" doesn't have any real meaning. It is a useful construct in analysis, but it contradicts the layman's notion of how numbers are supposed to work since it's ''almost'' a field, but not quite.

Related to the above, there's also a rather weird kind of mathematical structure called a [[https://en.wikipedia.org/wiki/Wheel_theory wheel]] where division is always possible. Some of the rules of algebra have to be sacrificed for this; for example, x/x=1 is not always true, and nor are 0x=0 or x-x=0. But it is a real thing.

All of this leads to fun mathematics mind-benders where you can mathematically prove that 1=2 and all sorts of other nonsense by neatly leaving out the important bit about equations of the form x/x=1 ''when x does not equal 0''[[note]]example: 1*0 = 2*0 = 0, therefore (1*0)/0 = (2*0)/0 = 1*(0/0) = 2*(0/0) (associativity -- see above), therefore 1*1 = 2*1 = 1 = 2[[/note]].

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